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Átírás

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4 ÓÐÓÞØ ÖÑÒÝ ÖÓÑ ØÑ Ö ÓÔÓÖØÓ ØØº Þ Ð ÀÓÔ¹ÐÖÓÓ Ø ÞØ Ò ÐÖ ÜÓÑØÙ Ø ÖÝÐ º Ñ Ó ÐÐÑÞ Ù ÒÑ ÓÑÑÙØØÚ ÐÓ ¹ ÐÑÐØÒº ÀÖÑÒØ ÓÐÝÒ ØÖÐÑÐØ ÖÑÒÝØ ÑÖØØÒ ÑÐÝ ÝÒ¹ ÀÓÔ¹ÐÖ ÓÞ Ô ÓÐ ÓÒ ØÖÙØ ÐÝÞÒ Ð Ý Ø ÓÐÓÑÖÒ Ð¹ ÐØÚ ÐØÐ ÒÓ Ø ÞÓØº ÃÐØÞ Ø ØÒØÚ Ú Ð ÞØÓØØ ÔÙÐ ÞÖÞ ÑÙÒ Ò ØÞ ÚØ ÐÐ Ð È ÓÓÞØ Ñ ÞÖÞ ØÐ ÐÙØ º Ã ÞÒØÒÝÐÚ ÒØ º Ã ÞÒØØ ÞÖØÒ ÑÓÒÒ ÑÒÒ Ø Ö ÞÖÞÑÒ Ö ÓÐÓÞØÒ ÞÖÔÐ ØÑ ÓÒ Ö Ñ ÔÖÓÐÑ ÓÒ ÓÐÓÞØÙÒ ÝØØº ÄÐ ÞÖ ËÞÐ ÒÝ ÃÓÖÒÐÒ ÝÓÖ ÓØÓÖ ØÑÚÞØÑÒ Ð Ø Ö ÞÖÞÑÒ ÚÐ ÑÒ¹ ÒÒÐ Ó Þ ÓÐÓÞØÑ ÝØØ ØÐ ÑÒÒÒÐ ØØ ØÒÙÐØÑº Ø¹ Ð ÑÒ ÖÑ ÚÓÐØ ÝØØ ÓÐÓÞÒ Ð ÒÖÓ ÖÞÞÓÒ ÒÓ ÓÐØ Ò ÌÓÑ Þ ÖÞÞ ËØÚ Ä ÐÙ ÅÒÒ ÖÓ ãøò ÊÓ ËØÖØ Ì ÓÖ ÂÓÓ Ø ÎÖÖÙÝ ÊÓÖØ Ï ÙÖº Ã ÞÒØØÐ ØÖØÓÞÓÑ Ñ Þ ÑÓ ÓÐÐ Ò Ð ÙÝÒ ØÐ Ò Ñµ ÒÑ ÔÙÐ ÐØÙÒ Þ Ò Ð ÚÐ ÞÐØ Ð ÒÝÓÒ ÓØ ØÒÙÐ¹ ØØÑº ËØ ÒÔÐ ÂÓ ÑÞ ÌÓÖÖÐÐ ÈÓØÖ Åº À ÄÖ Ã ÓÒ Ä Ð ÃÓÙØØ ÓÖÒ âó Î ÖÒÝ ÈØÖº ÆÝ¹ÒÝ ÞÒØØÐ ØÖØÓÞÓÑ Ð ÓÑÒ ÑÖØ ÞÖØ Ø ÑÓØ ÒÐÐ ÑÙÒ Ñ ÒÑ ÐÒÒ ÐØ º ÖÑÒ ÑÒ ØÖÑ ÞØ Ò ÞÚ Ò Ú ÐÐÐØ ÝÒÐ ÞÖÔØ Þ ÓØØÓÒ ÐØÓÒ ÞÐÑ¹ Ò ÖÒØ ØÖØ ÚØØ Ú ÞÒ Ð Ú ÐÐÑÖÐ ÚÐÑÒØ ÖÓÑ ÝÖÑÒ ÑÖØ Ö Öò Ò ÑÓ ÓÐÝÓÒ ÑÑ ÙÖ ÞÒÚÐÝÒ ÑØÑØ Ö ÒØººº

5 ÌÖØÐÓÑÝÞ ½º ÚÞØ ½º½º Æ ÒÝ Þ ÀÓÔ¹ÐÖ Ð ÞÙ ÐÑÐØÖÐ º º º º º º º º º º º º º º º ½º½º½º ÐÖ ÓµÑÓÙÐÙ º º º º º º º º º º º º º º º º º º º º º º º º º ½º½º¾º ÀÓÔ¹ÐÖ ÒØÖ ÐÐÑÐØ º º º º º º º º º º º º º º º º º º º º º º ½¼ ½º½º º ÀÓÔÐÓ ¹ÐÑÐØ º º º º º º º º º º º º º º º º º º º º º º º º º º º º ½½ ½º½ºº ÓÀÓÔ¹ÑÓÙÐÙ Ó º º º º º º º º º º º º º º º º º º º º º º º º º º º º ½ ½º½ºº ÀÓÔ¹ÐÖ ÓÒ ØÖÙ ÑÓÒ Ó ÓÖÑ Ð ÐÑÐØ º º º º º º º ½ ½º¾º ÀÓÔ¹ÐÖÓÓ º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º ½ ½º¾º½º Þ ÔØÚ R¹ÝòÖò R¹ÓÝòÖò ÐÖÓÓ º º º º º º º º º º ½ ½º¾º¾º ÀÓÔ¹ÐÖÓ ÜÑ ÚØÞÑÒÝ º º º º º º º º º º º º º º ¾¼ ½º º ÀÓÔ¹ÐÖÓÓ ÒØÖ ÐÐÑÐØ º º º º º º º º º º º º º º º º º º º º º º º º º ¾ ½º º½º ÁÒØÖ ÐÓ º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º ¾ ½º º¾º Å ¹ØÔÙ ØØÐ º º º º º º º º º º º º º º º º º º º º º º º º º º º ¾ ½º º º Ú Þ¹µÖÓÒÙ ¹ØÙÐÓÒ ÖÐ º º º º º º º º º º º º º º º º º º º º ¾ ½º ºº ÖÓÒÙ ¹µ ÀÓÔ¹ÐÖÓÓ ÙÐØ º º º º º º º º º º º º º º º º º ¾ ½ºº ÐÓ ¹ØÖ ÞØ ÀÓÔ¹ÐÖÓ ÞÑÑØÖ ÚÐ º º º º º º º º º º º º º º º ¾ ½ºº½º ÀÓÔ¹ÐÖÓÓ ÓÑÓÙÐÙ º º º º º º º º º º º º º º º º º º º º º º º ¾ ½ºº¾º Ý ÃÖÑÖÌÙ¹ØÔÙ ØØÐ º º º º º º º º º º º º º º º º º º º º ¾ ½ºº º Ö ÝÒ ØÖÙØÖ¹ØØÐ ÅÓÖØ¹ÐÑÐØÐ º º º º º º º º º º ¼ ½ºº ÝÒ ÀÓÔ¹ÐÖ ÓÀÓÔ¹ÑÓÙÐÙ º º º º º º º º º º º º º º º º º º º ¾ ½ºº½º ÝÒ ÀÓÔ¹ÐÖ º º º º º º º º º º º º º º º º º º º º º º º º º º º ¾ ½ºº¾º ÝÒ ÓÀÓÔ¹ÑÓÙÐÙ Ó º º º º º º º º º º º º º º º º º º º º º º º ½ºº ÑÓÒ Ó ÝÒ ÐÑÐØ º º º º º º º º º º º º º º º º º º º º º º º º º º º º ½ºº½º EM w (K) ÝÒ Ó ÞÓÖ ÞÓÖÞØ º º º º º º º º º º º º º º º º º º º ½ºº¾º ÝÒ ÐÞ º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º ¼ ½ºº ÝÒ ÑÓÒ Ó º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º ¾ ½ºº½º Þ ÜÑ ÝÒ ÑÓÒ Ó ÐÒÖÅÓÓÖ¹ØÖ º º º º º ½ºº¾º ÝÒ ÑÓÒ Ó ÑÒØ ÑÓÒ Ó º º º º º º º º º º º º º º º º º º º ½ºº º ÝÒ ÑÓÒ Ó ÝÒ ÐÖ º º º º º º º º º º º º º º º º º ½ººº ÝÒ ÀÓÔ¹ÑÓÒ Ó º º º º º º º º º º º º º º º º º º º º º º º º º º º ½ºº ÃÚÒØÙÑ ÖÙÔÓÓ ÐÐÑÞ º º º º º º º º º º º º º º º º º º º º º º º º º ÁÖÓÐÓÑÝÞ º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º º

6 6 TARTALOMJEGYZÉK 2. Doi-Hopf modules over weak Hopf algebras 57 Böhm, G. Comm. Algebra 28 (2000), no. 10, Hopf algebroids with bijective antipodes 71 Böhm, G.; Szlachányi, K. J. Algebra 274 (2004), no. 2, Integral theory for Hopf algebroids 117 Böhm, G. Algebr. Represent. Theory 8 (2005), no. 4, Erratum in press. 5. Galois theory for Hopf algebroids 161 Böhm, G. Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005), The weak theory of monads 193 Böhm, G. Adv. in Math. 225 (2010), no. 1, Weak bimonads and weak Hopf monads 227 Böhm, G.; Lack, S.; Street, R. J. Algebra in press. doi: /j.jalgebra

7 ½º ÞØ ÚÞØ ÞÒ ÚÞØ ÞØ ØØ ÐÐÐ ÖÓØØº ÝÐÐ ÓÝ ØÑÖ ØØÒØ Ø ÓÒ ÑÒÞÓÒ ÓÐÑÖÐ Ð ÑÖØÖÐ ÑÐÝÖ ÓÐÓÞØ ÞØ Ú Ð ÞØÓØØ ÔÙ¹ Ð µ ÔÐÒº Å ÐÐ ÞÝòØØØ ØØ ÞØÒ ÑÙØØ Ö ÖÐ ÐÓÒØÓ ÖÑÒÝØº ÞÒ ØØÐ ØÐ ÞÓÒÝØ ÑØÐ ÐØ ¾¹º ÞØÒ ÑÐÐÐØ ÓÐÓÞØÓÒº ÚÞØ ÞÓÒÝØ ÓØ ÒÑ ØÖØÐÑÞº ÐÝØØ ÝÞØÒ ØØ ÐÚ ÞÓÐÒ ÑÙÒ ÒØ ÑÓØÚ Ð Ö Ø ÞÓÒÝØ ÓÓÞ Ð ÞÒ ÐØ ÓÒÓÐØÓØ ÐÐÝÞÒ ÞØ Þ ÖÑÒÝØ ØÑÖ ÖÓÐÑ Òº ÞØÒ ÚØÞ ÐØÐ ÒÓ ÖÚÒÝò ÐÐ Ø ÞÒ ÐÙº ÅÒÚ k Ý ÓÑ¹ ÑÙØØÚ ÞÓØÚ Ý ÐÑ ÝòÖòº ÞØØÐÒ ÞÑÐÙÑ k¹ñóùðù Ó ØÒÞÓÖ ÞÓÖÞØ Ø ÐÐº ÅÒÒ ÐÓÖÙÐ ÝòÖò ÞÓØÚ Ý ÐÑ º Ý R ÝòÖò Ó¹ ÐÐØÚ Ð ÑÓÙÐÙ Ò ØÖ Ø M R ¹ÖÐ ÐÐØÚ R M¹ÖÐ ÐÐ P Q ÓÑÓÑÓÖÞ¹ ÑÙ ÐÑÞ Ø Hom R (P,Q)¹ÚÐ ÐÐØÚ R Hom(P,Q)¹ÚÐº ÑÓÙÐÙ ØÖ ÐÐ RM R P Q ÓÑÓÑÓÖÞÑÙ Ò ÐÑÞ R Hom R (P,Q)º Þ R¹ÑÓÙÐÙ ØÒÞÓÖ ÞÓÖÞ¹ ØÓØ R ÐÐº ½º½º Æ ÒÝ Þ ÀÓÔ¹ÐÖ Ð ÞÙ ÐÑÐØÖÐ Ò ÞØÒ ÖÚÒ ØØÒØ ÀÓÔ¹ÐÖ ÐÑÐØÒ ÞÓÒ Ö Ø ÑÐÝ ÐØÐ ÒÓ Ø ÞÖÔÐÒ Òº Ú ÑÖØ ÓÖÖ ÒØ Ú ÓÐÙ ¼ ÑÓÒÓÖ Øº ½º½º Òº Ý k¹ðö ØÓÚ Ò ÐÖµ Ý k¹ñóùðù A ÐÐ ØÚ η : k A Ý µ µ : A A A ÞÓÖÞ µ k¹ñóùðù ÐÔÞ Ð ÑÐÝÖ Þ Ð Ø ÖÑÑÐ Ö ÞÓÐØ ÞÓØÚØ Ý ÐØØÐ ØÐ ÐÒº A A A Id µ A A µ Id A A µ A µ η Id A A A Id η Id µ A A A µ A A f f A A µ A f A µ η k k A f A η Ý f : A A k¹ñóùðù ÐÔÞ Ø ÐÖ ÓÑÓÑÓÖÞÑÙ Ò ÑÓÒÙÒ ÒØ Ñ Ó Ø ÖÑ ÓÑÑÙØØÚº Þ a b ÐÑ ÞÓÖÞØ Ø ab¹úð ÐÐ Ñ k Ý ÐÑÒ η ÐØÐ ÔØ 1¹ÝÐº Ý ÐÖ ÐÐÒØØØ ÙÝÒÞÓÒ k¹ñóùðù A ÓÖØÓØØ ÓÖÖÒò a b ba ÞÓÖÞ Ðº

8 ½º ÂÌº ÎÌË ½º¾º Òº Ý A ÐÖ Ó ÑÓÙÐÙ M k¹ñóùðù Ó ÐÐ ØÚ Ý : M A M k¹ñóùðù ÐÔÞ Ð ÑÐÝÖ Þ Ð Ø ÖÑÑÐ Ö ÞÓÐØ ÞÓØÚØ Ý ÐØØÐ ØÐ ÐÒº M A A Id µ M A Id M A M M Id η M A Id M M A f Id M A M f M Ý f : M M k¹ñóùðù ÐÔÞ Ø A¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ò ÑÓÒÙÒ ÒØ ÖÑ ÖÑ ÓÑÑÙØØÚº ÌØ ÞÐ m M a A ÐÑÒ (m a) = m.a ÐÐ Ø ÞÒ ÐÙº ËÞÑÑØÖ¹ Ù Ò ÖØÐÑÞÞ Ð ÑÓÙÐÙ ÓØ ÓÑÓÑÓÖÞÑÙ Øº ÓÐÖ Þ ÐÖ ÓÐÓÑ Ù Ð ½º º Òº Ý k¹óðö ØÓÚ Ò ÓÐÖµ Ý k¹ñóùðù C ÐÐ ØÚ ε : C k ÓÝ µ δ : C C C Ó ÞÓÖÞ µ k¹ñóùðù ÐÔÞ Ðº Þ ÞÖ ÚÓÒØÓÞ Ó ÞÓØÚØ ÓÝ ÐØØÐØ Ý ÔÙ ÓÝ Þ ÐÖ ÜÑ ÖÑÒ ÑÒÒ ÒÝÐØ ÑÓÖØÙÒº ÓÐÖ ÓÑÓÑÓÖÞÑÙ Ó Ó ÞÓÖÞ Ð ÓÝ Ð ÓÑÔØÐ ÐÒ Ö ÐÔÞ º Ý c ÐÑ Ó ÞÓÖÞØ Ö ËÛÐÖ¹ÀÝÒÑÒÒ ÒÜ ÐÐ Ø ÞÒ ÐÙ δ(c) = c 1 c 2 Ñ ÐØØ Ú Ó Ø Þ ÖØÒ C C¹Òº Þ ÞÞ Ø ÒÑ ÐÐ ÜÔÐ¹ ØÒ ÖÖ Þ ÒÜ ÐÒÐØ ÙØÐº ÑÚØÐØ Ó ÞÓØÚØ ÑØØ Þ ÝÒÐ c 11 c 12 c 2 c 1 c 21 c 22 Þ ÐÝØØ ÖØÙÒ Ý ÞÖòÒ c 1 c 2 c 3 ¹Øº Ý ÓÐÖ ÐÐÒØØØ ÙÝÒÞÓÒ k¹ñóùðù C ÓÖØÓØØ ÓÖÖÒò c c 2 c 1 Ó ÞÓÖÞ Ðº ÖÑÐÝ C¹ÓÐÖ Hom(C,k) Ù Ð ÐÖ ØÖÒ ÞÔÓÒ ÐØ ØÖÙØÖ ÚÐº À Ý Ð¹ Ö A Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù Ý Hom(A A,k) = Hom(A,k) Hom(A,k)µ ÓÖ Hom(A,k) ÓÐÖ ØÖÒ ÞÔÓÒ ÐØ ØÖÙØÖ ÚÐº ½ºº Òº Ý C ÓÐÖ Ó ÓÑÓÙÐÙ M k¹ñóùðù Ó ÐÐ ØÚ Ý : M M C k¹ñóùðù ÐÔÞ Ð Þ Òº ÓØ Ð ÑÐÝÖ ÚÓÒØÓÞ Ó ÞÓØÚØ ÓÝ ÐØØÐØ Ý ÔÙ ÓÝ ÑÐÐ ÐÖ ÜÑ ÖÑÒ ÑÓÖ¹ ØÙ ÒÝÐØº Ý f : M M k¹ñóùðù ÐÔÞ Ø C¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ò ÑÓÒÙÒ ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ò Ò ÑÐÐ ÖÑ ÒÝÐ ÑÓÖ¹ Ø ÚÐ ÓÑÑÙØØÚº Ó C ÓÑÓÙÐÙ Ó ÓÑÓÑÓÖÞÑÙ ÐÓØØ ØÖ Ø M C ¹ÚÐ ÐÐ P Q ÓÑÓÑÓÖÞÑÙ Ó ÐÑÞ Ø Hom C (P,Q)¹ÚÐº ËÞÑÑØÖÙ Ò ÖØÐÑÞÞ ¹ ÐÐ Ð ÓÑÓÙÐÙ Ó C M ØÖ Øº À ÓÒÐÒ Ó ÞÓÖÞ ÓÞ Óµ ÓØ Ö ÑÔÐØ ÞÞ Ø ÐÒØ ÒÜ ÐÐ Ø ÞÒ ÐÙÒ (m) = m 0 m 1 º ÑÚØÐØ Ó ÞÓØÚØ ÑØØ Þ ÝÒÐ m 00 m 01 m 1 m 0 m 11 m 12 Þ ÐÝØØ ÖØÙÒ Ý ÞÖòÒ m 0 m 1 m 2 ¹Øº ½ºº Òº Ý k¹ðö ØÓÚ Ò ÐÖµ Ý k¹ñóùðù B ÐÐ ØÚ Ý (η,µ) ÐÖ ØÖÙØÖ ÚÐ Ý (ε,δ) ÓÐÖ ØÖÙØÖ ÚÐ Ý ÓÝ ε δ Ð¹ Ö ÓÑÓÑÓÖÞÑÙ Ó ÚÝ Ñ ÚÚÐ ÚÚÐÒ η µ ÓÐÖ ÓÑÓÑÓÖÞÑÙ Óµº ÐÖ ÓÑÓÑÓÖÞÑÙ Ó Ý ÞÖÖ ÐÖ¹ ÓÐÖ ÓÑÓÑÓÖÞÑÙ Óº

9 ½º½º ÆÀýÆ Ë ÀÇÈ¹ÄÊýÃ ÃÄËËÁÃÍË ÄÅÄÌÊÄ ÌØ ÞÐ a,b B ÐÑÒ ÖÚ ÐÖ ÜÑ ÚØÞ = 1 1, (ab) 1 (ab) 2 = a 1 b 1 a 2 b 2, ε(1) = 1, ε(ab) = ε(a)ε(b). Ý ÐÖ Ò Ö ÞÓÖÞ Ø Ö Ó ÞÓÖÞ Ø Þ ÐÐÒØØØÖ ÖÐÚ ÑØ ÐÖ Ø ÔÙÒº À Ý ÐÖ B Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÓÖ ÐÒ Ö Ù Ð Hom(B,k) ÐÖº Þ ÐÖ ØÖÙØÖ ÓÐÖ ØÖÙØÖ ØÖÒ ÞÔÓÒ ÐØÒØ ÓÐÖ ØÖÙØÖ Ô Þ ÐÖ ØÖÙØÖ ØÖÒ ÞÔÓÒ ÐØÒØ º ½ºº Òº Ý k¹àóô¹ðö ØÓÚ Ò ÀÓÔ¹ÐÖµ Ý ÐÖ H Ð¹ Ð ØÚ Ý ÒØÔÒ ÒÚÞØØ S : H H k¹ñóùðù ÐÔÞ Ð ÑÐÝÖ Þ Ð ÐØØÐ ØÐ ÐÒº δ H H H ε S Id H H δ k µ η H H Id S H H µ H ÀÓÔ¹ÐÖ ÓÑÓÑÓÖÞÑÙ Ó Þ f : H H ÐÖ ÓÑÓÑÓÖÞÑÙ Óº Þ Þ¹ ÔÔÒ ÖÞ Þ ÒØÔÓØ ÞÞ fs = S fº ÌØ ÞÐ a H ÐÑÒ ÖÚ ÀÓÔ¹ÐÖ ÜÑ ÚØÞ a 1 S(a 2 ) = ε(a)1 = S(a 1 )a 2. H H k¹ñóùðù ÓÑÓÑÓÖÞÑÙ Ó ÐÑÞ ÐÖ Ú ØØ Þ f g µ(f g)δ Òº ÓÒÚÓÐ ÞÓÖÞ Ð Þ ÒØÔ ÜÑ ÓÐÚ Ø Ý ÓÝ Ò Þ ÐÖ Ò S ÒÚÖÞ Þ ÒØØ ÐÔÞ Òº À Þ ÒØÔ ÐØÞ ÓÖ ÝÖØÐÑò ØÓÚ ÓµÐÖ ÓÑÓÑÓÖÞÑÙ H¹Ð Þ ÐÐÒØØØ ÓµÐÖ º À H Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÓÖ ÞÓÐØ ÓÝ Þ ÒØÔ ØÚ ÐÒ Ö Ù Ð Hom(H,k) ÀÓÔ¹ÐÖº Þ Ù Ð ÒØÔ H ÒØÔ Ò ØÖÒ ÞÔÓÒ ÐØÒØ º ½ºº ÈÐº ÄÝÒ G Ý ØØ ÞÐ ÓÔÓÖØº G ÐÑ ÐØÐ ÒÖ ÐØ Þ k¹ñóùðù kg ÐÐ ØØ ÀÓÔ¹ÐÖ ØÖÙØÖ ÚÐ ÚØÞÔÔÒº ÞÓÖÞ Ø ÓÔÓÖØ ÞÓÖÞ ÐÒ Ö ØÖ ÞØ ÒØ Ò ÐÙ Ó ÞÓÖÞ Ø Ô ÓÔÓÖØ ÐÑÒ ÓÒ Ð g g g ÐÔÞ ÐÒ Ö ØÖ ÞØ ÒØº Þ Ý ÐÑ Ø Ø ÓÔÓÖØ Ý Ð Þ ÓÝ ÓÔÓÖØ ÐÑÒ ÞÓÒÓ Ò 1 ÚÒÝ ÐÒ Ö ØÖ ÞØ Ñ Þ ÒØÔ ÓÔÓÖØ ÒÚÖÞ ÑòÚÐØÒ ÐÒ Ö ØÖ ÞØ º À G Ú ÓÔÓÖØ ÓÖ ÐÒ Ö Ù Ð k(g) ÞÞ G k ÚÒÝ ÐØÐ ÒÖ ÐØ k¹ñóùðù ÀÓÔ¹ÐÖ ØÖÒ ÞÔÓÒ ÐØ ØÖÙØÖ ÚÐº ½º½º½º ÐÖ ÓµÑÓÙÐÙ k¹ðö ÐÐÑÞØ ÑÒØ ÔÓÒØÓ Ò ÞÓ ÓµÐÖ ÑÐÝ ÓµÑÓÙÐÙ Ò ØÖ ÑÓÒÓ Ð ÑÓÞÞ Ý ÓÝ ÐØ ÙÒØÓÖ ÓµÑÓÙÐÙ ØÖ Ð k¹ñóùðù Ó ØÖ ÞÓÖÒ ÑÓÒÓ Ð º ÃÓÒÖØÒ ÑÚÐ ÓÝ ÐÖ ÓÑÓÑÓÖÞÑÙ ÖÑÐÝ B ÐÖ Ò Ó ÚÝ ÞÑÑØÖÙ Ò Ðµ ÑÓÙÐÙ Ñ Þ ÐÔÝòÖò κ.h = κε(h), κ k, h B.

10 ½¼ ½º ÂÌº ÎÌË ÓÝ ÓÐÚ Ø B Ø Ð k¹òº ÅÚÐ Ó ÞÓÖÞ ÐÖ ÓÑÓÑÓÖÞÑÙ ÖÑÐÝ M N Ó B¹ÑÓÙÐÙ Ó ØÒ M N Ó B¹ÑÓÙÐÙ Þ Òº ÓÒ Ð Ø ÖÚÒ (m n).h = m.h 1 n.h 2, m M, n N, h B. Ó ÞÓÖÞ ÓÐÚ Ø B Ø Ð B B¹Òº k¹ñóùðù Ó ØÖ Ò (M N) P = M (N P) ÞÓ ØÓÖ ÞÓÑÓÖÞÑÙ M k = M = k M Ý ÞÓÑÓÖÞÑÙ B¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó ÓÐÖ ÜÑ Ò ÞÒØÒº ÍÝÒÝ ÑÚÐ Þ Ý ÓÐÖ ÓÑÓÑÓÖÞÑÙ ÖÑÐÝ B ÐÖ Ò Ó ÚÝ ÞÑÑØÖÙ Ò Ðµ ÓÑÓÙÐÙ Ñ Þ ÐÔÝòÖò k k B = B κ κ1, κ k. Þ Ý ÓÐÚ Ø B ÓØ Ð k¹òº ÅÚÐ ÞÓÖÞ ÓÐÖ ÓÑÓÑÓÖÞÑÙ Ö¹ ÑÐÝ M N Ó B¹ÓÑÓÙÐÙ Ó ØÒ M N Ó B¹ÓÑÓÙÐÙ Þ Òº ÓÒ Ð ÓØ ÖÚÒ M N M N B m n (m n) 0 (m n) 1 := m 0 n 0 m 1 n 1, m M, n N. ÞÓÖÞ ÓÐÚ Ø B ÓØ Ð B B¹Òº k¹ñóùðù Ó ØÖ Ò (M N) P = M (N P) ÞÓ ØÓÖ ÞÓÑÓÖÞÑÙ M k = M = k M Ý ÞÓÑÓÖÞÑÙ B¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó Þ ÐÖ ÜÑ Ò ÞÒØÒº ½º½º¾º ÀÓÔ¹ÐÖ ÒØÖ ÐÐÑÐØ Þ ÒØÖ ÐÓ ÀÓÔ¹ÐÖ ØÒØØØØ ÐÑ ÑÐÝÒ ÚÞ ÐØ ÚÐ ÓÒØÓ ÒÓÖÑ ÒÝÖØ Þ ÐÖ ÞÖÞØÖÐº ½ºº Òº ÄÝÒ B Ý ÐÖ M Ý Ð B¹ÑÓÙÐÙ º M ÒÚÖ Ò Ò ÚÙ Þ M inv := {n M a B, a.n = ε(a)n} = B Hom(k,M) k¹ö ÞÑÓÙÐÙ ÐÑØº ËÞÑÑØÖÙ Ò ÖØÐÑÞÞ Ó ÑÓÙÐÙ Ó ÒÚÖ Ò Øº Ð ÒØÖ ÐÓ ÐØØ B¹Ò Ð ÖÙÐ Ö Ö ÞÓÐ ÒÚÖ Ò Ø ÖØ ÞÞ B inv := {l B a B, al = ε(a)l} Ó Ð ÐÑØº ËÞÑÑØÖÙ Ò Ó ÒØÖ ÐÓ Ó ÖÙÐ Ö Ö ÞÓÐ ÒÚÖ Ò º ½ºº ÈÐº Ý Ú G ÓÔÓÖØ ÐÑ ÐØÐ ÒÖ ÐØ kg ÀÓÔ¹ÐÖ Ò Þ ÒØÖ ÐÓ g ÀÖ ÐÑ Þ Ñ ÞÓÖÓ º g G Å Ð ÞÙ ÓÔÓÖØ ÐÖ Ð Ý ÞÖò Ö ÚÓÒØÓÞ ØØÐØ ÐØÐ ÒÓ ØÚ ÚØÞ ÞÓÐØº ½º½¼º ÌØÐ ËÛÐÖ ¼ ÒÔÐÅÐØÖÙ ¾ µº Ý H ÀÓÔ¹ÐÖ Ö ÚÓÒØÓÞ ¹ ÚØÞ ÐÐØ Ó ÚÚÐÒ º H ÞÔÖ Ð k¹ðö H Ð Ý ÞÖò k ÐØØ ÞÞ Ý Ð ÚÝ Ó H¹ÑÓÙÐÙ ÔÑÓÖÞÑÙ ÔÓÒØÓ Ò ÓÖ Ð ÑÒØ k¹ñóùðù ÔÑÓÖÞÑÙ Ð µ ÓÝ ε Ð Ó ÚÝ Ðµ H¹ÑÓÙÐÙ ÔÑÓÖÞÑÙ

11 ½º½º ÆÀýÆ Ë ÀÇÈ¹ÄÊýÃ ÃÄËËÁÃÍË ÄÅÄÌÊÄ ½½ ÐØÞ H¹Ò Ý Ó ÚÝ Ðµ ÒØÖ Ð l Ñ ÒÓÖÑ ÐØ Þ ε(l) = 1 ÖØÐÑÒº Ý Ñ ÓÒØÓ ÚÓÒ ÑÐÝØ Þ ÒØÖ ÐÓ Ø ÚÐ ÚÞ ÐØÙÒ Ú Þµ ÖÓ¹ ÒÙ ¹ØÙÐÓÒ º Ý ÐÖ A ÖÒÐÞ ÖÓÒÙ ¹ØÙÐÓÒ Ð Ú Ò ¹ ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÞÓÑÓÖ Hom(A,k)¹ÚÐ ÑÒØ ÚÚÐÒ ÑÓÒ Ó ÚÝ Ðµ A¹ÑÓÙÐÙ º ÂÓ ÐÐØÚ Ð Ú Þ ÖÓÒÙ ÐÖ ÖÐ ÞÐÒ A Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù Ú Þ¹ÞÓÑÓÖ Hom(A,k)¹ÚÐ ÑÒØ Ó ÐÐØÚ Ð A¹ÑÓÙÐÙ ºµ ÅÒØ ÈÖ ÞÓÐØ Ý ÀÓÔ¹ÐÖ ÔÓÒØÓ Ò ÓÖ Ó Ú Þ¹ÖÓÒÙ ¹ ÐÖ Ð Ú Þ¹ÖÓÒÙ ¹ÐÖ Ñ ÔÓÒØÓ Ò ÓÖ ØÐ Ð Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ ÑÓÙÐÙ ½ º ÖÓÒÙ ¹ØÙÐÓÒ ÖÐ ÚØÞ ØÙØº ½º½½º ÌØÐ ÄÖ ÓÒËÛÐÖ ÈÖ ½ µº Ý H ÀÓÔ¹ÐÖ Ö ÚÓÒØÓÞ ÚØÞ ÐÐØ Ó ÚÚÐÒ º H ÖÓÒÙ ¹k¹ÐÖ Hom(H,k) ÖÓÒÙ ¹k¹ÐÖ ÐØÞ H¹Ò Ý Ó ÚÝ Ðµ ÒØÖ Ð l ÑÐÝ ÒÑ ÒÖ ÐØ Ò Þ ÖØÐÑÒ ÓÝ ÚØÞ ÐÔÞ ØÚº Hom(H,k) H φ (Id φ)δ(l) l 1 φ(l 2 ) Hom(H,k) H φ (φ Id)δ(l) φ(l 1 )l 2 À k ÈÖ ÓÔÓÖØ ØÖÚ Ð ÔÐ ÙÐ k Ý Ø Øµ ÓÖ ÒØ ØÙÐÓÒ Ó ÔÓÒØÓ Ò ÓÖ ØÐ ÐÒ H Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ú ÑÒÞ µº ½º½º º ÀÓÔÐÓ ¹ÐÑÐØ ÆÑ ÓÑÑÙØØÚ ÐÖ ÀÓÔ¹ÐÖ Ð ÚÐ ØÖ ÞØ Ø ÑÒØ ÓÑÑÙØØÚ ÝòÖò ÓÔÓÖØØÐ ÚÐ ØÖ ÞØ Ò ÐØÐ ÒÓ Ø Ø Ð ÞÖ ËÛÐÖ ØÒÙÐÑ ÒÝÓÞØ ¼ ÑÙÒ Ò ÐÐØÚ ÃÖÑÖ ÌÙ ¹Òº ÀÓÔÐÓ ¹ÐÑÐØ Ý Ð¹ Ö Ø ÓÖ Ò ØÐÒÐ ÚÞ ÐØ ØÖÙØÖ Ò ÔÐº ÓÔÓÖØ Ö ÐØ ÐÖ Òµº ÐÖ ÚÓÒØÓÞ ÑÐÐØØ ÀÓÔÐÓ ¹ØÖ ÞØ ÒÑ ÓÑÑÙØØÚ ÓÑØÖ ÒÞÔÓÒØ Ð Ö ÓÐ ÔÖÒÔ Ð ÒÝÐ Ó ÒÑ ÓÑÑÙØØÚ ÑÐÐÒØ ¹ ÐÒÒ Ñº Þ ½º½º½ ÞØÒ Ð ØØÙ ÓÝ Ý H ÐÖ Ó ÓÑÓÙÐÙ Ò M H ØÖ ÑÓÒÓ Ð º ÑÓÒÓÓØ Ò ØÖ Ò H¹ÓÑÓÙÐÙ ÐÖ Ò ÚÙº ÜÔÐ¹ ØÒ Ý H¹ÓÑÓÙÐÙ ÐÖ Ý ÐÖ A ÑÒ ÞÓÖÞ Ý H¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ º ÞÞ = 1 1; (a a) 0 (a a) 1 = a 0a 0 a 1a 1, a,a A. A ÓÑÓÙÐÙ ÒÚÖ Ò ÖÚÒ ÓÒÚÖ Ò µ ÐØØ A coh := {b A b 0 b 1 = b 1} Ö ÞÐÖ ÐÑØ ÖØº ÓÐÓÞØÙÒÒ ÑÒÚ Ó ÓÑÓÙÐÙ ÐÖ Ð ÓÐÓ¹ ÞÙÒº Þ ÒÑ ÐÒØ ÞÑÑØÖ Ñ ÖØ Ø ÞÒ B Ð ÓÑÓÙÐÙ Ó ÓÑÓÙÐÙ ÒÒ ÐÖ Ò ÑÐÝÒ Þ ÐÖ ØÖÙØÖ Ø ÑÓÖØÚ ÚÝ Ú ÐØÓÞØÐÒÙÐ ÝÚµ B ÓÐÖ ØÖÙØÖ Ø Þ ÐÐÒØØØÖ ÖÐº

12 ½¾ ½º ÂÌº ÎÌË ½º½¾º Òº ÄÝÒ H Ý ÐÖº Ý B A ÐÖ ØÖ ÞØ Ø Ó H¹ÐÓ ¹ ØÖ ÞØ Ò ÑÓÒÙÒ A Ó H¹ÓÑÓÙÐÙ ÐÖ B = A coh Þ Ð ÒÓÒÙ ÐÔÞ ÞÓÑÓÖÞÑÙ º can : A B A A H a B a a a 0 a 1. ½º½µ ½º½ º ÈÐº Ð ÞÙ ÐÓ Ø Ø ÚØ Ú ÓÔÓÖØÓÐ ÀÓÔÐÓ ¹ØÖ ÞØ Ðº ÜÑÔÐ ºº º ÄÝÒ G Ý Ú ÓÔÓÖØ F Ý Ø Ø ÑÒ G ÙØÓÑÓÖÞÑÙ ÓÐ Ø ÞÞ ÐØÞ Ý ÓÔÓÖØ ÓÑÓÑÓÖÞÑÙ G Aut(F) g α g º ÖÑÐÝ k F Ø ØÖ F k(g)¹óñóùðù ÐÖ Þ a g G α g(a) δ g ÓØ ÖÚÒ ÓÐ δ g k(g) ÖØ g¹ò 1 Þ Þ Ø ÓÔÓÖØ ÐÑÒ 0º ÓÒÚÖ Ò Ö ÞÐÖ F co k(g) ÐÑ ÔÔÒ G¹Ø ÒÚÖ Ò F cok(g) F Ý k(g)¹ðó ¹ØÖ ÞØ º ½º½º ÈÐº Þ ÔÐ ÀÓÔÐÓ ¹ØÖ ÞØ ÓÑØÖ ÒØÖÔÖØ ÚÐ Ô¹ ÓÐØÓ º À ÓÒ Ý Ú G ÓÔÓÖØ ÐÖÐ Ý Ú X ÐÑÞÓÒº Þ X¹Ò ÖØÐÑ¹ ÞØØ ÚÒÝ k(x) ÐÖ ÔÓÒØÓÒÒØ ÞÓÖÞ Ðµ k(g)¹óñóùðù ÐÖ Þ f g G f(g. ) δ g ÓØ ÖÚÒº k(x) co k(g) ÓÒÚÖ Ò Ö ÞÐÖ ÐÑ ÔÓÒØÓ Ò G¹ Ô ÐÝ ÓÒ ÓÒ ØÒ ÚÒÝº k(x) co (G) k(x) ÔÓÒØÓ Ò ÓÖ k(g)¹ðó ¹ØÖ ÞØ G¹Ø X¹Ò Þ ØÖÒÞØÚ ÞÞ X ÖÑÐÝ x,y ÐÑÖ ÐØÞ ÔÓÒØÓ Ò Ý g ÐÑ G¹Ò ÑÖ y = x.gº Þ ØÒ ÔÖ Þ k(x) co k(g) = kºµ ½º½º ÈÐº ÖÑÐÝ B ÐÖ Ñ B¹ÓÑÓÙÐÙ ÐÖ Ó ÞÓÖÞ ÑÒØ ÓØ ÖÚÒº B cob ÓÒÚÖ Ò Ö ÞÐÖ ÐÑ ÔÓÒØÓ Ò Þ Ý ÐÑ Þ Ñ ÞÓÖÓ º k B ØÖ ÞØ ÓÖ ÓÖ B¹ÐÓ B ÀÓÔ¹ÐÖº ËÞ ÑÓ ÓÐÝÒ ØØÐ ÑÖØ ÑÐÝ ÒÓÒÙ ÐÔÞ ØÚØ Ö ÚÓÒØÓÞ Ð ¹ ÐØØÐØ ÓÐÑÞ Ñº ÈÐÒØ ÐÐÓÒ ØØ Ý Ð ÞÙ º ½º½º ÌØÐ ÃÖÑÖÌÙ µº ÄÝÒ H Ý Ú ÑÒÞ ÀÓÔ¹ÐÖ ØØ ÞÐ¹ Ø Ø ÐØØ ÐÝÒ A Ý Ó H¹ÓÑÓÙÐÙ ÐÖº ÞÒ ÐØÚ ÑÐÐØØ Þ ½º½µ ÒÓÒÙ ÐÔÞ ÔÓÒØÓ Ò ÓÖ ØÚ ÞÖØÚº ÌÒØ Ò Ý H ÐÖ Ø Ý A Ó H¹ÓÑÓÙÐÙ ÐÖ Øº Ò ÞÖÒØ A Ý ÑÓÒÓ M H ¹Ò Ý ØÒØØ Ó A¹ÑÓÙÐÙ Ó ØÖ Ø M H ¹Òº ÞÒ ÑÓÙÐÙ ÓØ ÖÐØÚ ÀÓÔ¹ÑÓÙÐÙ ÓÒ ÚÙº ÜÔÐØÒ Ý (A,H) ÖÐØÚ ÀÓÔ¹ÑÓÙÐÙ Ý Ó A¹ÑÓÙÐÙ Ó H¹ÓÑÓÙÐÙ M Ý ÓÝ Þ A¹Ø H¹ÓÑÓÙÐÙ ÓÑÓÑÓÖ¹ ÞÑÙ ÞÞ (m.a) 0 (m.a) 1 = m 0.a 0 m 1 a 1, m M, a A. Þ (A,H) ÖÐØÚ ÀÓÔ¹ÑÓÙÐÙ Ó ÓÑÓÑÓÖÞÑÙ Ó A¹ÑÓÙÐÙ Ó H¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Óº ÖÐØÚ ÀÓÔ¹ÑÓÙÐÙ Ó ÓÑÓÑÓÖÞÑÙ ØÖ Ø M H A ¹ÚÐ ÐÐº A ÓÒÚÖ Ò Ò Ö ÞÐÖ Ø B¹ÚÐ ÐÐÚ Þ ( ) B A : M B M A Ð ÙÒ ÐØ ÙÒØÓÖ ÖÒÐÞ Ý ( ) B A : M B M H A ÐÞ Ð ÞÞ ÖÑÐÝ P Ó B¹ÑÓÙÐÙ ØÒ P B A ÖÐØÚ ÀÓÔ¹ÑÓÙÐÙ A ÒÝÐÚ ÒÚÐ ÖÐØÚ ÀÓÔ¹ÑÓÙÐÙ ØÖÙØÖ ÖÚÒµº ÐÞÓØØ ÙÒØÓÖ Ó ÙÒ ÐØ ( ) coh : M H A M Bº ÀÓÔ ÐÓ ¹ÐÑÐØ ÞÔÓÒØ Ö ÓÝ Þ ÙÒØÓÖ ÑÓÖ ØÐ ò ÝÒ ØÖÙØÖ¹ØØÐµ ÐÐØÚ ÚÚÐÒ Ö ØÖÙØÖ ØØÐµº ÌÖØÒØÐ Þ Ý Ð ÐÝÒ ØØÐ ÚØÞ ÚÓÐØº ½º½º ÌØÐ ÀÓÔ¹ÑÓÙÐÙ Ó ÐÔØØÐ ¼ µº ÖÑÐÝ H ÀÓÔ¹ÐÖ Ö ( ) H : M k M H H ÚÚÐÒ ÞÞ ÖÙÐ Ö ÓÑÓÙÐÙ ÐÖ Ö Þ Ö ØÖÙØÖ¹ØØÐ ØÐ Ðº

13 ½º½º ÆÀýÆ Ë ÀÇÈ¹ÄÊýÃ ÃÄËËÁÃÍË ÄÅÄÌÊÄ ½ À H Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÓÖ ÖÑÐÝ A Ó H¹ÓÑÓÙÐÙ Ð¹ Ö ØÒ ØÐ ÐØ Ý ÐÐÑ ÐÖ ÑÐÝÒ ÑÓÙÐÙ ØÖ ÞÓÑÓÖ ÖÐØÚ (A,H)¹ÀÓÔ¹ÑÓÙÐÙ Ó ØÖ ÚÐº Þ Ö ØØ ÐÖ Ù Ð H := Hom(H,k) A ÐÖØ ÞÓÖÞØÒØ ÖØº ÅÒØ k¹ñóùðù ÑÝÞ A H ¹ÚÐº ÞÓÖÞ (a φ)(b ψ) = a 0 b (a 1.ψ)φ, ÓÐ h.ψ = φ( h) H Ð Ø H ¹ÓÒº Ò Þ ØÒ Ø Ø Þ Ö ÝÒ ØÖÙØÖ¹ ØØÐ ÑÓÙÐÙ ØÖ Þ ÓÒÐØ Ö ÚÞØÒ Ý ÅÓÖØ¹ÐÑÐØ ÐÐÑÞ ÚÐ ÞÓÒÝØØº ½º½ºº ÓÀÓÔ¹ÑÓÙÐÙ Ó Þ ½º½º ÞØÒ Ý B ÐÖ Ó ÚÝ Ðµ ÓÑÓÙÐÙ ÐÖ Ø Ý Ò ÐØÙ ÑÒØ ÑÓÒÓÓØ Ó ÚÝ Ðµ B¹ÓÑÓÙÐÙ Ó ÑÓÒÓ Ð ØÖ Òº ËÞÑÑØÖÙ Ò Ò ÐØÙÒ Ó ÚÝ Ðµ B¹ÑÓÙÐÙ ÓÐÖ Ø ÑÒØ ÓÑÓÒÓÓØ Ó ÚÝ Ðµ B¹ÑÓÙÐÙ Ó ÑÓÒÓ Ð ØÖ Òº ½º½º Ò Ó ÃÓÔÔÒÒ µº ÂÓ¹Óµ ÓÀÓÔ¹ØÓ ÐØØ ÓÐÝÒ (A,B,C) ÖÑ ÓØ ÖØÒ ÓÐ B Ý ÐÖ A Ý Ó B¹ÓÑÓÙÐÙ ÐÖ C Ý Ó B¹ÑÓÙÐÙ ÓÐÖº ÐÒ Ó¹Ó ÓÀÓÔ¹ØÓØ ÚÞØÒÒ ÞÒ Ø ÐØ Ø Ñ¹ ÔÙ B ÓµÐÖ ØÖÙØÖ Ø Þ ÐÐÒØØØÖ ÖÐ Þ Ý ÒÝÖØ ÐÖ Ó¹Ó ÓÀÓÔ¹ØØ ØÒØº ÖÑÐÝ (A,B,C) ÓÀÓÔ¹Ø ØÒ ØÒØØ Þ Ð k¹ñóùðù ÐÔÞ Ø ψ : C A A C c a a 0 c.a 1, ÑÐÝ ÓÑÑÙØØÚÚ Ø Þ Þ Ð ÖÑÓØº Id µ C A A C A ψ Id A C A ψ C Id η C A ψ C A δ Id C C A ψ Id ψ C A C C A ε Id A ψ Id ψ A A C µ Id A C C η Id A C ψ Id A C Id δ A C C A C Id ε A ØÖÐÑÐØÒ ÞÓ Ó ØÖÑÒÓÐ ÚÐ Þ ÞØ ÐÒØ ÓÝ ψ Ý ÚÝ ÑÓÒ ÓØ ÓÑÓÒ ÓØ ÞÙÐ ÓÐµ ÞØÖÙØÚ Þ ÐÝ º ÍÝÒÖÖ ÀÓÔ¹ÐÖ ÞÖÓ¹ ÐÓÑ Þ ÞÙÐ ÓÐ ØÖÙØÖ ÒØÛÒÒ ØÖÙØÙÖµ ÐÒÚÞ Ø ÞÒ Ð ¾ º ÞÖÒØ ÑÒÒ C A A C ÚÝ ÞØÖÙØÚ Þ ÐÝ ÒÙ Ð Ý ( ) A ÑÓÒ ÓØ C¹ ÓÑÓÙÐÙ Ó ØÖ Ò Þ (M k,( ) A) ÑÓÒ ÐÞ Øµ Ý ( ) C ÓÑÓÒ ÓØ Þ A¹ÑÓÙÐÙ Ó ØÖ Ò (M k,( ) C) ÓÑÓÒ ÐÞ Øµº ÞÒ ÑÓÒ ÓÑÓÒ ÐÒÖÅÓÓÖ¹ØÖ ÞÓÑÓÖ ÞÓ Ó ÐÐ M C A º ÜÔÐØÒ ÓØÙÑ A : M A M Ó A¹ÑÓÙÐÙ Ó C : M M C Ó C¹ÓÑÓÙÐÙ Ó ÑÖ Þ Ð ÖÑ ÓÑÑÙØØÚº M A A M C Id M C A Id ψ M A C A Id M C C

14 ½ ½º ÂÌº ÎÌË ÑÓÖÞÑÙ Ó A¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó C¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Óº À A ØÖÚ Ð B¹ÓÑÓÙÐÙ ÐÖ k ÓÖ M C A ÑÝÞ C¹ÓÑÓÙÐÙ Ó ØÖ ÚÐº À C ØÖÚ Ð B¹ÑÓÙÐÙ ÓÐÖ k ÓÖ M C A ÑÝÞ Þ A¹ÑÓÙÐÙ Ó ØÖ ÚÐº À C ÖÙÐ Ö B¹ÑÓÙÐÙ ÓÐÖ B ÓÖ M C A ÑÝÞ Þ (A,B)¹ÖÐØÚ ÀÓÔ¹ ÑÓÙÐÙ Ó ØÖ ÚÐº ËÞ ÑÓ Ö Ø ÑÔØ Ñ A C ÐÐÑ Ú Ð ÞØ ¹ ÚÐ ÔÐ ÙÐ Þ Òº ÄÓÒ ÑÓÙÐÙ Ó ÚÝ ØØÖÖÒÐ³¹ÑÓÙÐÙ Ó ØÖ º À C Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÓÖ ÓÒ ØÖÙ ÐØ Ý ÐÐÑ ÐÖ ÑÐÝÒ ÑÓÙÐÙ ØÖ ÞÓÑÓÖ M C A ¹ÚÐº Þ Ö ØØ ÐÖ Ù Ð C := Hom(C,k) A ÐÖØ ÞÓÖÞØÒØ ÖØº Þ ÐÒÖÅÓÓÖ¹ØÖ ÐØÐ ÒÓ ØÙÐÓÒ ÞÖÒØ Þ M C A MC ÐØ ÙÒØÓÖ Ð ÙÒ ÐØ ( ) A : M C M C A ÓÐ Þ A¹Ø M A¹Ò M µ : M A A M A C¹ÓØ M A C Id M C A Id ψ M A C. ËÞÑÑØÖÙ Ò Þ M C A M A ÐØ ÙÒØÓÖ Ó ÙÒ ÐØ ( ) C : M A M C A ÓÐ C¹ÓØ M C¹Ò M δ : M C M C C Þ A¹Ø M C A Id ψ M A C A Id M C. ½º½ºº ÀÓÔ¹ÐÖ ÓÒ ØÖÙ ÑÓÒ Ó ÓÖÑ Ð ÐÑÐØ ÅÒØ ÖÖ ÒÓÐ ÙØÐØÙÒ Þ ½º½º½ Þ ½º½º ÞØÒ ÞÖÔÐ ÐÐØ Ó ÞÐ Ó ÚØÞ ÑÓÒ Ó ÚÐ ÐØÐ ÒÓ Òº ÓÖÑ Ð ÐÑÐØÐ º ÖÑÐÝ K ØÖ ÓÞ ÓÞÞ ÖÒÐØ K¹Ð ÑÓÒ Ó EM(K) ØÖ Ø º ÒÒ ÓØÙÑ ÚÝ 0¹ÐÐ µ K¹Ð ÑÓÒ Ó ÞÞ t : A A ½¹ÐÐ ÐÐ ØÚ η : Id A t Ý µ µ : tt t ÞÓÖÞ µ ¾¹ÐÐ Ð ÑÐÝÖ ÞÓ Ó ÞÓØÚØ Ý ÐØØÐ ØÐ ÐÒ µid ttt Id µ tt tt µ t µ η Id Id η t tt Id tt µ t µ ½º¾µ Þ (A,t) (A,t ) ½¹ÐÐ EM(K)¹Ò K¹Ð x : A A ½¹ÐÐ ÐÐ ØÚ Ý ψ : t x xt ¾¹ÐÐ ÚÐ ÑÐÝÖ Þ Ð ÖÑÓ ÓÑÑÙØØÚº t t x Id ψ t xt ψ Id xtt µ Id t x ψ xt Id µ η Id x x t x ψ xt Id η ½º µ Þ (x,ψ) (y,φ) ¾¹ÐÐ EM(K)¹Ò K¹Ð : x yt ¾¹ÐÐ ÑÐÝÖ Þ Ð ÐØØÐ ØÐ Ðº t x Id t xt ψ Id xtt ψ xt Id xtt Id µ xt Id µ ½ºµ

15 ½º½º ÆÀýÆ Ë ÀÇÈ¹ÄÊýÃ ÃÄËËÁÃÍË ÄÅÄÌÊÄ ½ EM(K) ÑÓÒ Ø Ó ÞÓÖÒ ÛÖØµ Ú º ÑÓÒÓØØ ÞÖÒØ Ó ÞÓÖ Ý (s,ψ) : (A,t) (A,t) ½¹ÐÐ EM(K)¹Ò ÐÐ ØÚ ÞÓÖÞ Ð ÑÐÝ ν : ss st ¾¹ÐÐ Ý Ð Ñ ϑ : Id st ¾¹ÐÐ K¹Òº À ((A,t),(s,ψ)) Ó ÞÓÖ ÓÖ (A,st) ÑÓÒ K¹Ò Þ (A,t) ÑÓÒ Þ s ½¹ÐÐ Òº Ó ÞÓÖ ÞÓÖÞØ º ½º½º ÈÐº ÄÝÒ (A,B,C) Ý Ó¹Ó ÓÀÓÔ¹Øº Þ ÑØ ÖÓÞ Ý ((k,a),(c, ψ)) Ó ÞÓÖØ ÝòÖò ÑÓÙÐÙ Ó ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó Bim¹ÑÐ ÐÐØ Ø¹ Ö Ò ÓÐ ψ : A C C A, a φ φ( a 1 ) a 0 ; Ó ÞÓÖ ÞÓÖÞ Ý Ô ν : C C C A, φ φ φ φ 1 ϑ : k C A, 1 ε 1, ÓÐ C ÞÓÖÞ φ φ φ φ := (φ φ)δº Þ Ó ÞÓÖ ÞÓÖÞØ ÞÓÑÓÖ C A ÐÖØ ÞÓÖÞØ ÚÐ Ðº ½º½º ½º½º ÞØµº ÄÝÒ (x,ψ) (y,φ) Ô ÖÙÞÑÓ ½¹ÐÐ EM(K)¹Òº ÐØØ ÞØ Ö Ø ÓÝ ÑÐÝ K¹Ð x ω y ¾¹ÐÐ ØÒ Ð Þ x ω y Id η yt ¾¹ÐÐ EM(K)¹Ò (x,ψ)¹ð (y,φ)¹º Ú Ð Þ Þ ÓÝ ÔÓÒØÓ Ò ÓÖ ÚØÞ ÖÑ ÓÑÑÙØØÚº t x Id ω t y ½ºµ ψ xt ω Id yt φ K¹Ð ÑÓÒ Ó ÑÒØ ¼¹ÐÐ Þ ½º µ¹ò Ò ÐØ ½¹ÐÐ Þ ½ºµ¹ØÐ ÐÐÑÞØØ K¹ Ð ¾¹ÐÐ ÑÒØ ¾¹ÐÐ ØÖ Ø ÐÓØÒ ÑÐÝÒ ÞÓ Ó ÐÐ Mnd(K)º ÞÒ ØÖ ÓÒØÓ ÞÖÔØ Ø Þ Þ Òº ÐÞ ÔÖÓÐÑ Òº ½º¾¼º Òº ÖÑÐÝ K ØÖ ØÒ ØÒØ ÓÒ Ð K Mnd(K) Þ¹ ÓÖµ ÙÒØÓÖØ ÑÐÝ Ý A ÓØÙÑÓØ Þ (A,Id) ÑÓÒ Ý x : A B ½¹ÐÐ Ø Þ (x,id) : (A,Id) (B,Id) ½¹ÐÐ Ý ω : x y ¾¹ÐÐ Ø ÒÑ ÔÞº À ÒÒ ÚÒ Ó ÙÒ ÐØ ÓÖ ÞØ ÑÓÒÙ ÓÝ ÐØÞÒ K¹Ò Þ ÐÒÖÅÓÓÖ ÓØÙÑÓº ÜÔÐØÒ A t ÓØÙÑ K¹Ò Þ (A,t) ÑÓÒ ÐÒÖÅÓÓÖ¹ÓØÙÑ ÐØÞ Ý B¹Ò ØÖÑ ÞØ ÞÓÑÓÖÞÑÙ º K(B,A t ) = Mnd(K)((B,Id),(A,t)) ½º¾½º ÈÐº Þ ÐÒÖÅÓÓÖ¹ÓØÙÑ Ò Ø Cat µ ØÖ ÙÒØÓÖÓ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ¾¹ØÖ µ ÔÐ ÑÓØÚ Ðº À t : A A ÑÓÒ Cat¹Ò ÓÖ A t ÐØÞ Þ Òº ÐÒÖÅÓÓÖ¹ÐÖ ØÖ º ÜÔÐØÒ ÓØÙÑ (a,α) Ô ÖÓ ÓÐ a ÓØÙÑ A ØÖ Ò α : t(a) a Ô ÑÓÖÞÑÙ A¹Ò ÑÐÝÖ ÞÓ Ó ÞÓØÚØ Ý ÐØØÐ ØÐ ÐÒº ½º¾¾º ÈÐº ÝòÖò ÑÓÙÐÙ Ó ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó Bim¹ÑÐ ÐÐØ Ø¹ Ö Ò ÑÒÒ ÑÓÒ Ò ÐØÞ ÐÒÖÅÓÓÖ ÓØÙÑº Ò Þ ØÒ Ý (A,t) ÑÓÒ ÐÖØ Ý η : A t ÝòÖò ÓÑÓÑÓÖÞÑÙ Ð Ðº ÚÒ Þ ½º¾ ÞØÒµº Þ ÐÒÖÅÓÓÖ ÓØÙÑ Ñ t ÝòÖòº

16 ½ ½º ÂÌº ÎÌË ½º¾ º ÌØÐ ËØÖØ µº À Ý ØØ ÞÐ K ØÖ Ò Þ (A,t) ÑÓÒ Ò ÐØÞ A t ÐÒÖÅÓÓÖ¹ÓØÙÑ ÓÖ ÚÒ K¹Ò Ý f v : A t A ÙÒ ÑÐÝÖ t = vf ÑÐÝÒ Ý ÑÓÒ η Ý ÑÐÝÒ ǫ ÓÝ Ø ÚÐ ÑÓÒ ÞÓÖÞ µ = vǫf ÐÒ ÖØº ½º¾º ÈÐº Cat¹Ò v : A t A ÐØ ÙÒØÓÖº Ð ÙÒ ÐØ f : A A t Ý a ÓØÙÑÓØ (t(a),µ(a))¹ Ý h ÑÓÖÞÑÙ Ø t(h)¹ Ú Þº Bim¹Ò v¹ø f¹ø Ý ÔÙ ÓÝ t¹ø ÑÒØ A¹t ÑÓÙÐÙ Ø ÐÐØÚ t¹a ÑÓÙÐÙ Ø ØÒØº ÒØ ÐÒÖÅÓÓÖ¹ÓØÙÑÓ Ø ÚÐ ÐÞ ÔÖÓÐÑ ÚØÞÔÔÒ ÓÐÑÞØ Ñº ÄÝÒ ÓØØ Ý K ØÖ Ò Ý (A,t) Ý (B,s) ÑÓÒ Ñ¹ ÐÝÒ ÐØÞ A t ÐÐØÚ B s ÐÒÖÅÓÓÖ¹ÓØÙÑº Ý x : A B ½¹ÐÐ ÐÞ ÐØØ ÓÐÝÒ x : A t B s ½¹ÐÐ Ø ÖØÒ ÑÖ Ð ÓÐÐ ½¹ÐÐ Ö ÚÓÒØÓÞµ ÖÑ ÓÑÑÙØØÚº A t x B s vx xv v A x B v À ÓÒÐÒ Ý ω : x y ¾¹ÐÐ ÐÞ ÐØØ ÓÐÝÒ ω : x y ¾¹ÐÐ Ø ÖØÒ ÑÖ Ó ÓÐÐ ¾¹ÐÐ Ö ÚÓÒØÓÞµ ÖÑ ÓÑÑÙØØÚº ½º¾º ÌØÐ ËØÖØ µº Ý ÓØØ x : A B ½¹ÐÐ ÐÞ ØÚ Ô ÓÐØÒ ÚÒÒ Mnd(K) (x,ψ) Ð ½¹ÐÐ ÚÐº Ý ω : x y ¾¹ÐÐ Ò ÔÓÒØÓ Ò ÓÖ ÚÒ ω : x y ÐÞ ω ¾¹ÐÐ Mnd(K)¹Ò ÑÐÐ (x,ψ) (y,φ) ½¹ÐÐ ÞØØ ÓÖ ÐÞ ÝÖØÐÑòº ËØÖØ ØØÐÐ ÞÓÒÒÐ Þ ÖÑÞØØØ Ò ÒÝ Ð ÞÙ ÀÓÔ¹ÐÖ ÖÑÒÝº ½º¾º ÈÐº ÖÑÐÝ ÓØØ B ÐÖ ØÒ ØÚ Ô ÓÐØ ÚÒ Þ Ð ØÖÙØÖ ÞØØº B ÐØ ÐÖ ØÖÙØÖ Id ω M k ÑÓÒÓ Ð ØÖÙØÖ Ò ÐØ ÐÞ M B ¹Öº ½º¾º ÈÐº ÖÑÐÝ A ÐÖ C ÓÐÖ ØÒ ØÚ Ô ÓÐØ ÚÒ Þ Ð ØÖÙØÖ ÞØØº Þ (M k,( ) A) ÑÓÒ ÐØ ÐÞ M C ¹Ö Þ (M k,( ) C) ÓÑÓÒ ÐØ ÐÞ M A ¹Ö Þ A¹Ø C¹Ø ÞÙÐ ÓÐ ÚÝ ÞØÖÙØÚ Þ ÐÝÓ ÞÞ ((k,a),(c,ψ)) ÓÑÓ¹ Ò Ó Mnd(Bim)¹Òµº vy yv ω Id ½º¾º ÀÓÔ¹ÐÖÓÓ ÐÖ ÓÐÓÑ ÐØÐ ÒÓ Ø ÒÑ ÓÑÑÙØØÚ ÐÔÝòÖò ØÖ ÌÙ ÒÚÞ òþ Ø ÑÒØ ÖÑÒ Ú ÐÐÐØ ¾ º ÌÙ ÖØÐ R ¹ÐÖ ÐÒ¹ ÚÞ Ø ÞÒ ÐØ Ð ÓÖÒ ÂºÀº ÄÙ ½ ÑÙÒ ÒÝÓÑ Òµ ÒÔÒÒ ÐÖÓ

17 ½º¾º ÀÇÈ¹ÄÊÇÁÇÃ ½ ÐÒÚÞ ÐÓÓØØº ÐÖ Þ ÑÓ ÚÚÐÒ ÐÖ ÞÐ Ó ÞÔÒ ÐØÐ ÒÓ¹ ØØ ÐÖÓÓÖº ÈÐ ÙÐ Ý R ÝòÖò ÐØØ ÐÖÓÓ ÐÐÑÞØ ÑÓÙÐÙ ØÖ Ù ÑÓÒÓÐØ ÚÐ Ý Þ R ÑÓÙÐÙ Ó ØÖ ÑÒ ÐØ ÙÒØÓÖ ÞÓÖ ÑÓÒÓÐØ ÚÐº Þ ÖÓÐÓÑÒ Ø ÐØÖÒØÚ Ú ÐØ ØÐ ÐØ ÖÖ ÑÐÝÒ ØÓÚ ØÙÐÓÒ ÓØ ÚØÐÒ Ñ Ý ÀÓÔ¹ÐÖÓØÐº ÀÓÔ¹ÐÖ ÒØÔ H H ÒØØ Ð¹ ÔÞ ÒÚÖÞ Þ (f g)(h) := f(h 1 )g(h 2 ) ÓÒÚÓÐ ÞÓÖÞ Ö ÒÞÚº Ý R ÝòÖò ÐØØ B ÐÖÓ ØÒ ÞÓÒÒ Þ ÒØØ ÐÔÞ ÒÑ ÐÑ Þ ÐÔ R¹ÓÝòÖò Þ ÐÔ R¹ÝòÖò ÞØØ ÑÓÙÐÙ ÐÔÞ ÐÓØØµ ÓÒÚÓÐ ÐÖ Òº ÞÒ ÔÖÓÐÑ Ø¹ Ð Ö ÂºÀº ÄÙ Þ ÒØÔ ÜÑ ÑÓÐÑÞ ÓÞ B B B R B ÔÑÓÖÞÑÙ Ý ÞÐ Ø ÞÒ ÐØ ½ Ñ ÞÓÒÒ Ó ÞÑÔÓÒØÐ ÒÑ ØÖÑ ÞØ º ËÞÐ ÒÝ ÃÓÖÒÐÐÐ Þ Ú ÐÐÐÓÞ ÙÒ Ð Ý ÓÐÝÒ ÀÓÔ¹ÐÖÓ ÓÐÓÑ ÑÐ¹ ÓØ ÚÓÐØ ÑÐÝ ØÖÐÑÐØ ÞÑÔÓÒØÐ ØÖÑ ÞØ ÑÐÝ ÐÐÑ ÞÑÑØ¹ Ö ÐÖ Ö ÑÒÐ Øµ ÓÐÝÒ ÐÝÞØÒ ÑÓÖ ÀÓÔ¹ÐÖ Ñ Ö ÒÑ ÞÒ ÐØ ÑÐÝÖ ÀÓÔ¹ÐÖ Ö ÚÓÒØÓÞ ÖÑÒÝ ÑÒÐ ÞÐ Ö ØÖ ÞØØº Þ ÐØÐÙÒ Ú ÓÐØ Ò ÞÖÒØ Ý ÀÓÔ¹ÐÖÓÒ Ý ÓØØ ÐÖ Ò ÒÑ Ý ÒÑ Ø ÓÑÔØÐ ÐÖÓ ØÖÙØÖ ÚÒ ÐÒ R ÐÐØÚ R op ÐØØº ÒÒ ÑÐÐÒ ÚÒ Ø ÓÒÚÓÐ ÐÖ ÑÐÝØ Ý ÅÓÖØ¹ Þ Ø Þº Þ ÒØØ ÐÔÞ ÅÓÖØ¹ Þ Ý ÑÓÙÐÙ Ò ÐÑ Þ ÒØÔ Ñ Ò Þ ÝÑ ÒÚÖÞ ÅÓÖØ¹ Þ ÞÓÖÞ Ö ÒÞÚ ½ º ÀÓÔ¹ÐÖÓÓÖÐ ÖØ Ð ½ ¹ Ò ÖÖ Þ ØÖ ÞÓÖØÓÞØÙÒ ÑÓÖ Þ Ý Ò ÐØ ÒØÔ Ö ÙÐ ØÚ Ý Þ Ý ÐÖÓ ØÖÙØÖ ÖÙÒ Ò ÒÓÖÑ Ø ÐÒØ ÑÔØ Ñ Ð Þ ÒØÔ ÒÒ ÒÚÖÞ ÐØÐº ÅÒØ Ý ËØÖØ ¹Ò Ø ÖÝÐØ Þ Þ ÜÑ ÒÒ ÐÐÒ Ñ ÓÝ ÞÓÖÒ ÑÓÒÓ Ð ÐØ ÙÒØÓÖ B¹ÑÓÙÐÙ Ó ØÖ Ð Þ R¹ÑÓÙÐÙ Ó Ø¹ Ö Ö ÙÐ ÖÞ ¹ÙØÓÒÑ ØÖÙØÖ Ø º ËÙÒÙÖ ¹Ò Ý ÐØÐ ÒÓ ÀÓÔ¹ÐÖÓ ÓÐÑØ Ú ÓÐØ ÑÐÝÒ ÐØ ÙÒØÓÖ Þ ÖØ ÓØ ÚÝ Ð ÓÑ¹ ÓØµ ÖÞº ËÙÒÙÖ Ð ÒÐ ÒÑ ÚØÞ ÞÓÒÒ Ý H H ÒØÔ ÐØÞ º ½º¾º½º Þ ÔØÚ R¹ÝòÖò R¹ÓÝòÖò ÐÖÓÓ Ý k ÓÑÑÙØØÚ ÝòÖò ÐØØ ÓµÐÖ ÓµÑÓÒÓÓ k¹ñóùðù Ó ÑÓÒÓ Ð ¹ ØÖ Òº Ý ØØ ÞÐ ÒÑ ÐØØÐÒÐµ ÓÑÑÙØØÚ R k ÐØØµ ÐÖ ØÒ Ó ÚÝ Ðµ R¹ÑÓÙÐÙ Ó ØÖ ÒÑ ÑÓÒÓ Ð º ÌÒØØ ÞÓÒÒ Þ R¹ ÑÓÙÐÙ Ó ØÖ Ø ÑÐÝ ÑÓÒÓ Ð Þ R¹ÑÓÙÐÙ ØÒÞÓÖ ÞÓÖÞ ÖÚÒ ÑÓÒÓ Ð Ý ÖÙÐ Ö ÑÓÙÐÙ Rº ÀÒ ÐÝÓÞÞÙ ÞÓÒÒ ÓÝ Þ ÖØ ÑÓ Ø Ø ÐÒØ ØÖÚ Ð R = k ØÒ k¹ñóùðù Ó ØÖ ØÐ Ö ÞØÖÒØ ØÖØÐÑÞÞ k¹ñóùðù Ó ØÖ Øºµ ½º¾º Òº ÑÓÒÓÓØ Þ R¹ÑÓÙÐÙ Ó ØÖ Ò R¹ÝòÖòÒ ÚÙº ÜÔ¹ ÐØÒ Ý R¹ÝòÖò Ý R¹ÑÓÙÐÙ A ÐÐ ØÚ µ : A R A A ÞÓÖÞ µ η : R A Ý µ R¹ÑÓÙÐÙ ÐÔÞ Ð ÑÐÝ ÞÓ Ó ÞÓØÚØ Ý ÐØØ¹ ÐÒ Ø ÞÒ ÐØº R¹ÝòÖò ÓÑÓÑÓÖÞÑÙ ÞÓÖÞ Ð Ý Ð ÓÑÔØÐ R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Óº ÎÝ ÞÖ ÓÝ Ý R¹ÝòÖò ÐÖØ ÝØÐÒ R A ÐÖ ÓÑÓÑÓÖÞÑÙ Ðº ÙÝÒ A Ý R¹ÝòÖò ÓÖ ÐÐ ØØ Ý ÐÖ ØÖÙØÖ ÚÐº Þ Ý Ø Þ R ÐÖ k R Ý Ø Þ A R¹ÝòÖò R A Ý ÚÐ ÓÑÔÓÒ ÐÚ ÔÙ ÞÓÖÞ Ø

18 ½ ½º ÂÌº ÎÌË Ô Þ A A A R A ÔÑÓÖÞÑÙ Ø Þ R¹ÝòÖò A R A A ÞÓÖÞ ÚÐ ÓÑÔÓ¹ Ò ÐÚº Ð ÞÒ ÐÚ ÓÝ ÑÒ η ÑÒ µ Óµ R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ η Óµ Ý ÒÒÝÒ Ð ØØ ÓÝ η ÐÖ ÓÑÓÑÓÖÞÑÙ º ÓÖØÚ Ý η : R A ÐÖ ÓÑÓÑÓÖÞÑÙ R¹ÑÓÙÐÙ ØÖÙØÖ Ø ÒÙ Ð A¹Òº A ÞÓÖÞ ÝÒ ÐÝÓÞÓØØ R¹Ò Ý ÐÖØ ÑÒØ Þ A A A R A ÔÑÓÖÞÑÙ Ý ÝÖØÐÑò µ : A R A A ÐÔÞ ÓÑÔÓÞº µ ÞÓÖÞ Þ η Ý ÖÚÒ A Ý R¹ÝòÖòº ½º¾º Òº Ý A R¹ÝòÖò Ó ÑÓÙÐÙ Ý Ó R¹ÑÓÙÐÙ M ÐÐ ØÚ Ý M R A M Ó R¹ÑÓÙÐÙ ÐÔÞ Ð ÑÐÝ ÞÓ Ó ÞÓØÚØ Ý ÐØØÐÒ Ø Þ ÐØº Þ ÐÞ Þ ÓÒÓÐØÑÒØÞ ÓÒÐÒ ÒÒÝò Ð ØÒ ÓÝ Þ ÔÔÒ ÙÝÒÞ ÑÒØ Þ A ÐÖ Ó ÑÓÙÐÙ º Ý A R¹ÝòÖò Ó ÑÓÙÐÙ Ò ÓÑÓ¹ ÑÓÖÞÑÙ Ó R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó ÑÐÝ ÓÑÔØÐ Ø ÓÐº Þ ÑÝÞ Þ A ÐÖ Ó ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÚÐº ËÞÑÑØÖÙ Ò Ò ÐÙ R¹ÝòÖò Ð ÑÓÙÐÙ Ø ÞÓ ÓÑÓÑÓÖÞÑÙ Øº ½º ¼º ÈÐº ÖÑÐÝ R ÐÖ ØÒ Þ R R k¹ñóùðù ÓÑÓÑÓÖÞÑÙ Ó Ý End(R)¹ÖÐ ÐÐØ R¹ÝòÖòØ ÐÓØÒº Þ ÐÖ ØÖÙØÖ Ø ÐÔÞ ÓÑÔÓÞ R End(R) r r( ) ÐÖ ÓÑÓÑÓÖÞÑÙ º Ó¹ Ð Ø Ó ÞÖÔÒ Ð ÖÐ ØÚ Ô ÓÐØÓØ ØÖÑØ Þ R¹ÑÓ¹ ÙÐÙ Ó Þ R op ¹ÑÓÙÐÙ Ó ÞØØº Þ ÒÙ ÐØ R M R R opm R op ÞÓÑÓÖÞÑÙ ÒØ¹ ÑÓÒÓ Ð º Ý R¹ÝòÖò ÐÐÒØØØÒ ÞØ Þ R op ¹ÝòÖòØ ÖØ ÑÐÝ ÞÒ ÒØ¹ÑÓÒÓ Ð ÞÓÑÓÖÞÑÙ Ú Þ Øº Ý η : R A ÐÖ ÓÑÓÑÓÖÞÑÙ Ð ÐÐÑÞØØ R¹ÝòÖò ØÒ Þ Þ η : R op A op ÐÖ ÓÑÓÑÓÖÞÑÙ Ð ÐÐÑÞØØ R op ¹ÝòÖòØ ÐÒØº ÚØÞÒ ÐÒ Ò ÓÒØÓ Ð ÞÒ Þ ÑÙÒÖ Þ R R op ¹ÝòÖòº ÒØ ÞÖÒØ Ý R R op ¹ÝòÖò ÐÖØ Ý η : R R op A ÐÖ ÓÑÓÑÓÖÞÑÙ Ðº Ú¹ ÚÐÒ ÑÓÒ η ÐÝØØ ØÒØØ Þ s := η( 1) : R A t := η(1 ) : R op A ÐÖ ÓÑÓÑÓÖÞÑÙ ÓØ ÑÐÝ ÖØ ÞÐØ ÓÑÑÙØ ÐÒ A¹Òº Ý A R R op ¹ ÝòÖòÒ ÒÝ ÓÑÑÙØ Ð R¹Ø ÚÒ ÐÒ Þ R A¹ÚÐ ÐÐØ Ð R¹ÑÓÙÐÙ ÓÒ Ø R A A r h s(r)h Þ A R ¹ÚÐ ÐÐØ Ó R¹ÑÓÙÐÙ ÓÒ Ø A R A h r t(r)h Þ R A¹ÚÐ ÐÐØ Ð R¹ÑÓÙÐÙ ÓÒ Ø R A A r h ht(r) Þ A R ¹ÚÐ ÐÐØ Ó R¹ÑÓÙÐÙ ÓÒ Ø A R A h r hs(r)º ÒØ Ø Ó ÓÑÒ ÚÐ ÒÝ R¹ÑÓÙÐÙ ØÖÙØÖ Ø Ò ÐØÙÒ A¹Òº Þ R A R R R A R R¹ÑÓÙÐÙ ÒØÖÙÑ Ø A R A¹ÚÐ ÐÐ A ÒÑ ÚÐ ÚØØ ÌÙ ÞÓÖÞØ Ò ÒÚÞÞº A R A Ñ R R op ¹ÝòÖòº ÐÖ ØÖÙØÖ Ø Ð Ò ÐØµ ØÓÖÓÒÒØ ÞÓÖÞ R R op A R A r r s(r) R t(r ) ÐÖ ÓÑÓÑÓÖÞÑÙ º ËÞÑÑØÖÙ Ò ÒØ ÐÝØØ ØÒØØ Þ R A R R RA R R¹ÑÓÙÐÙ ÒØÖÙÑ Ø ÑÐÝÖ ÐÐ Ò ÐÝÒ A R Aº

19 ½º¾º ÀÇÈ¹ÄÊÇÁÇÃ ½ ½º ½º Òº ÓÑÓÒÓÓØ Þ R¹ÑÓÙÐÙ Ó ØÖ Ò R¹ÓÝòÖòÒ ÚÙº ÜÔÐØÒ Ý R¹ÓÝòÖò Ý R¹ÑÓÙÐÙ C ÐÐ ØÚ δ : C C R C Ó ÞÓÖÞ µ ε : C R ÓÝ µ R¹ÑÓÙÐÙ ÐÔÞ Ð ÑÐÝ ÞÓ Ó Ó ÞÓØÚØ ÓÝ ÐØØÐÒ Ø ÞÒ ÐØº R¹ÓÝòÖò ÓÑÓÑÓÖÞÑÙ Ó ÞÓÖÞ Ð ÓÝ Ð ÓÑÔØÐ R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Óº Ý R¹ÓÝòÖò ÐÐÒØØØÒ ÞØ Þ R op ¹ÓÝòÖòØ ÖØ Ñ Ø Þ R M R R opm R op ÒØ¹ÑÓÒÓ Ð ÞÓÑÓÖÞÑÙ Ú Þº ÜÔÐØÒ Ý (C,δ,ε) ÓÝòÖò ØÒ C¹Ø ÑÒØ R op ¹ ÑÓÙÐÙ Ø ØÒØ Ó ÞÓÖÞ δ : C C R opc c c 2 R op c 1 ÓÝ ε : C R op º ½º ¾º Òº Ý C R¹ÓÝòÖò Ó ÓÑÓÙÐÙ Ý Ó R¹ÑÓÙÐÙ M ÐÐ ØÚ Ý M M R C Ó R¹ÑÓÙÐÙ ÐÔÞ Ð ÑÐÝ ÞÓ Ó Ó ÞÓØÚØ ÓÝ ÐØØÐÒ Ø Þ ÐØº Ý C R¹ÓÝòÖò Ó ÓÑÓÙÐÙ Ò ÓÑÓÑÓÖÞÑÙ Ó R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó ÑÐÝ ÓÑÔØÐ ÓØ ÓÐº ËÞÑÑØÖÙ Ò Ò ÐÙ R¹ÓÝòÖò Ð ÓÑÓÙÐÙ Ø ÞÓ ÓÑÓÑÓÖÞÑÙ Øº ½º º Òº Ý Ð R¹ÐÖÓ B ÐÐ Ý (s : R B,t : R op B) R R op ¹ÝòÖòÐ Ý ( R B R,δ,ε) R¹ÓÝòÖòÐ ÑÐÝÖ Þ Ð ÜÑ ØÐ ÐÒ δ ÐÖØ ÑÒØ Ý B B R B ÐÖ ÓÑÓÑÓÖÞÑÙ B R B B R B ÒÐÞ ÓÑÔÓÞ B End(R) b ε(bs( )) ÐÔÞ ÐÖ ÓÑÓÑÓÖÞÑÙ º Ý Ð ÐÖÓ Ó ÞÓÖÞ Ö ÞÓ Ó δ(h) = h 1 R h 2 ËÛÐÖ¹ÀÝÒÑÒÒ ÒÜ ÐÐ Ø ÞÒ ÐÙº B ØØ ÞÐ h,h ÐÑÒ R ØØ ÞÐ r ÐÑÒ ÖÚ ÐÖÓ ÜÑ ÚØÞ ÐÓØ ÐØº h 1 t(r) R h 2 = h 1 h 2 s(r); 1 1 R 1 2 = 1 R 1; (h h) 1 R (h h) 2 = h 1 h 1 R h 2 h 2; ε(1) = 1; ε(h h) = ε(h sε(h)). Ñ Ó ÓÖÒ ÞÖÔÐ Ñ Ó ÜÑ Ó ÓÐÐ Ò ÐÐ Þ Þ Ð ÓÖÒ ÞÖÔÐ ÜÑ ÑØØ Ð Ò ÐØº ÌÒØ Ò Ý B Ð R ÐÖÓÓØº Þ s : R B ÐÖ ÓÑÓÑÓÖÞÑÙ ÖÚÒ R BR R¹ÑÓÙÐÙ ÖÒÐÞ Ý R¹ÝòÖò ØÖÙØÖ ÚÐº ÅÚÐ Ò ÞÖÒØ R B R R¹ ÑÓÙÐÙ ÖÒÐÞ Ý R¹ÓÝòÖò ØÖÙØÖ ÚÐ ÑÓÙÐÙ ÐÔÞ Hom( R B R, R B R ) ÐÑÞ ÐÖ (f g)(h) := f(h 1 )g(h 2 ), f,g Hom( R B R, R B R ), h B, ÓÒÚÓÐ ÞÓÖÞ ÖÚÒ sε Ý Ðº À (s : R B,t : R op B) Ý R R op ¹ÝòÖò ÓÖ ÑÒÒ Ð B¹ÑÓÙÐÙ ØÖÑ Þ¹ Ø Ò R¹ÑÓÙÐÙ Þ r.m.r := s(r)t(r ).m Ø ÖÚÒº B¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó R ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ó º Å ÞÚÐ ÐØÞ Ý B M R M R ÐØ ÙÒØÓÖº ½º º ÌØÐ ËÙÒÙÖ µº ÖÑÐÝ (s : R B,t : R op B) R R op ¹ÝòÖò ØÒ ØÚ Ô ÓÐØ ÚÒ Þ Ð ØÖÙØÖ ÞØØº Ð ÐÖÓ ØÖÙØÖ Þ ÓØØ R R op ¹ÝòÖòÒ

20 ¾¼ ½º ÂÌº ÎÌË ÑÓÒÓ Ð ØÖÙØÖ Ð B¹ÑÓÙÐÙ Ó ØÖ Ò Ý ÓÝ B M R M R ÐØ ÙÒØÓÖ ÞÓÖÒ ÑÓÒÓ Ð º À B Ý Ð R ÐÖÓ ÓÖ (t : R B op,s : R op B op ) R R op ¹ÝòÖòº Ð ÐÖÓ ÜÑ ÐÝØØ ÞÓÒÒ Ñ ÓÒÐ Ð ÜÑ Ò Ø Þ ÐØ ½º º Òº Ý Ó R¹ÐÖÓ B ÐÐ Ý (s : R B,t : R op B) R R op ¹ ÝòÖòÐ Ý ( R B R,δ,ε) R¹ÓÝòÖòÐ ÑÐÝÖ Þ Ð ÜÑ ØÐ ÐÒ δ ÐÖØ ÑÒØ Ý B B R B ÐÖ ÓÑÓÑÓÖÞÑÙ B R B B R B ÒÐÞ ÓÑÔÓÞ B End(R) op b ε(s( )b) ÐÔÞ ÐÖ ÓÑÓÑÓÖÞÑÙ º Ý Ó ÐÖÓ Ó ÞÓÖÞ Ö ÞÓ Ó δ(h) = h 1 R h 2 ËÛÐÖ¹ÀÝÒÑÒÒ ÐÐ Ø ÞÒ ÐÙ ÞØØÐ Ð ÐÝØØ Ð ÒÜÐº Ó ÐÖÓÓ ÔÓÒØÓ Ò ÞÓ Þ R R op ¹ÝòÖò ÑÐÝ Ó ÑÓÙÐÙ ØÖ ÑÓÒÓ Ð ÐØ ÙÒØÓÖ Þ R¹ ÑÓÙÐÙ Ó ØÖ ÞÓÖÒ ÑÓÒÓ Ð º ½º º ÈÐº ÖÑÐÝ B ÐÖ Ð Ó ÐÖÓ R := k ÐØØº Þ R R op ¹ ÝòÖòØ ÐÖ s : k B t : k B ÐÖ ÓÑÓÑÓÖÞÑÙ Ó ÑÒÝ ÑÝÞ B Ý ÚÐº Þ R¹ÓÝòÖò ØÖÙØÖ Ø B ÓÐÖ ØÖÙØÖ º À Ý B Ð R¹ÐÖÓÒ R B Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ð R¹ÑÓÙÐÙ ÓÖ R Hom(B,R) Ð R¹Ù Ð ÐÐ ØØ Ý Ó R¹ÐÖÓ ØÖÙØÖ ÚÐº À B R Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ó R¹ÑÓÙÐÙ ÓÖ Hom R (B,R) Ó R¹Ù Ð Ð ØØ Ð Ý Ó R¹ ÐÖÓ ØÖÙØÖ ÚÐº ËÞÑÑØÖÙ Ò Ý ÑÐÐ ÓÐÐÓÒ Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ó ÐÖÓ Ð¹ ÐÐØÚ Ó Ù Ð ØÖÑ ÞØ ÑÓÒ Ð ÐÖÓ Ðº º ½º¾º¾º ÀÓÔ¹ÐÖÓ ÜÑ ÚØÞÑÒÝ ÀÓÔ¹ÐÖÓ ÜÑ Ò Ð ÑÓÐÑÞ ½ ¹Ð ÚÐº ÄÝÒ L R ØØ ÞÐ ÐÖ º ÌÒØ Ò Ý H ÐÖ Ø ÑÐÝ ÖÒÐÞ Ý Ð L¹ÐÖÓ ØÖÙØÖ ÚÐ s L : L H t L : L op H ÓÑÑÙØ Ð ÖØ ÞÐØò ÐÖ ÓÑÓÑÓÖÞÑÙ ÓÐ δ L : L H L L H L L L H L Ó ÞÓÖÞ Ð ÐÐØÚ ε L : L H L L ÓÝ Ð ÚÐÑÒØ Ý Ó R¹ÐÖÓ ØÖÙØÖ ÚÐ s R : R H t R : R op H ÓÑÑÙØ Ð ÖØ ÞÐØò ÐÖ ÓÑÓÑÓÖÞÑÙ ÓÐ δ R : R H R R H R R RH R Ó ÞÓÖ¹ Þ Ð ÐÐØÚ ε : R H R R ÓÝ Ðº Ø ÐÖÓ ÞØØ ÚØÐ Ñ Þ Ð ÓÑÔØÐØ ÜÑ Ø s L ε L t R = t R ; t L ε L s R = s R ; s R ε R t L = t L ; t R ε R s L = s L. ½ºµ ÃÒÒÝÒ Ð ØØ ÓÝ ÞÒ ÜÑ ÚØÞØÒ ε R s L : L R op ÐÖ ÞÓÑÓÖÞÑÙ ÑÐÝÒ ÒÚÖÞ ε L t R º ÍÝÒÝ ε L s R : R L op ÐÖ ÞÓÑÓÖÞÑÙ ÑÐÝÒ ÒÚÖÞ ε R t L º ½ºµ ØÐ Ð ØÒ ÖØÐÑ ÑÚØÐÒ ÚØÞ ÜÑ Øº (δ R L Id)δ L = (Id R δ L )δ R ; (δ L R Id)δ R = (Id L δ R )δ L, ½ºµ ÑÒØ H R H R R R H L L L H L ÐÐØÚ H L H L L L H R R RH R ÐÔÞ º ÃØ ÒØ Ô ÓÐØÒ ÐÚ ÐÖÓ ÑØ ÖÓÞ Ý ÅÓÖØ¹ Þ Ø Hom( L H L, L H L ) Hom( R H R, R H R ) ÓÒÚÓÐ ÐÖ ÞØØº Ø ÞÖÔÐ ÑÓÙÐÙ Hom( L H R, L H R )

21 ½º¾º ÀÇÈ¹ÄÊÇÁÇÃ ¾½ Hom( R H L, R H L )º ÅÓÖØ¹ Þ ÚÐÑÒÒÝ ÑòÚÐØ ÑÐÐ ÓÒÚÓÐ ÞÓÖ¹ Þ Ð ÓØØ ÞÞ H ÖÑÐÝ h ÐÑÖ (φ φ )(h) = φ(h 1 )φ (h 2 ), φ,φ Hom( L H L, L H L ); ½ºµ (ψ ψ )(h) = ψ(h 1 )ψ (h 2 ), ψ,ψ Hom( R H R, R H R ); (φ α)(h) = φ(h 1 )α(h 2 ), φ Hom( L H L, L H L ), α Hom( L H R, L H R ); (α ψ)(h) = α(h 1 )ψ(h 2 ), ψ Hom( R H R, R H R ), α Hom( L H R, L H R ); (ψ β)(h) = ψ(h 1 )β(h 2 ), ψ, Hom( R H R, R H R ), β Hom( R H L, R H L ); (β φ)(h) = β(h 1 )φ(h 2 ), φ Hom( L H L, L H L ), β Hom( R H L, R H L ); (α β)(h) = α(h 1 )β(h 2 ), α Hom( L H R, L H R ), β Hom( R H L, R H L ); (β α)(h) = β(h 1 )α(h 2 ), α Hom( L H R, L H R ), β Hom( R H L, R H L ), ÓÐ ÓÖ Ò ÚÞØØØ δ L (h) = h 1 L h 2 δ R (h) = h 1 R h 2 ÐÐ Ø ÞÒ ÐØÙº ÆÝÐÚ ÒÚÐÒ H H ÒØØ ÐÔÞ Hom( L H R, L H R ) ÑÓÙÐÙ ÐÑº ½º º Òº Ý ÀÓÔ¹ÐÖÓ ÐØØ ÚØÞ ØÖÙØÖ Ø ÖØº Ý H ÐÖ Ø ÐÐ ØÚ Ý Ð L¹ÐÖÓ Ý Ó R¹ÐÖÓ ØÖÙØÖ ÚÐ ÑÐÝÖ ½ºµ ½ºµ ÜÑ ØÐ ÐÒ ØÓÚ Þ ½ºµ ÅÓÖØ¹ Þ Ò Id Hom( L H R, L H R ) ÒÚÖ¹ Ø ÐØº ÜÔÐØÒ Þ ÙØ ÚØÐÑÒÝ ÞØ ÐÒØ ÓÝ ÐØÞ S Hom( R H L, R H L ) ÑÐÝÖ h 1 S(h 2 ) = s L ε L (h); S(h 1 )h 2 = s R ε R (h), h H. ½ºµ ½º º ÌØÐ ½¼ ÈÖÓÔÓ ØÓÒ ¾º µº Ý H ÀÓÔ¹ÐÖÓ S ÒØÔ Ö ÚØÞ ÐÐØ Ó ØÐ ÐÒº S ÓÑÓÑÓÖÞÑÙ Þ (s R : R H,t R : R op H) R R op ¹ÝòÖòÐ Þ (s R = t L ε L s R : R H,s L ε L s R : R op H) R R op ¹ÝòÖò ÐÐÒØØØ S ÓÑÓÑÓÖÞÑÙ Þ (s L : L H,t L : L op H) L L op ¹ÝòÖòÐ Þ (s L = t R ε R s L : L H,s R ε R s L : L op H) L L op ¹ÝòÖò ÐÐÒØØØ S ÓÑÓÑÓÖÞÑÙ ( L H L,δ L,ε L ) L¹ÓÝòÖòÐ ÓÝòÖò ÑÐÝ Þ ε L s R : R op L ÐÖ ÞÓÑÓÖÞÑÙ ÐØÐ ÒÙ ÐØ R opm R op = L M L ÑÓÒÓ Ð ÞÓÑÓÖÞ¹ ÑÙ Ú Þ ( R H R,δ R,ε R ) R¹ÓÝòÖò ÐÐÒØØØØ S ÓÑÓÑÓÖÞÑÙ ( R H R,δ R,ε R ) R¹ÓÝòÖòÐ ÓÝòÖò ÑÐÝ Þ ε R s L : L op R ÐÖ ÞÓÑÓÖÞÑÙ ÐØÐ ÒÙ ÐØ L opm L op = R M R ÑÓÒÓ Ð ÞÓÑÓÖÞ¹ ÑÙ Ú Þ ( L H L,δ L,ε L ) L¹ÓÝòÖò ÐÐÒØØØØº Þ ½º ÌØÐÐ ÒÝÐÚ ÒÚÐ ÓÝ Ò Þ ØÒ Ý ÀÓÔ¹ÐÖÓ ÒØÔ ØÚ Ó ÐÖÓÓØ Ò Ð ÚÐÑÒÒÝ ÐÔÞ ÞØ Ð ÐÖÓ ØÖÙØÖ Ø ÐÖ ÐÔÞ Ð Þ ÒØÔÐ ÐÐØÚ ÒÒ ÒÚÖÞÚÐº ÀÓÔ¹ÐÖÓÓ ÜÑ ÞÒ ØÓÐ ÑÓÐÑÞÚ ½ º ÞØÒ ØÐ ÐØº ½º º ÅÝÞ º ÆÝÓÒ ÓÒØÓ Ò ÐÝÓÞÒÙÒ ÓÝ Þ ½ºµ ÜÑ H R H R R RH L L L H L ÐÐØÚ H L H L L L H R R R H R ÐÔÞ Ö ÚÓÒØÓÞÒº ÌÙÙ ÙÝÒ ÓÝ Þ Ð ÝÒÐ Ð ÓÐÐ Ò ÞÖÔÐ ÐÔÞ ØÓÖÞ Ð (H R H) L H¹Ò Ö ÞØÐ Ó ÓÐÐÓÒ ÐÐ Ô H R (H L H)¹Ò Ö ÞØÐº Þ ÞÓÒÒ ÒÑ Ö ÞÐ¹ ÑÞ R H R R RH L L L H L ¹Ò ÑÐÝÒ ÐØÒ ÑØ ÞØØ ÚÒÒº ÆÓ ÒÝÐÚ ÒÚÐ

22 ¾¾ ½º ÂÌº ÎÌË (H R H) L H (H R H) L H ÐÔÞ ÒØÚ (H R H) L H (H R H) L H ÒÑ Þ ÐÐ ØÓÚ ÐØÚ ÒÐÐ ÒÑº Ý Þ ÓÐÝÒ ϕ ÐÔÞ ØÒ ÑÐÝ ÑÓÒÙ (H R H) L H¹Ò ÚÒÒ ÖØÐÑÞÚ ÒÑ ØÖ ÞØØ H R H L H¹Ö ÔÐº H R ε R L H : (H R H) L H H L Hµ ϕ(δ R L Id)δ L Þ ØÐØ Ö ÒÑ ÐÐÑÞØ Þ ½ºµ ÜÑº ËÛÐÖ ÒÜØ ÞÒ ÐÚ ÒÓ h 1 1 R h 1 2 L h 2 = h 1 h 2 1 L h 2 2 ÑÒØ R H R R RH L L L H L ÐÑ ϕ(h 1 1 R h 1 2 L h 2 ) = ϕ(h 1 h 2 1 L h 2 2) ÝÒÐ ÑÚÐ Ó ÓÐÐ ÖÓ ÞÙÐ Ò ÐØº ËÒÓ ÐÝÒ ØÔÙ Ø ÑÑ ÐØÐ ÚÝ ÚØÓÞÓØØ Ò ÐÚØØØ ÑØØ Ø Ñ ÚØ Ö ÞÓÖÙÐØº ÀÓÔ¹ÐÖÓÓ ÝÒ ÀÓÔ¹ÐÖ Ý ÀÓÔ¹ÐÖ µ ÐØÐ ÒÓ Ø ¹ ÚØÞ ÖØÐÑÒº ½º¼º ÌØÐ ½ ÜÑÔÐ ºµº ÄÝÒ B Ý ÝÒµ ÐÖ B B B b b 1 b 2 Ó¹ ÞÓÖÞ Ð ε : B k ÓÝ Ð Ðº ½º Òµº ÞÒ ØÓ ÑØ ÖÓÞÒ Ý Ó ÐÖÓ ØÖÙØÖ Ø B¹Ò Þ {1 1 ε(b1 2 )} b B Ö ÞÐÖ ÐØØ Ý Ð ÐÖÓ ØÖÙØ¹ Ö Ø Þ {ε(1 1 b)1 2 } b B Ö ÞÐÖ ÐØØº À ØÓÚ B ÝÒµ ÀÓÔ¹ÐÖ S ÒØÔÐ Ðº ½º Òµ ÓÖ ÒØ ÐÖÓÓ S ÀÓÔ¹ÐÖÓ ØÖÙØÖ Ø Ò ÐÒ B¹Òº ÀÓÔ¹ÐÖÓÓ ÓÒØÓ ÐÐÑÞ ÓÝ ÐÖ ÐÖ ØØ ÑÐÝ ò ÖÓÒÙ ¹ ØÖ ÞØ Ò ÞÑÑØÖ Øº ÖÓÒÙ ¹ØÙÐÓÒ ÞØ ÐÒØ ÓÝ M Ú Ò Ò¹ Ö ÐØ ÔÖÓØÚ Ó ÚÝ Ðµ N¹ÑÓÙÐÙ Ó ÚÝ Ðµ ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÓÐ ÐÐ Hom N (M,N) ÞÓÑÓÖ M¹ÑÐ ÑÒØ N¹M ÚÝ M¹Nµ ÑÓÙÐÙ º ØØ ÑÐÝ ÐØ¹ ØÐ ÞØ ÐÒØ ÓÝ M N M ÖØ ÞÒ M Ú Ó ÔÐ ÒÝ Ò ÖØ ÞÒ ÑÒØ N¹M ÑÓÙÐÙ M¹N ÑÓÙÐÙ º ½º½º ÌØÐ ½ ËØÓÒ µº ÌÒØ Ò Ý N M ØØ ÑÐÝ ò ÖÓÒÙ ¹ØÙÐÓÒ ÐÖ ØÖ ÞØ Øº ÞÒ ÐØÚ ÑÐÐØØ ÑÒ M N¹ÑÓÙÐÙ ÒÓÑÓÖÞÑÙ Ò Ð¹ Ö ÑÒ Þ M N M M¹ÑÓÙÐÙ ÒØÖÙÑ ÑÐÝÒ ÐÖ ØÖÙØÖ ØÓÖÓÒ¹ ÒØ ÞÓÖÞ Ð ÐÐº ÐÐÒØØØ ÞÓÖÞ Ð ÓØØµ ÀÓÔ¹ÐÖÓÓ ØÚ ÒØÔÐº Þ ½º½ ÌØÐ ÐØÐ ÒÓ Ø ÒØ ÖÑÐÝ k¹ðò Ö ØÖ ÖÓÒÙ ¹ØÙÐÓÒ ØØ ÑÐÝ ò ½¹ÐÐ ÑØ ÖÓÞ Ø ÑÐÐ ÖØÐÑÒ Ù Ð µ ÀÓÔ¹ÐÖÓÓØ Ðº ¾¼ º ÌÓÚ ÔÐ ÀÓÔ¹ÐÖÓÓÖ ½ º ÞØÒ ØÐ ÐØº ½º¾º ÌØÐ ½ ÈÖÓÔÓ ØÓÒ º¾µº Ý L R ÐÖ ÐØØ ÀÓÔ¹ÐÖÓÒ H L L L H H R R R H h L h h h 1 R h 2 ÐÔÞ ØÚ ÞÞ s L : L H ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ó ÐÖÓÐ Ðº ½º ÞØµº ÎÝ Þ ÞÒ ÐØ ÚÐ ÑÒÒ ÀÓÔ¹ÐÖÓ R ¹ÀÓÔ¹ÐÖº ÞØ ÑÝÐ Ø ÓÑÒ ÐÚ ËÙÒÙÖ ÌÓÖÑ Ò ÒØÓÒ ºµ ÖÑ¹ ÒÝÚÐ ÞØ Ð ØÙ ÓÝ Þ M H R M R ÐØ ÙÒØÓÖ ÖÞ ÓÖÐ Þ ÖØ ØÖÙØÖ Øº Ð ÚØÞ ÓÝ ÞÓÒ Ó H¹ÑÓÙÐÙ ÓÒ ÑÐÝ Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ð R¹ÑÓÙÐÙ Ó Ý ÚÒ Ð Ù Ð Ù R M R ¹Òµ ÚÒ Ð Ù Ð Ù M H ¹Ò ÑÐÝØ Þ M H R M R ÐØ ÙÒØÓÖ ÖÞº Ý ËØÖØ ¹Ò ÔÖÓÐÑ Ø ÐØÐ ÒÓ Ò ÑÓÒÓ Ð ØÖ Ò Ñ¹ ÓÐÑÞÚµ ÞØ Ö Ø ÚÞ ÐØ ÓÝ Þ M H R M R ÐØ ÙÒØÓÖ ÑÐÝ H Ó R¹ÐÖÓÓÖ ÖÞ ÓÖÐ Þ ÖØ Ò Ð Ö ¹ÙØÓÒÑ ØÖÙØÖ Øº Þ Þ Þ¹ ØÓÚ ØÖÙØÖ Ø H¹Ò ¹ÙØÓÒÑ ØÖÙØÖ Ò ÒÚÞØ ËØÓÒ º

23 ½º º ÀÇÈ¹ÄÊÇÁÇÃ ÁÆÌÊýÄÄÅÄÌ ¾ ½º º ÌØÐ ½ ÌÓÖÑ ºµº ÄÝÒ B Ý Ó ÐÖÓº Ý Ö ¹ÙØÓÒÑ ØÖÙ¹ ØÖ ¹Ò Ø ÖÝÐØ ÖØÐÑÒµ B¹Ò ÚÚÐÒ Ý ØÚ ÒØÔÐ ÑÐÝ B¹Ø ÀÓÔ¹ ÐÖÓ Ø Þº ½º º ÀÓÔ¹ÐÖÓÓ ÒØÖ ÐÐÑÐØ ÓÝÒ ÀÓÔ¹ÐÖ ØÒ ÙÝÒÝ ÀÓÔ¹ÐÖÓÓ ÚÞ ÐØ Ò ÓÒØÓ ÒÓÖÑ ÒÝÖØ Þ ÐÖ ÞÖÞØÖÐ Þ ÒØÖ ÐÓ ØÒÙÐÑ ÒÝÓÞ ÚÐº ¹ ÞØ Þ ÐÔ ÙÐ ÞÓÐ Ð ½¼ ÑÙÒ Ð ÒÒ Ö ÖÒ ÚÞ ÐØ Ò¹ Ò ÑÐÐ ÒØÖ Ð ÓÐÓÑÒ ÑØÐ Ð ÑÐÝ ÐØ Ø ØÖÑØ ÓÐÝÒ ÐÖ ØÙÐÓÒ Ó ÐÖ Ö ÑÒØ Ð ÓµÝ ÞÖò Óµ ÞÔÖÐØ ÚÝ Ú Þ¹µ ÖÓÒÙ ¹ØÙÐÓÒ º ÑÐÐ ÒØÖ Ð ÓÐÓÑ Ø ÚÐ ÀÓÔ¹ÐÖÓÓÖ ÀÓÔ¹ÐÖ Ð ÑÒÝÙØØ ÑÓÒ ÒÐ ØØÐ ÞÓÐØº ÐÒÝ ÐÒ ÓÝ Ñ ÀÓÔ¹ÐÖ ØÒ ÝØÐÒ ÓÑÑÙØØÚ ÝòÖò ÐØØ ÐÖ ÐÐº ÓÐÖµ ØÙÐÓÒ ÖÐ ÚÒ Þ Ý ÀÓÔ¹ÐÖÓÒ ÒÝ Ô ÖÓÒÒØ ÞÓÑÓÖµ ÝòÖòØ ÐÐº Ø ÓÝòÖòØµ ÐÐ ÚÞ ÐÒÙÒ ÒÑ ÓÑÑÙØØÚ ÐÖ ÐØØº ½º º½º ÁÒØÖ ÐÓ ËÙÒÙÖ ½º ÌØÐÒ ÞØØ ÞÖÚØÐ ÞÖÒØ H Ý Ð L¹ÐÖÓ ÓÖ L Ð H¹ÑÓÙÐÙ h.l := ε(hs(l)) Ø ÖÚÒº Þ Ð Ò ÞØ ØÒÝØ ÞÒ Ð Ðº ½ºº Òº ÄÝÒ B Ý Ð L¹ÐÖÓº Ý M Ð B¹ÑÓÙÐÙ ÒÚÖ Ò Ò ÒÚÞÞ BHom(L,M) = {n M h B, h.n = sε(h).n} k¹ö ÞÑÓÙÐÙ ÐÑØº Ð ÒØÖ ÐÓ B¹Ò Ð ÖÙÐ Ö B¹ÑÓÙÐÙ ÒÚÖ Ò º ËÞÑÑØÖÙ Ò Ò ÐÙ Ý Ó ÐÖÓ Ó ÑÓÙÐÙ Ò ÒÚÖ Ò Ø Þ ÒØÖ ÐÓØ ÑÒØ Ó ÖÙÐ Ö ÑÓÙÐÙ ÒÚÖ Ò Øº Ý ÀÓÔ¹ÐÖÓÒ Ý ÞÖÖ ÚÒ ÐÒ Ý Ó Ý Ð ÐÖÓ ØÖÙØÖ Ý Þ ØÒ ÑÒ Ð¹ ÑÒ Ó ÒØÖ ÐÓØ ÖØÐÑÞØÒº Ý Ð ÚÝ Óµ ÒØÖ Ð ÒØÔ ÐØÐ Ô Ó ÚÝ Ðµ ÒØÖ Ðº Ý ÐÖÓ ÓÑÓÙÐÙ Ò Þ ÐÔ ÓÝòÖò ÓÑÓÙÐÙ Ø ÖØº ÖÑÐÝ B Ð L¹ÐÖÓ ØÒ L Ó B¹ÓÑÓÙÐÙ Þ L L L B = B l s(l) ÓØ ÖÚÒ Ð B¹ÓÑÓÙÐÙ Þ L B L L = B l t(l) ÓØ ÖÚÒº Ý B¹Ð ÒØÖ ÐÓ Ù Ð ÒØ ÚÞØØ Þ Ð ÓÐÑØ º ½ºº Òº ÄÝÒ B Ý Ð L¹ÐÖÓº Ý s¹òøö Ð B¹Ò Ý B L Ð B¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÞÞ Ý : B L ÐÔÞ ÑÖ (s(l)h) = l (h); t (h 2 )h 1 = s (h), l L, h H. Ý t¹òøö Ð B¹Ò Ý B L Ó B¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÞÞ Ý : B L ÐÔÞ ÑÖ (t(l)h) = (h)l; s (h 1 )h 2 = t (h), l L, h H. ËÞÑÑØÖÙ Ò Ò ÐÙÒ ÒØÖ ÐÓØ Ý B Ó R¹ÐÖÓÓÒ ÑÒØ Ð¹ ÐÐØÚ Ó ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÓØ B Rº

24 ¾ ½º ÂÌº ÎÌË À Ý B Ð L¹ÐÖÓÒ L B Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ð L¹ÑÓÙÐÙ Ý LHom(B,L) Ó L ÐÖÓ ÓÖ Ý s¹òøö Ð B¹Ò ÔÓÒØÓ Ò ÙÝÒÞ ÑÒØ Ý Ó Ò¹ ØÖ Ð L Hom(B,L)¹Òº À B L Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ó L¹ÑÓÙÐÙ Ý Hom L (B,L) Ó L ÐÖÓ ÓÖ Ý t¹òøö Ð B¹Ò ÔÓÒØÓ Ò ÙÝÒÞ ÑÒØ Ý Ó ÒØÖ Ð Hom L (B,L)¹Òº Ý ÀÓÔ¹ÐÖÓÒ Ý ÞÖÖ ÚÒ ÐÒ Ý Ó Ý Ð ÐÖÓ ØÖÙØÖ Ý Ý ÀÓÔ¹ÐÖÓÓÒ ÒÝÐ ÒØÖ Ð ÖØÐÑÞØ s¹ t¹òøö ÐÓ Þ ÐÔ Ð¹ Ó ÐÖÓÓÓÒº À s¹òøö Ð Ð ÐÖÓÓÒ ÓÖ S t¹òøö Ð ÙÝÒÞÒ Ð ÐÖÓÓÒº ½º º¾º Å ¹ØÔÙ ØØÐ ÄÝÒ H Ý ÀÓÔ¹ÐÖÓ L R Þ ÔÔÒ ÒØ¹ÞÓÑÓÖµ ÐÖ ÐØØº Ñ¹ ÐÐ ÖØÐÑÒ Ðº ½º ÌØÐµ ÒÓÖÑ ÐØ ÒØÖ ÐÓ ÐØÞ Ò H¹Ò ÒÝ s L : L H t L : L op H s R : R H ÐÐØÚ t R : R op H ÐÖ ØÖ ÞØ Ð Ý ÞÖò ¹ Þ ÐÐØÚ ÞÔÖÐØ ÓÞ ÚÒ Þº Ý R H ÐÖ ØÖ ÞØ Ø ÞÔÖ Ð Ò ÑÓÒÙÒ H R H H ÞÓÖÞ Ð H¹ÑÓÙÐÙ ÔÑÓÖÞÑÙ º Þ R H ÐÖ ØÖ ÞØ ÓÖÐ ÐÐØÚ ÐÖÐµ Ð Ý ÞÖò ÑÒÒ ÓÐÝÒ Ó ÐÐØÚ Ðµ H¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ð ÑÐÝ Ð Ó ÐÐØÚ Ðµ R¹ÑÓÙÐÙ ÔÑÓÖ¹ ÞÑÙ ºµ ½ºº ÌØÐ ½¼ ÌÓÖÑ º½µº ÄÝÒ H Ý ÀÓÔ¹ÐÖÓ L R ÐÖ ÐØØº ÚØÞ ÐÐØ Ó ÚÚÐÒ º Þ s R : R H ØÖ ÞØ ÞÔÖ Ð t R : R op H ØÖ ÞØ ÞÔÖ Ð Þ s L : L H ØÖ ÞØ ÞÔÖ Ð Þ t L : L op H ØÖ ÞØ ÞÔÖ Ð Þ s R : R H ØÖ ÞØ ÓÖÐ Ð Ý ÞÖò t R : R op H ØÖ ÞØ ÓÖÐ Ð Ý ÞÖò Þ s L : L H ØÖ ÞØ ÐÖÐ Ð Ý ÞÖò t L : L op H ØÖ ÞØ ÐÖÐ Ð Ý ÞÖò ÄØÞ Ý Ð ÒØÖ Ð l Þ ÐÔ Ð L¹ÐÖÓÒ Ñ ÒÓÖÑ ÐØ Þ ε L (l) = 1 ÖØ¹ ÐÑÒ ÄØÞ Ý Ó ÒØÖ Ð l Þ ÐÔ Ó R¹ÐÖÓÒ Ñ ÒÓÖÑ ÐØ Þ ε R (l) = 1 ÖØÐÑÒ ε L : H L Ð ÔÑÓÖÞÑÙ Ð H¹ÑÓÙÐÙ Ó ØÖ Ò ε R : H R Ð ÔÑÓÖÞÑÙ Ó H¹ÑÓÙÐÙ Ó ØÖ Òº À ÓÒÐÔÔÒ ÑÐÐ ÖØÐÑÒ Ðº ½º ÌØÐµ ÒÓÖÑ ÐØ ÒØÖ ÐÓ ÐØÞ Ò H¹Ò Ø L ÐÐØÚ R ÐØØ ÓÝòÖò Ð ÓÝ ÞÖò Þ ÐÐØÚ Ó ÞÔÖÐØ ÓÞ ÚÒ Þº Ý R ÐØØ H ÓÝòÖòØ Ó ÞÔÖ Ð Ò ÑÓÒÙÒ H H R H Ó ÞÓÖÞ Ð

25 ½º º ÀÇÈ¹ÄÊÇÁÇÃ ÁÆÌÊýÄÄÅÄÌ ¾ H¹ÓÑÓÙÐÙ ÑÓÒÓÑÓÖÞÑÙ º H R¹ÓÝòÖò ÓÖÐ ÐÐØÚ ÐÖÐµ Ð ÓÝ ÞÖò ÑÒÒ ÓÐÝÒ Ó ÐÐØÚ Ðµ H¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ð ÑÐÝ Ð Ó ÐÐØÚ Ðµ R¹ÑÓÙÐÙ ÑÓÒÓÑÓÖÞÑÙ ºµ ½ºº ÌØÐ ½¼ ÌÓÖÑ º¾µº ÌÒØ Ò Ý H Ý ÀÓÔ¹ÐÖÓÓØ L R ÐÖ ÐØØº ÚØÞ ÐÐØ Ó ÚÚÐÒ º H ÑÒØ R¹ÓÝòÖò Ó ÞÔÖ Ð H ÑÒØ L¹ÓÝòÖò Ó ÞÔÖ Ð H ÑÒØ R¹ÓÝòÖò ÓÖÐ Ð ÓÝ ÞÖò H ÑÒØ R¹ÓÝòÖò ÐÖÐ Ð ÓÝ ÞÖò H ÑÒØ L¹ÓÝòÖò ÓÖÐ Ð ÓÝ ÞÖò H ÑÒØ L¹ÓÝòÖò ÐÖÐ Ð ÓÝ ÞÖò ÄØÞ Ý s¹òøö Ð λ Þ ÐÔ R¹ÐÖÓÓÒ Ñ ÒÓÖÑ ÐØ λ(1) = 1 ÖØÐÑÒ ÄØÞ Ý t¹òøö Ð λ Þ ÐÔ R¹ÐÖÓÓÒ Ñ ÒÓÖÑ ÐØ λ(1) = 1 ÖØÐÑÒ ÄØÞ Ý s¹òøö Ð λ Þ ÐÔ L¹ÐÖÓÓÒ Ñ ÒÓÖÑ ÐØ λ(1) = 1 ÖØÐÑÒ ÄØÞ Ý t¹òøö Ð λ Þ ÐÔ L¹ÐÖÓÓÒ Ñ ÒÓÖÑ ÐØ λ(1) = 1 ÖØÐÑÒ s R : R H Ð ÑÓÒÓÑÓÖÞÑÙ Þ ÐÔ Ó R¹ÐÖÓ Ó ÓÑÓÙÐÙ Ò ØÖ Ò t R : R H Ð ÑÓÒÓÑÓÖÞÑÙ Þ ÐÔ Ó R¹ÐÖÓ Ð ÓÑÓÙÐÙ Ò ØÖ Ò s L : L H Ð ÑÓÒÓÑÓÖÞÑÙ Þ ÐÔ Ð L¹ÐÖÓ Ð ÓÑÓÙÐÙ Ò ØÖ Ò t L : L H Ð ÑÓÒÓÑÓÖÞÑÙ Þ ÐÔ Ð L¹ÐÖÓ Ó ÓÑÓÙÐÙ Ò ØÖ Òº ½º º º Ú Þ¹µÖÓÒÙ ¹ØÙÐÓÒ ÖÐ ÄÝÒ H Ý ÀÓÔ¹ÐÖÓ L R ÐÖ ÐØØº ÑÐÐ ÖØÐÑÒ Ðº ½º ÌØÐµ ÒÑ ÒÖ ÐØ ÒØÖ ÐÓ ÐØÞ ÒÝ s L : L H t L : L op H s R : R H ÐÐØÚ t R : R op H ÐÖ ØÖ ÞØ ÖÓÒÙ ¹ØÙÐÓÒ ÚÐ Ðº ½º½ ÌØÐØ ÑÐÞ Þ µ Ô ÓÐØÓ º ½ºº ÌØÐ ½¼ ÖÖØÙÑ ÓÖÓÐÐÖÝ µº ÌÒØ Ò Ý H Ý ÀÓÔ¹ÐÖÓÓØ L R ÐÖ ÐØØº ÚØÞ ÐÐØ Ó ÚÚÐÒ º Þ s R : R H t R : R op H ØÖ ÞØ ÑÒÝ ÖÓÒÙ ¹ØÖ ÞØ Þ s L : L H t L : L op H ØÖ ÞØ ÑÒÝ ÖÓÒÙ ¹ØÖ ÞØ H R Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ó R¹ÑÓÙÐÙ ÐØÞ Ý s¹òøö Ð λ Þ ÐÔ Ó R¹ÐÖÓÓÒ ÑÖ H Hom R (H,R) h λ(h ) ÐÔÞ ØÚ

26 ¾ ½º ÂÌº ÎÌË Þ S ÒØÔ ØÚ RH Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ð R¹ÑÓÙÐÙ ÐØÞ Ý t¹ ÒØÖ Ð λ Þ ÐÔ Ó R¹ÐÖÓÓÒ ÑÖ H R Hom(H,R) h λ(h ) Ð¹ ÔÞ ØÚ L H Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ð L¹ÑÓÙÐÙ ÐØÞ Ý s¹òøö Ð λ Þ ÐÔ Ð L¹ ÐÖÓÓÒ ÑÖ H L Hom(H,L) h λ( h) ÐÔÞ ØÚ Þ S ÒØÔ ØÚ LH Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ó L¹ÑÓÙÐÙ ÐØÞ Ý t¹ ÒØÖ Ð λ Þ ÐÔ Ð L¹ÐÖÓÓÒ ÑÖ H Hom L (H,L) h λ( h) ÐÔÞ ØÚ ÄØÞ Ý Ð ÒØÖ Ð l Þ ÐÔ Ð L¹ÐÖÓÒ Ñ ÒÑ ÒÖ ÐØ Ò Þ ÖØÐÑÒ ÓÝ ÑÒØ Hom R (H,R) H, ϕ l 1 s R ϕ(l 2 ) RHom(H,R) H, ψ l 2 t R ψ(l 1 ) ÐÔÞ ØÚ ÄØÞ Ý Ó ÒØÖ Ð l Þ ÐÔ Ó R¹ÐÖÓÒ Ñ ÒÑ ÒÖ ÐØ Ò Þ ÖØÐÑÒ ÓÝ ÑÒØ LHom(H,L) H, ϕ s L ϕ(l 1 )l 2 Hom L (H,L) H, ψ t L ψ(l 2 )l 1 ÐÔÞ ØÚº ÒØ ÚÚÐÒ ØÙÐÓÒ ÓÐ ÖÒÐÞ ÀÓÔ¹ÐÖÓÓØ ÖÓÒÙ ¹ ÀÓÔ¹ÐÖÓÒ ÑÓÒÙº ÒØ ØØÐ ½¼ ÌÓÖÑ ºµ¹Ò ÑÐÒØ ÓÖÑ ÒÑ ÐÝ Ý ½º Å¹ ÝÞ Ò ÑÖØØØØ ÑØØº Þ ÒØÖ ÐÓ ÚÞ ÐØ ÚÐ Þ Ð ÐØØÐ Ø Ñ ÒÝ s L : L H t L : L op H s R : R H ÐÐØÚ t R : R op H ÐÖ ØÖ ÞØ Ú Þ¹ÖÓÒÙ ¹ ØÙÐÓÒ Ö Ðº ½¼ ÌÓÖÑ º¾µº ½º ºº ÖÓÒÙ ¹µ ÀÓÔ¹ÐÖÓÓ ÙÐØ ÅÒØ ÞØ Þ ½º¾º½ ÞØÒ Ø ÖÝÐØÙ Ý Ð ÚÝ Óµ R¹ÐÖÓÓÒ ÚÐÑÐÝ R¹ ÑÓÙÐÙ Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÓÖ ÑÐÐ Ù Ð ÖÒÐÞ Ý Ó ÚÝ Ðµ ÐÖÓ ØÖÙØÖ ÚÐº À Ø Ø Ý ÀÓÔ¹ÐÖÓ Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÚÐÑÐÝ ÖØÐÑÒ ÓÖ ÑÐÐ Ù Ð ÖÒÐÞ Ý ÐÖÓ ØÖÙØÖ ÚÐº ÆÑ ÑÖØ ÞÓÒÒ Þ ÐØÐ ÒÓ Ò ÞÒ ÞÒØÒ ÓÝ Ù Ð ÀÓÔ¹ÐÖÓ¹º ØÖÑ ÞØ ÐÐØ ÀÓÔ¹ÐÖÓ ÒØÔ ÚÐ ÚÐ ÓÑÔÓÞ ÙÝÒ ÒÑ Ò Ð ÒØÔÓØ Ý Ù Ð ÓÒ Ñ ÑÖØ ÖÖ ÑòÚÐØÖ ÒÞÚ Ý Ñ Þ ÖØº Þ ÒØÔÐ ÚÐ ÓÑÔÓÞ Þ Ý Ù Ð Ð Ý Ñ ÚÐ ÐÔÞ º ÀÓÔ¹ÐÖÓÓ Ý Þò Ó ÞØ ÐÝ ÖÐ ØÙØ ÓÝ Þ ÖØ ÙÐØ Ö ÒÞÚº ÌÒØ Ò Ý H ÖÓÒÙ ¹ÀÓÔ¹ÐÖÓÓØ L R ÐÖ ÐØØº Þ ØÒ H Þ L¹ ÐÐØÚ R¹ ÑÓÙÐÙ ØÖÙØÖ Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ø Ø ÑÒ ÒÝ Ù Ð ÖÒÐÞ Ð ÚÝ Óµ ÐÖÓ ØÖÙØÖ ÚÐº ËØ Þ ½º ÌØÐÒ Ð ØÓØØ ÞÓÑÓÖÞÑÙ Ó ÞÒ Ù Ð Ó ÐÐº ÐÐÒØØØ ÞØØ ÐÖÓ ÞÓÑÓÖÞÑÙ Ó ÓÑÒ ÐØ Ðº ½ ÌÓÖÑ º½µº ÞÐ ÓÑÔÓÒ ÐÚ Þ ÖØ ØÚµ ÒØÔ ØÖÒ ÞÔÓÒ ÐØ Ø ÖÑÐÝ Ù Ð ÐÐ ØØ Ý ØÚ ÒØÔÐº Å Ø Þ Ð ØØÐ ÐÐ ÒÒº ½ºº ÌØÐ ½ ÌÓÖÑ º½ Ò ÈÖÓÔÓ ØÓÒ º½µº Ý ÖÓÒÙ ¹ÀÓÔ¹ÐÖÓ ÒÝ Ù Ð ÒØ¹µÞÓÑÓÖ ÖÓÒÙ ¹ÀÓÔ¹ÐÖÓº

27 ½ºº ÄÇÁË¹ÃÁÌÊÂËÌËÃ ÀÇÈ¹ÄÊÇÁ ËÁÅÅÌÊÁýÎÄ ¾ ½ºº ÐÓ ¹ØÖ ÞØ ÀÓÔ¹ÐÖÓ ÞÑÑØÖ ÚÐ ÅÒØ ÞØ Þ ½º½º ÞØÒ Ð ØØÙ ÀÓÔÐÓ ¹ÐÑÐØ ÐÐÑÞ Ò ÞÐ ÓÔÓÖØ Ö ÐØ ÐÖ Ð Ò Ø ÖÝÐ ØÐ ÒÑ ÓÑÑÙØØÚ ÖÒ ÐÓÑØÖ ÔÖÒÔ Ð ÒÝÐ Ò ÑÓÐÑÞ º ÂÐÒ ÞØ Ð Þ Ð ÐÔ ÑÙØØ ÐÓ ¹ ÐÑÐØ ÀÓÔ¹ÐÖÓÓÖ ÚÐ ÐØÐ ÒÓ Ø Ðº Þ ÐÑÐØ Ñ Ò ÓÐÐ ÖÙÔÓ Ö ÐØ ÐÖ ÖÙÔÓ ÒÝÐ Ó ÒÑ ÓÑÑÙØØÚ ÑÐÐÒ ÐÖ Øº Ý Ð ÞÙ ÐÓ ¹Ø ØÚØ ÐÐÑÞØ ÐÓ ¹ ÓÔÓÖØÖ ÚÐ ÚØÓÞ ÒÐ¹ Ð ÑÒØ Ú ÞÔÖ Ð ÒÓÖÑ Ð ÚØ º ÚÚÐ ÞÑÒ ÒÑ ÑÖÒ ÓÐÝÒ ÒÖÒ ØÙÐÓÒ ÓØ ÑÐÝÐ ÀÓÔ¹ÐÖ Ö ÚÐ ÜÔÐØ ÚØÓÞ ÒÐÐ ÐÐÑÞØ¹ ÒÒ ÀÓÔÐÓ ¹ØÖ ÞØ Øº ÆÑ ÑÖØ ØÓÚ Ô ÓÐØ Ý ÓØØ ÀÓÔÐÓ ¹ ØÖ ÞØ ÞÑÑØÖ Ø ÐÖ ÐØ ÀÓÔ¹ÐÖ ÞØØº ÅÒØ ÐÒØ ËÞÐ ÒÝ Ú Ð ÑÙÒ Ò ÑÑÙØØØ ÑÒÞÒ Ö ÑÚ Ð ÞÓÐØ ÖÓÒÙ ¹µ ÀÓÔ¹ ÐÖÓÓÐ ÚØØ ÐÓ ¹ØÖ ÞØ Ø ÖÒº ÅÒØ ÞØ Þ ½º½º ÞØÒ ØØÒØØØ Ý ÀÓÔ¹ÐÖ ÚÐ ÚÐ ÐÓ ¹ØÖ ÞØ ÐØØ ØÙÐÓÒÔÔÒ Þ ÐÔ ÐÖ ÚÐ ÚÐ ØÖ ÞØ Ø ÖØº Å ÀÓÔ¹ÐÖ ØÓÚ ØÙÐÓÒ Ø ÞÒ ÐÚ ÀÓÔ¹ÐÖ Ð ÚÐ ØÖ ÞØ Ö Ö ÐÐØ ¹ Ó ÞÓÐØ ÑÒØ ÐÖ Ð ÚÐ ØÖ ÞØ Öº ÐÖÓÓÐ ÐÐØÚ ÀÓÔ¹ ÐÖÓÓÐ ÚÐ ØÖ ÞØ ÞØØ ÓÐ ÐÔÚØ ÐÚ ÐÒ ÚÒÒ Ð Ò ÓÝ Ý ÀÓÔ¹ÐÖÓÒ Ý ÞÖÖ Ø ÐÖÓ Ý Ð Ý Óµ ÚÒ ÐÒº ÌÒØØÒ ÐÓ ¹ØÖ ÞØ Ø Þ ÖÑÐÝÚÐ Þ Ð Ö Þ Ú ÞÓ¹ ÒÝ Ò Ø ÞØ Þ º ÓÒØÓ Ð ØÒÙÒ ØÓÚ ÓÝ ÀÓÔÐÓ ¹ØÖ ÞØ Ö ÚÓÒØÓÞ ØØÐ ÞÐ ÑÐÝ ØÖ ÞØØ ÀÓÔ¹ÐÖÓÓÖº ÒÓÒÙ ÐÔÞ ØÚØ ¹ Ö ÞÞ ÐÓ ¹ØÙÐÓÒ Öµ ÚÓÒØÓÞ Ð ÐØØÐØ ÑÓÐÑÞ Ð ÞÙ ØØÐ ÞÐ ÃÖÑÖ ÌÙ ½º½ ÌØÐØ ½½ ¹Ò ËÒÖ Ö Áº ØØÐØ ¹Ò ÖÞÞÓÒÚÐ ÅÒÒÚÐ ÝØØÑòÚµ ÐØÐ ÒÓ ØÓØØÑ ÀÓÔ¹ÐÖÓÓÖº Ö ÝÒ ØÖÙØÖ¹ØØÐØ ÅÓÖØ¹ÐÑÐØ Ø ÚÐµ ÞÒØÒ ½½ ¹Ò ÞÓÐØÑº ½ºº½º ÀÓÔ¹ÐÖÓÓ ÓÑÓÙÐÙ ÖØÐÑ ÞÖòÒ Ý B ÑÓÒÙ Óµ R¹ÐÖÓ Ð ÚÝ Óµ ÓÑÓÙÐÙ ÐØØ Þ ÐÔ R¹ÓÝòÖò ÓÑÓÙÐÙ Ø ÖØº Þ ½º ¾ Ò ÞÖÒØ Ø Ø Ý ÑÓÒÙµ Ó B¹ÓÑÓÙÐÙ Ý Ó R¹ÑÓÙÐÙ M ÐÐ ØÚ Ý : M M R RB R m m 0 R m 1 Ó R¹ÑÓÙÐÙ ÐÔÞ Ð ÓÐ ÑÔÐØ ÞÞ ÖØÒµ ÑÐÝÖ ÞÓ Ó Ó ÞÓ¹ ØÚØ ÓÝ ÐØØÐ ØÐ ÐÒº ÆÓ Ý Ó B¹ÓÑÓÙÐÙ Ò ÞÖÒØ Ó R¹ÑÓÙÐÙ ØÖÙØÖ ÚÐ ÖÒÐÞ ÄÑÑ ½ºº½µ ÞÖÒØ R¹ÑÓÙÐÙ ØØ Þ r.m := m 0.ε(s(r)m 1 ) Ð R¹Ø ÚÞØ ÚÐº ÖÖ Ø Ö ÒÞÚ ÑÒÒ Ó¹ ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ º Å Ø Ó B¹ÓÑÓÙÐÙ Ó M B ØÖ ÑÓÒÓ Ð ÒØ ÓÒ ØÖÙ ÐØ M B R M R ÙÒØÓÖ ÞÓÖÒ ÑÓÒÓ Ð º À ÓÒÐÒ B Ð ÓÑÓÙÐÙ Ò B M ØÖ ÑÓÒÓ Ð ÚÒ Ý ÞÓÖÒ ÑÓ¹ ÒÓ Ð ÙÒØÓÖ B M R opm R opº Ý B Ð L¹ÐÖÓ Ð ÚÝ Ó ÓÑÓÙÐÙ Ò ØÖ ÑÓÒÓ Ð Ð ÚÒ Ð ØÚ B M L opm L op ÐÐØÚ M B L M L ÞÓÖÒ ÑÓÒÓ Ð ÙÒØÓÖÖÐº Ý B Ó ÐÖÓ Ó ÓÑÓÙÐÙ ÐÖ Ø Ø Ø Ò ÐØÙ ÑÒØ ÑÓÒÓÓØ Ó B¹ÓÑÓÙÐÙ Ó ÑÓÒÓ Ð ØÖ Òº Ý ØØ ÞÐ M B¹ÓÑÓÙÐÙ ÓÒÚ¹ Ö Ò ÐØØ Þ M cob := {n M n 0 R n 1 = n R 1} ÐÑÞ ÐÑØ ÖØº Ý A Ó B¹ÓÑÓÙÐÙ ÐÖ ÓÒÚÖ Ò A cob Ö ÞÐÖ Ø ÐÓØÒº Ý ØÒØØ can : A A cob A A R B a A cob a a a 0 R a 1

28 ¾ ½º ÂÌº ÎÌË ÒÓÒÙ ÐÔÞ Øº À Þ ØÚ ÓÖ Þ A cob A ÐÖ ØÖ ÞØ Ø B¹ÐÓ ¹ ØÖ ÞØ Ò ÑÓÒÙº Ý H ÀÓÔ¹ÐÖÓÒ Ý ÞÖÖ ÚÒ ÐÒ Ý Ó R¹ÐÖÓ Ý Ð L¹ ÐÖÓº Ý ÓÑÓÙÐÙ Ø Ò Ó ÓÙÒ ÀÓÔ¹ÐÖÓ ÓÑÓÙÐÙ Ò ÚÒº ÐÝØØ ØÒØØ Þ Ð ÞÑÑØÖÙ ÓÐÑØº ½º¼º Ò ½½ ÒØÓÒ º¾µº Ý L R ÐÖ ÐØØ H ÀÓÔ¹ÐÖÓ Ó ÓÑÓÙÐÙ ÐØØ Ý M k¹ñóùðù Ø ÖØÒ ÑÐÝ Ý ÞÖÖ ÓÑÓÙÐÙ Þ ÐÔ Ó R¹ ÐÖÓÒ ÚÐÑÐÝ Ó R¹Ø Ý R : M M R RM R ÓØ ÖÚÒ ÓÑÓÙÐÙ Þ ÐÔ Ð L¹ÐÖÓÒ ÚÐÑÐÝ Ó L¹Ø Ý L : M M L L M R ÓØ ÖÚÒ ØÓÚ R Ó L¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ L Ó R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Ø ÓØ Ö Þ Ð ÝÒÐ ØÐ ÐÒº (Id R δ ) R L = ( R L Id) L (Id L δ ) L R = ( L R Id) R. ÞÞ R ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Þ ÐÔ Ð L¹ÐÖÓÒ L ÓÑÓÙÐÙ ÓÑÓ¹ ÑÓÖÞÑÙ Þ ÐÔ Ó R¹ÐÖÓÒº H¹ÓÑÓÙÐÙ Ó ÓÑÓÑÓÖÞÑÙ ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÑÒ Þ ÐÔ Ð L¹ÐÖÓÒ ÑÒ Þ ÐÔ Ó R¹ÐÖÓÒº Ó H¹ÓÑÓÙÐÙ Ó ÓÑÓÑÓÖÞÑÙ ØÖ Ø M H ¹ÚÐ ÐÐº ÅÐÒ¹ ÞØØ Ð Þ ÐÔ Ð L¹ÐÖÓ ÓÑÓÙÐÙ Ò ØÖ Ø M H L¹ÚÐ Ñ Þ ÐÔ Ó R¹ÐÖÓ ÓÑÓÙÐÙ Ò ØÖ Ø M H R¹ÚÐ ÐÐº ÃÓÖ ÓÒÚÒÒ¹ ÓÞ ÓÒÐÒ Þ ÐÔ Ó R¹ÐÖÓ ÓØ Ö Þ m m 0 R m 1 ÒÜ ÐÐ Ø ÐÐÑÞÞÙ Ñ Þ ÐÔ Ð L¹ÐÖÓ ÓØ Ö m m 0 L m 1 ¹Ø ÓÐ ÑÒØ ØÒ ÑÔÐØ ÞÞ ÖØÒº ËÞÑÑØÖÙ Ò ÖØÐÑÞÞ Ý ÀÓÔ¹ÐÖÓ Ð Ó¹ ÑÓÙÐÙ Øº À M Ý L R ÐÖ ÐØØ H ÀÓÔ¹ÐÖÓ Ó ÓÑÓÙÐÙ ÓÖ ØÒØØ M ÓÒÚÖ Ò Ø Þ ÐÔ Ó R¹ÐÖÓ ÓØ Ö Þ M ÞÓÒ m ÐÑ ÑÐÝÖ m 0 R m 1 = m R 1 ÐÐØÚ M ÓÒÚÖ Ò Ø Þ ÐÔ Ð L¹ÐÖÓ ÓØ Ö Þ M ÞÓÒ m ÐÑ ÑÐÝÖ m 0 L m 1 = m L 1º ½ ÓÖÖÒÙÑ ÈÖÓÔÓ ØÓÒ µ ÐÔ Ò Þ Ð ÖØÐÑÒ ÚØØ ÓÒÚÖ Ò Ó ÓÒÚÖ Ò Ó Þ ÙØ ÖØÐÑÒ Ø ÓÒÚÖ Ò ÓÐÓÑ Ý ÑÒÞÓÒ ØÒ ÑÓÖ Þ ÒØÔ ØÚº ½º½º ÌØÐ ½ ÓÖÖÒÙÑ ÌÓÖÑ µº ÖÑÐÝ L R ÐÖ ÐØØ H ÀÓÔ¹ ÐÖÓ ØÒ H¹ÓÑÓÙÐÙ Ó M H ØÖ ÑÓÒÓ Ð Þ Ð ÖÑ ÓÑÑÙØ¹ ØÚ ÒÒ ÞÖÔÐ ÐØµ ÙÒØÓÖÓ ÞÓÖÒ ÑÓÒÓ Ð º M H G R G L M H L M H R L opm L op = RM R ÀÒ ÐÝÓÞÞÙ ÓÝ Ö ÐÐÒÔÐ Ø ÒÑ ÑÖÒ ØØ ÞÐ ÀÓÔ¹ÐÖÓÓ ¹ ØÒ ÒÑ ÞÓÒÝØÓØØ ÓÝ ÒØ ÖÑÒ ÞÖÔÐ G L G R ÙÒØÓÖÓ ÞÓÑÓÖÞÑÙ Óº Þ ÞØ ÐÐØ ½½ ÌÓÖÑ º½µ ÞÓÒÝØ Ò Þ ½º ÅÝÞ Ò ÐÖØ ØÐ ÐØº G L G R ÙÒØÓÖÓ ÞÓÑÓÖÞÑÙ Ó ÞÓÒÝÓ ØÓÚ ÐØÚ ÑÐÐØØ ÔÐ ÙÐ H ÐÔÓ Ð L¹ÑÓÙÐÙ ÐÔÓ Ð R¹ÑÓÙÐÙ º Þ ½º½ ÌØÐÖ ÐÔÓÞÚ ÑÓÐÑÞØ ÚØÞº ½º¾º Òº Ý H ÀÓÔ¹ÐÖÓ Ó ÚÝ Ðµ ÓÑÓÙÐÙ ÐÖ ÑÓÒÓÓ Ó ÚÝ Ðµ H¹ÓÑÓÙÐÙ Ó ØÖ Òº

29 ½ºº ÄÇÁË¹ÃÁÌÊÂËÌËÃ ÀÇÈ¹ÄÊÇÁ ËÁÅÅÌÊÁýÎÄ ¾ ÜÔÐØÒ H ÀÓÔ¹ÐÖÓ L R ÐÖ ÐØØ ÓÖ Ý Ó H¹ÓÑÓÙÐÙ ÐÖ Ý R¹ÝòÖò ÑÐÝÒ R H Ý H R H H ÞÓÖÞ H¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Óº Ý H¹ÓÑÓÙÐÙ ÐÖ ÓÑÓÙÐÙ ÐÖ ÑÒØ ÐÔ ÐÖÓ¹ Òº À A Ý L R ÐÖ ÐØØ H ÀÓÔ¹ÐÖÓ Ó ÓÑÓÙÐÙ ÐÖ ÓÖ Ø ÐÔ ÐÖÓ ÓØ Ò ÑÐÐÒ Ø ÒÓÒÙ ÐÓ ¹ÐÔÞ Ø ØÒØØÒ can R : A BR A A R RH R, a BR a a a 0 R a 1 ½º½¼µ can L : A BL A A L L H L, a BL a a 0 a L a 1, ÓÐ B R Þ A ÓÒÚÖ Ò Ò Ö ÞÐÖ Ø ÐÐ Þ ÐÔ Ó R¹ÐÖÓ ÓØ Ö Ñ B L Þ A ÓÒÚÖ Ò Ò Ö ÞÐÖ Ø ÐÐ Þ ÐÔ Ð L¹ÐÖÓ ÓØ Öº À H ÒØÔ ØÚ ÓÖ B R = B L can R ÔÓÒØÓ Ò ÓÖ ØÚ can L ØÚº ÞÞ Ò Þ ØÒ Ý Ó H¹ÓÑÓÙÐÙ ÐÖ A ÔÓÒØÓ Ò ÓÖ ÐÓ ¹ ØÖ ÞØ B R = B L ¹Ò Þ ÐÔ Ó R¹ÐÖÓÐ ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ð L¹ÐÖÓÐº ½ºº¾º Ý ÃÖÑÖÌÙ¹ØÔÙ ØØÐ ÃÖÑÖ ÌÙ Ð ÞÙ ½º½ ÌØÐ ÚØÞÔÔÒ ÐØÐ ÒÓ ØØ ÀÓÔ¹ÐÖÓ¹ ÓÖº ½º º ÌØÐ ½½ ÄÑÑ º Ò ÓÖÓÐÐÖÝ º µº ÌÒØ Ò Ý H ÀÓÔ¹ÐÖÓÓØ L R k¹ðö ÐØØ ÑÒ H R RH LH H L ÑÓÙÐÙ Ó ÑÒÝ Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÑÒ Þ ÒØÔ ØÚº ÄÝÒ A Ý Ó H¹ÓÑÓÙÐÙ ÐÖ Ñ ÐÒ ØÒ ÙÝÒÞ ÑÒØ ÖÑÐÝ ÐÔ ÐÖÓ ÓÑÓÙÐÙ ÐÖ µ ÐÝÒ B ÓÒÚÖ Ò Ö ÞÐÖ Ó¹ ÚÝ ÚÚÐÒ ÑÓÒ Ð ÐÔ ÐÖÓ ÓØ Ö ÒÞÚµº À (A A) coh = A B ÔÐº ÑÖØ A ÐÔÓ k¹ñóùðù µ ÓÖ ÚØÞ ÐÐØ Ó ÚÚÐÒ º B A ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ó R¹ÐÖÓÐ ÞÞ Þ ½º½¼µ¹Ò Ò ÐØ can R ÒÓÒÙ ÐÔÞ ØÚ B A ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ð L¹ÐÖÓÐ ÞÞ Þ ½º½¼µ¹Ò Ò ÐØ can L ÒÓÒÙ ÐÔÞ ØÚ can R ÞÖØÚ can L ÞÖØÚº ½½ ÑÙÒ Ò ÒØ ØØÐ Ý ÓÐ ÐØÐ ÒÓ ØØÐÐ ÌÓÖÑ º½µ ÐØÐ ¹ ÒÓ Ø Ðµ ÚØÞº Þ ½º½º ÞØÒ Ð ØÓØØÓÞ ÓÒÐÒ Ý B Ó R¹ÐÖÓ A Ó ÓÑÓÙÐÙ ÐÖ ÑØ ÖÓÞ Ý ψ : B R A A R B b R a a 0 R ba 1 ½º½½µ ÚÝ ÞØÖÙØÚ Þ ÐÝØ Þ A R¹ÝòÖò B R¹ÓÝòÖò ÞØØ ÑÐÝÒ 1 ÓÔÓÖØ ÞÖò ÐÑ ÞÞ 1 1 R 1 2 = 1 R 1 ε(1) = 1µº À B Þ ÐÔ Ó ÐÖÓ Ý ÀÓÔ¹ ÐÖÓÒ ÑÐÝÒ ÒØÔ ØÚ ÓÖ ÒØ ψ ØÚº À Ö ÙÐ Ò ÀÓÔ¹ÐÖÓÒ Þ Þ ÐÐÔ R¹ÑÓÙÐÙ Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÓÖ Þ ÑÓÙÐÙ Ó ÐÔÓ ØÓÚ B ÔÖÓØÚ ÑÒØ Ö Ð Ö Ó B¹ÓÑÓÙÐÙ Ò Þ ÖØÐÑÒ ÓÝ Hom B (B, ) ÙÒØÓÖ B¹ÓÑÓÙÐÙ Ó ØÖ Ð k¹ñóùðù Ó ØÖ ÖÞ Þ ÔÑÓÖÞÑÙ ÓØµ ÞÞ ½½ ÌÓÖÑ º¾µ Þ ÐØÚ ØÐ Ðº

30 ¼ ½º ÂÌº ÎÌË ½ºº º Ö ÝÒ ØÖÙØÖ¹ØØÐ ÅÓÖØ¹ÐÑÐØÐ À ÓÒÐÒ Þ ½º½º ÞØÒ Ð ØÓØØÓÞ ÓÝ ÞØ ½½ º ÞØ Ø ÖÝÐ ÖÑÐÝ H Ó R¹ÐÖÓ A Ó ÓÑÓÙÐÙ ÐÖ ØÒ ØÒØØ Þ Òº ÖÐØÚ ÀÓÔ¹ ÑÓÙÐÙ Ó M H A ¹ÚÐ ÐÐØ ØÖ Øº ÞØ Ý Ò ÐÙ ÑÒØ Ó A ÑÓÙÐÙ Ó Ø¹ Ö Ø Ó H¹ÓÑÓÙÐÙ Ó M H ØÖ Ò ÞÒ Ò ÞÖÒØ A Ý ÑÓÒÓ M H ¹Òµº Ð ÞÒ ÐÚ ÞØ Þ ÞÖÚØÐØ ÓÝ A ÑØ ÖÓÞ Ý ÚÝ ÞØÖÙØÚ Þ¹ ÐÝØ Ðº ½º½½µµ M H A ¹Ö ØÒØØÒ Ý ÑÒØ Ý A R H A¹ÓÝòÖò ÓÑÓÙÐÙ Ò ØÖ Öº Þ A ÓÒÚÖ Ò Ò Ö ÞÐÖ Ø B¹ÚÐ ÐÐÚ ØÒØØ Þ ( ) B A M B M H A M H ( ) coh A M B ½º½¾µ ÙÒ ÐØ ÙÒØÓÖ Ô ÖØº Ò ÞÚÖÒÝÞØÒ Þ Ö ÝÒ ØÖÙØÖ ØØÐ Ó ÙÒ ÐØ ò ØÐ Ö ÐÐØÚ ÚÚÐÒ ÚÓÐØ Ö ÚÓÒØÓÞ ÐØØÐØ ÓÐÑÞÒ Ñº À H Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ð R¹ÑÓÙÐÙ ÓÖ M H A ÞÓÑÓÖ Þ A RH A¹ÓÝòÖò Ù Ð ÒØ A¹ÝòÖò ÑÓÙÐÙ Ò ØÖ ÚÐº ÍÝÒÝ ÑÒØ ÐÖ ¹ ØÒ Þ Ù Ð ÝòÖò ÑÓ Ø A¹Ò H Ð R¹Ù Ð Ò H := R Hom(H,R)¹Ò H A ÐÖØ ÞÓÖÞØÒØ ÖØ ÑÐÝ ÑÓ Ø Ý ÐÐÑ Ó ÞÓÖ ÞÓÖÞØµº Ò Þ ØÒ Ø Ø Þ Ö ÝÒ ØÖÙØÖ¹ØØÐ Ú ÞÚÞØØ ÚØÞ ÅÓÖØ¹ Þ ÚÞ ÐØ Öº Ø ÞÖÔÐ ÐÖ B ÐÐØÚ H A ÐÐÔ ÑÓÙÐÙ Ó Ô A¹Ò ÐÐØÚ ( H A) coh ¹Ò ÚÒÒ Ò ÐÚº Þ ÅÓÖØ¹ Þ Ô Ð ¹ Ø ÞÓÒ ÅÓÖØ¹ Þ Ò ÑÐÝØ ÒÔÐ Ø Ö ¾ ¹Ò ÓÔÓÖØ ÞÖò ÐÑÑÐ ÖÒÐÞ A¹ÓÝòÖòÞ ÖÒÐØ ÞÓÒ ÐØÚ ÑÐÐØØ ÓÝ ÓÝòÖò Ú Ò ÒÖ ÐØ ÔÖÓØÚ Ð A¹ÑÓÙÐÙ º ËÞÑÑØÖÙ Ò ÞÐØ Ý Ð ÐÖÓ Ó ÓÑÓÙÐÙ ÐÖ Øº ÚØÞ ÝÒ ØÖÙØÖ¹ØØÐØ Ý ÔÙ ÓÝ ÅÓÖØ¹ÐÑÐØ ÖÑ¹ ÒÝØ ÓÑÒ ÐÙ ÞÞÐ Þ ÐÞ ÞØÒ Ð ØÓØØ ØÒÒÝÐ ÓÝ Þ ÐÔ Ó¹ ÐÐØÚ Ð ÐÖÓÐ ÚØØ ÐÓ ¹ØÖ ÞØ Ý Ò ÞÓÒ ÀÓÔ¹ÐÖÓÓ ØÒ ÑÐÝ¹ Ò ÒØÔ ØÚ ÑÐÝÒ Þ Þ ÖÐÚ Ò ÑÓÙÐÙ ØÖÙØÖ Ú Ò ÒÖ ÐØ ÔÖÓØÚº ½ºº ÌØÐº ½½ ÌÓÖÑ ºµ ÌÒØ Ò Ý H ÀÓÔ¹ÐÖÓÓØ L R ÐÖ ÐØØ ÑÒ H R RH LH H L ÑÓÙÐÙ Ó ÑÒÝ Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÑÒ Þ ÒØÔ ØÚº ÄÝÒ A Ý Ó H¹ÓÑÓÙÐÙ ÐÖ Ñ ÐÒ ØÒ ÙÝÒÞ ÑÒØ ÖÑÐÝ ÐÔ ÐÖÓ ÓÑÓÙÐÙ ÐÖ µ ÐÝÒ B ÓÒÚÖ Ò Ö ÞÐÖ Ó¹ ÚÝ ÚÚÐÒ ÑÓÒ Ð ÐÔ ÐÖÓ ÓØ Ö ÒÞÚµº ÞÒ ÐØÚ ÑÐÐØØ ÚØÞ ÐÐØ Ó ÚÚÐÒ º B A ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ó ÐÖÓÐ A ÔÖÓØÚ Ð B¹ÑÓÙÐÙ ÒÓÒÙ H A B End(A) ÐÖ ÒØ¹ÓÑÓÑÓÖ¹ ÞÑÙ ÞÓÑÓÖÞÑÙ A ÒÖ ØÓÖ Ó H A¹ÑÓÙÐÙ Ó ØÖ Ò ( ) coh : M H A M B ò ØÐ ( H A) coh B A H A ÅÓÖØ¹ÐÔÞ ÞÖ Ý µ B A ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ð ÐÖÓÐ

31 ½ºº ÄÇÁË¹ÃÁÌÊÂËÌËÃ ÀÇÈ¹ÄÊÇÁ ËÁÅÅÌÊÁýÎÄ ½ A ÔÖÓØÚ Ó B¹ÑÓÙÐÙ ÒÓÒÙ H A End B (A) ÐÖ ÒØ¹ÓÑÓ¹ ÑÓÖÞÑÙ ÞÓÑÓÖÞÑÙ ÓÐ H = L Hom(H,L) Þ ÐÔ Ð ÐÖÓ Ð Ù Ð µ A ÒÖ ØÓÖ Ð H A¹ÑÓÙÐÙ Ó ØÖ Ò ( ) coh : A M H B M ò ØÐ ÓÐ A M H A Ð ÑÓÙÐÙ Ò ØÖ Ó H¹ÓÑÓÙÐÙ Ó ØÖ Òµ Þ A B ( H A) coh H A ÅÓÖØ¹ÐÔÞ ÞÖ Ý µº Ý C R¹ÓÝòÖò ÚÐÑÐÝ M Ó ÓÑÓÙÐÙ Ø ÖÐØÚ ÒØÚÒ ÑÓÒÙ ÑÒ¹ ÞÓÒ f : P Q Ó ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÓÖ ÑÐÝ Ð Ó R¹ÑÓÙÐÙ ÑÓÒÓÑÓÖÞÑÙ Ó Hom C (f,m) : Hom C (Q,M) Hom C (P,M) ÞÖº ÝÐÐ ½ ÈÖÓÔÓ ØÓÒ º½µ Ñ ÐÐ ÈÖÓÔÓ ØÓÒ ºµ ÞÖÒØ Þ ½º ÌØÐÒ Ø ÖÝÐØ ÐÓ ¹ ØÖ ÞØ Ò A ÔÓÒØÓ Ò ÓÖ ÖÐØÚ ÒØÚ Ó»Ð ÓÑÓÙÐÙ Þ ÐÔ Ó ÚÝ Ðµ ÐÖÓÒ B A ÝÞ Ð Ó»Ð B¹ÑÓÙÐÙ ÑÓÒÓÑÓÖÞÑÙ ÚÝ Ñ ÚÚÐ ÚÚÐÒ A òò ÐÔÓ Ó»Ð B¹ÑÓÙÐÙ º ÌÓÖÑ º½µ ÞÖÒØ Þ ÒØÔ ØÚØ Ò ÞÒØÒ A ÔÓÒØÓ Ò ÓÖ ÖÐØÚ ÒØÚ Ó ÓÑÓÙÐÙ ÖÐØÚ ÒØÚ Ð ÓÑÓÙÐÙ º ÅÒÞÒ ÞÖÚØÐØ ØÚÞÚ ¾ ÌÓÖÑ ºµ¹ØÐ ÐÐØÚ ÅÓÖØ¹ÐÑÐØ ÖÑÒÝÚÐ ÚØÞ Ö ØÖÙØÖ¹ØØÐÖ ÙØÙÒº ½ºº ÌØÐ ½½ ÈÖÓÔÓ ØÓÒ º ÈÖÓÔÓ ØÓÒ º¾µº ÌÒØ Ò Ý H ÀÓÔ¹ÐÖÓÓØ L R ÐÖ ÐØØ ÑÒ H R RH LH H L ÑÓÙÐÙ Ó ÑÒÝ Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÑÒ Þ ÒØÔ ØÚº ÄÝÒ A Ý Ó H¹ÓÑÓÙÐÙ ÐÖ Ñ ÐÒ ØÒ ÙÝÒÞ ÑÒØ ÖÑÐÝ ÐÔ ÐÖÓ ÓÑÓÙÐÙ ÐÖ µ ÐÝÒ B ÓÒÚÖ Ò Ö ÞÐÖ Ó¹ ÚÝ ÚÚÐÒ ÑÓÒ Ð ÐÔ ÐÖÓ ÓØ Ö ÒÞÚµº ÞÒ ÐØÚ ÑÐÐØØ ÚØÞ ÐÐØ Ó ÚÚÐÒ º B A ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ó ÚÝ Ðµ ÐÖÓÐ A òò ÐÔÓ Ð B¹ÑÓÙÐÙ A ÔÖÓØÚ ÒÖ ØÓÖ Ð B¹ÑÓÙÐÙ Ó ØÖ Ò ÒÓÒÙ H A BEnd(A) ÐÖ ÒØ¹ÓÑÓÑÓÖÞÑÙ ÞÓÑÓÖÞÑÙ ( ) coh : M H A M B ÚÚÐÒ B H A ÞØØ ÅÓÖØ¹ Þ ÞÓÖ B A ÐÓ ¹ØÖ ÞØ Þ ÐÔ Ó ÚÝ Ðµ ÐÖÓÐ A òò ÐÔÓ Ó B¹ÑÓÙÐÙ A ÔÖÓØÚ ÒÖ ØÓÖ Ó B¹ÑÓÙÐÙ Ó ØÖ Ò Ý ÒÓÒÙ H A End B (A) ÐÖ ÒØ¹ÓÑÓÑÓÖÞÑÙ ÞÓÑÓÖÞÑÙ ( ) coh : A M H B M ÚÚÐÒ B H A ÞØØ ÅÓÖØ¹ Þ ÞÓÖº

32 ¾ ½º ÂÌº ÎÌË ½ºº ÝÒ ÀÓÔ¹ÐÖ ÓÀÓÔ¹ÑÓÙÐÙ À R ØØ ÞÐ ÐÖ Ý k ÓÑÑÙØØÚ ÝòÖò ÐØØ ÓÖ ÑÒØ Ð ØØÙ ÑÒÒ R¹ ÝòÖò k¹ðö º ÍÝÒÓÖ Ý R¹ÓÝòÖò ÒÑ ÐØØÐÒÐ k¹óðö R ÒÑ ÖÒÐÞ ØÓÚ ØÙÐÓÒ ÓÐº ÅÒØ Þ ½º½º¾ ÞØÒ ÐÞØ R¹Ø ÖÓÒÙ ¹k¹ÐÖ Ò ÑÓÒÙ Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ØÓÚ R Hom(R,k) ÞÓÑÓÖ ÑÒØ Ó ÚÝ Ðµ R¹ÑÓÙÐÙ Óº ÆÑ ÒÞ Ð ØÒ ÓÝ Þ ÚÚÐÒ Ý ψ Hom(R,k) Òº ÖÓÒÙ ¹ ÙÒÓÒ Ð Ý i e i f i R k R Òº Ù Ð Þ ÐØÞ ÚÐ ÑÐÝ i ψ(re i)f i = r = e i ψ(f i r) ÝÒÐ Ò Ø ÞÒ ÐØ R ÑÒÒ r ÐÑÖº Ð ÚØÞ ÓÝ i re i f i = i e i f i r R ÑÒÒ r ÐÑÖº Ý R ÖÓÒÙ ¹ÐÖ ÒÜ i e if i ÐÑ R ÒØÖÙÑ Òº À Þ ÝÒÐ R Ý ÐÑÚÐ ÓÖ ÖÑÐÝ M N R¹ÑÓÙÐÙ Ó ØÒ Þ M R N M k N m R n i m.e i k f i.n R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Þ M k N M R N ÒÓÒÙ ÔÑÓÖÞÑÙ ÞÐ º ÞÖÒØ R ÞÔÖ Ð ÞÒ Þ R R R r i re i f i = i e i f i r R¹ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ ÞÓÖÞ Ý ÞÐ º À B Ý R¹ÐÖÓ ÓÐ R 1 ÒÜò ÖÓÒÙ ¹ÐÖ k ÐØØ ÓÖ B k¹ðö k¹óðö º ÐÖÓ B B R B Ó ÞÓÖÞ Ø ÓÑÔÓÒ ÐÚ B R B B k B ÞÐ Ð ÒÝÖ ÓÐÖ Ó ÞÓÖÞ Øº ÓÝ ÞÖÔØ ÖÓÒÙ ¹ÙÒÓÒ Ð Ø Þº Þ Ý ÒÝÖØ ÐÖ ÓÐÖ ØÖÙØÖ ÐØ ÝÒ ÐÖ ÜÑ Ø Ø ÑÒÒ ÝÒ ÐÖ Ð ÐÐ ÐÝÒ ÑÓÒ Ý 1 ÒÜò ÖÓÒÙ ¹ÐÖ ÐØØ ÐÖÓ¹ Ðº ÝÒ ÀÓÔ¹ÐÖ ÔÓÒØÓ Ò ÞÓ H L R ÒØ¹ÞÓÑÓÖ 1 ÒÜò ÖÓÒÙ ¹ ÐÖ ÐØØ ÀÓÔ¹ÐÖÓÓ ÑÐÝÒ Ø ÐÔ L¹ ÐÐØÚ R¹µ ÓÝòÖò ÙÝÒÞÓÒ ÓÐÖ Ð Ø H k H H R H ÐÐØÚ H k H H L H ÔÑÓÖÞÑÙ ÞÐ Ò Ø ÚÐº ÁÐÝØÒ ÑÓÒ ÝÒ ÐÖ Ö ÐÐØÚ ÝÒ ÀÓÔ¹ÐÖ Ö ÚÓÒØÓÞ ÖÑ¹ ÒÝ Ý Ö Þ ÑÔØ ÐÖÓÓÖ ÐÐØÚ ÀÓÔ¹ÐÖÓÓÖ ÚÓÒØÓÞ ÐÐØ Ó Ô Ð ØÒØº ÌÖØÒØÐ ÞÓÒÒ Þ ÑÓ ÐÝÒ ØØÐ ÓÖ Ò ØÐÒÐ ÞÐØØØ Ñº ÎÒÒ ØÓÚ ÓÐÝÒ Ö ØØÐ ÑÐÝ ÐØÐ ÒÓ Ø ØØ ÞÐ ÀÓÔ¹Ð¹ ÖÓÓÖ ÒÑ ÐØ ÚÝ Ñ ØÒ ÐÐ ÒÑ ÑÖØº ½ºº½º ÝÒ ÀÓÔ¹ÐÖ ÝÒ ÀÓÔ¹ÐÖ ÜÑ Ò Ð Ú ÐØÓÞØ ½ ¹Ò ÐÒØ Ñ ÝÒ ÐÖ Ø Ô ÑÐØ Ð ÞÖº ØÖÙØÖ Ö ÞÐØ ÚÞ ÐØ ½ ¹Ò ØÐ ÐØº ÝÒ ÐÞ ÖÖ ÙØÐ ÓÝ ÐÖ ÜÑ ÞÐ Ó ÞÓÖÞ Þ Ý ÓÑÔØÐØ Ø ÑÚØÐ δ(1) = 1 1 ÚÐÑÒØ ÞÓÖÞ ÓÝ ÓÑÔØÐØ Ø ÑÚØÐ ε(ab) = ε(a)ε(b) ÜÑ ÝÒØØØÒº ½ºº Ò ½ ÒØÓÒ ¾º½µº Ý ÝÒ ÐÖ ÚÐÑÐÝ k ÓÑÑÙØØÚ Ýò¹ Öò ÐØØµ Ý k¹ñóùðù B ÐÐ ØÚ Ý (η,µ) ÐÖ ØÖÙØÖ ÚÐ Ý (δ,ε) ÓÐÖ ØÖÙØÖ ÚÐ ÑÐÝÖ Þ Ð ÖÑÓÐ ÑÓÐÑÞÓØØ ÓÑÔØÐØ ÐØØÐ ØÐ¹ ÐÒ ÓÐ tw : V W W V v w w v Ð ÖÐ Ø ÞÞ k¹ñóùðù Ó

33 ½ºº Æ ÀÇÈ¹ÄÊýÃ ÇÁÀÇÈ¹ÅÇÍÄÍËÁ ØÖ Ò ÞÑÑØÖ ÓÔÖ Ø ÐÐµº B 2 δ δ B 4Id tw Id B 4 B 3 Id δ Id B 4 Id tw Id Id δ Id µ 2 µ B B 4 µ µ B B 2 ε k B 4 µ µ B 2 ε ε ε ε δ η η B 2 µ µ η η k B η 2 δ δ B 4 B δ 2 B 4 B 3 B 2 δ δ B 4 Id µ Id Id tw Id Id µ Id ÌØ ÞÐ B¹Ð a b c ÐÑÒ ÖÚ ÝÒ ÐÖ ÜÑ Þ Ð ÐÓØ ÐØº (ab) 1 (ab) 2 = a 1 b 1 a 2 b 2 ε(ab 1 )ε(b 2 c) = ε(abc) = ε(ab 2 )ε(b 1 c) = = , ÓÐ = δ(1) B B¹Ð ÐÑ ÔÐ ÒÝØ ÐÐº ÎÝ ÞÖ ÓÝ ÝÒ ÐÖ ÜÑ ÒÙ Ð ÞÞ Þ Ø ÖÔÖÞÒØ Ð ÖÑÓ Þ ÒÚÖ Ò ÒÝÐ ÑÓÖØ Öº ÃÚØÞ ÔÔÒ Ý ÝÒ ÐÖ B Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÞÞ ÐØÞ Ù Ð k¹ñóùðù Ó Ø¹ Ö Òµ ÓÖ Hom(B,k) Ù Ð ÝÒ ÐÖ ØÖÒ ÞÔÓÒ ÐØ ØÖÙØÖ ÚÐº Ý ÝÒ ÐÖ Ò Óµ ÞÓÖÞ Ø Þ ÐÐÒØØØÖ ÖÐÚ ÑØ ÝÒ ÐÖ Ø ÔÙÒº ½ºº ýððø º ½ ÈÖÓÔÓ ØÓÒ ¾º Ò ÈÖÓÔÓ ØÓÒ ¾º½½µ ÖÑÐÝ B ÝÒ ÐÖ Ò L : B B b ε(1 1 b)1 2 R : B B b 1 1 ε(b1 2 ) ÐÔÞ ÑÔÓØÒ ØÓÚ B L := L (B) B R := R (B) Þ Òº Ð¹ ÐÐØÚ Ó Ö ÞÐÖ µ ÝÑ Ð ÓÑÑÙØ Ð ÒØ¹ÞÓÑÓÖ ½¹ÒÜò Ý ÞÔÖ Ð µ ÖÓÒÙ ¹ ÐÖ ÑÐÝÖ δ(1) B R B L º ½ºº ÄÑÑ ½ Ô ½ºµº ÖÑÐÝ B ÝÒ ÐÖ Ö ÚØÞ ÐÐØ Ó ÚÚ¹ ÐÒ B ÐÖ δ(1) = 1 1 ε(ab) = ε(a)ε(b) a,b B R (a) = ε(a)1 a B L (a) = ε(a)1 a Bº ½º¼ ÌØÐ ÞÖÒØ ÑÒÒ B ÝÒ ÐÖ ÑØ ÖÓÞ Ý Ó ÐÖÓÓØ B R ÐØØ Ý Ð ÐÖÓÓØ B L ÐØØº Ó ÐÖÓ B R (B R ) op ¹ÝòÖò ØÖÙØÖ B R B (B R ) op B r ε(r1 1 )1 2 ÐÖ ÓÑÓÑÓÖÞÑÙ ÓÐ ÓØØº Ó ÞÓÖ¹ Þ Ø ÝÒ ÐÖ B B k B Ó ÞÓÖÞ Ø B k B B R R RB ÔÑÓÖÞÑÙ Ð

34 ½º ÂÌº ÎÌË ÓÑÔÓÒ ÐÚ ÒÝÖ ÓÝ Ô R º ËÞÑÑØÖÙ Ò Ð ÐÖÓ B L (B L ) op ¹ ÝòÖò ØÖÙØÖ B L B (B L ) op B l 1 1 ε(1 2 l) ÐÖ ÓÑÓÑÓÖÞÑÙ ÓÐ ÓØØº Ó ÞÓÖÞ Ø ÝÒ ÐÖ B B k B Ó ÞÓÖÞ Ø B k B B L L L B ÔÑÓÖÞÑÙ Ð ÓÑÔÓÒ ÐÚ ÒÝÖ ÓÝ Ô L º Ð ÑÝÐ Ð ÞÓÒÒÐ ÚØÞ ÓÝ B Ó ÐÐØÚ Ð ÑÓÙÐÙ Ò ØÖ ÑÓÒÓ Ð B R ÐÐØÚ B L ÐØØ ÑÓÙÐÙ ØÒÞÓÖ ÞÓÖÞ ÖÚÒº ÅÚÐ B L B R ½ ÒÜò ÖÓÒÙ ¹ÐÖ B R ÐÐØÚ B L ÐØØ ÑÓÙÐÙ ØÒÞÓÖ ÞÓÖÞØ ÞÓÑÓÖ k¹ñóùðù ØÒÞÓÖ ÞÓÖÞØ ÑÐÐ ÖØ¹ ÖØÙÑ ÚÐµº ÁÞÓÐØ ÓÝ Ý ÝÒ ÐÖ ÞÞ ÐÔ ÓÐÖ µ ÓÑÓÙÐÙ Ò ØÖ ÞÓÑÓÖ ÑÐÐ Ö Ð Ö Óµ ÐÖÓ ÓÑÓÙÐÙ Ò ØÖ ¹ ÚÐ Ý Þ ÑÓÒÓ Ð ÑÐÐ ÑÓÙÐÙ ØÒÞÓÖ ÞÓÖÞ ÖÚÒ Ñ ÞÓÑÓÖ k¹ñóùðù ØÒÞÓÖ ÞÓÖÞØ ÑÐÐ ÖØÖØÙÑ ÚÐµº ½ºº Ò ½ ÒØÓÒ ¾º½µº Ý ÝÒ ÀÓÔ¹ÐÖ Ý ÝÒ ÐÖ H ÐÐ ØÚ Ý H H ÒØÔÒ ÒÚÞØØ ÐÒ Ö ÐÔÞ Ð ÑÐÝ Þ Ð ÖÑÓÐ ÑÓÐÑÞÓØØ ÜÑ Ò Ø Þ ÐØº H δ H 2 Id S H 2 µ L H H δ H 2 S Id H 2 µ R H H δ2 H S 3 S Id S H 3 µ 2 H ÌØ ÞÐ h H ÐÑÒ ÖÚ ÝÒ ÀÓÔ¹ÐÖ ÜÑ ÚØÞº h 1 S(h 2 ) = ε(1 1 h)1 2 S(h 1 )h 2 = 1 1 ε(h1 2 ) S(h 1 )h 2 S(h 3 ) = S(h). ÝÒ ÀÓÔ¹ÐÖ ÜÑ ÒÙ Ð Ý Ú Ò ÒÖ ÐØ ÝÒ ÀÓÔ¹ÐÖ Ù Ð ÝÒ ÀÓÔ¹ÐÖ ØÖÒ ÞÔÓÒ ÐØ ØÖÙØÖ ÚÐº Ý ÝÒ ÀÓÔ¹ÐÖ Ò ÑÒ ÞÓÖÞ Ø ÑÒ Ó ÞÓÖÞ Ø Þ ÐÐÒØØØÖ ÖÐÚ ÑØ ÝÒ ÀÓÔ¹ÐÖ Ø ÔÙÒ Þ ÖØ ÒØÔÐº À Þ ÒØÔ ÞÓÑÓÖÞÑÙ ÓÖ ÚÝ ÞÓÖÞ Ø ÚÝ Ó ÞÓÖÞ Ø Þ ÐÐÒØØØÖ ÖÐÚ ÝÒ ÀÓÔ¹ÐÖ Ø ÔÙÒ ÓÐ Þ ÒØÔ Þ ÖØ ÒØÔ ÒÚÖÞº ½º¼º ýððø ½ ÄÑÑ ¾º Ò ÌÓÖÑ ¾º½¼µº Ý ÝÒ ÀÓÔ¹ÐÖ ÒØÔ Ý¹ ÖØÐÑò ÓØØ ÝÒ ÐÖ ØÒµº Þ ÒØÔ ÐÖ ÒØ¹ÓÑÓÑÓÖÞÑÙ ÓÐÖ ÒØ¹ÓÑÓÑÓÖÞÑÙ º Ý Ø Ø ÐØØ Ú ÑÒÞ ÝÒ ÀÓÔ¹ÐÖ ÒØÔ ØÚº ÖÑÐÝ ÝÒ ÀÓÔ¹ÐÖ ÑØ ÖÓÞ Ý ÀÓÔ¹ÐÖÓÓØ ÒØ ÐÖØ ÐÖÓ ØÖÙØÖ Ð Ð¹ ÐÐØÚ Ó Ö ÞÐÖ ÐØØ ÝÒ ÀÓÔ¹ÐÖ ÒØÔÐº ½º½º ÈÐº ÆÝÐÚ ÒÚÐÒ ÑÒÒ ÐÖ ÝÒ ÐÖ ÑÒÒ ÀÓÔ¹ÐÖ ÝÒ ÀÓÔ¹ÐÖº ÝÒ ÐÖ ÝÒ ÀÓÔ¹ÐÖµ ØÓÚ ÐÖ ÀÓÔ¹ ÐÖ µ Ú ÖØ Þº Ð¹ Ó Ö ÞÐÖ Ø ÖØ ÞÒ Ý ÐÑ ÞØ º ½º¾º ÈÐº ÖÑÐÝ B ÝÒ ÐÖ Ò B R B L Ö ÞÐÖ ÐØÐ ÒÖ ÐØ Ö Þ¹ ÐÖ ÝÒ ÀÓÔ¹ÐÖº ÓÐÖ ØÖÙØÖ B ÓÐÖ ØÖÙØÖ Ò Ñ ÞÓ¹ ÖØ ÚÐ Þ ÒØÔ S(rl) := ε(r1 1 )1 2 ε(1 3 l) r B R l B L º Þ ½º¼ ýððø ÞÖÒØ ÞÒ S Ñ ÞÓÖØ B L (B R ) op ÐÐØÚ B R (B L ) op ÐÖ ÞÓÑÓÖÞÑÙ Óº ½º º ÈÐº ÄÝÒ C Ý ØÖ Ú Ó ÓØÙÑÑÐ ÐÝÒ k Ý ÓÑÑÙØØÚ ÝòÖòº C ÑÓÖÞÑÙ ÐØÐ ÒÖ ÐØ Þ k¹ñóùðù ÐÐ ØØ Ý ÝÒ ÐÖ

35 ½ºº Æ ÀÇÈ¹ÄÊýÃ ÇÁÀÇÈ¹ÅÇÍÄÍËÁ ØÖÙØÖ ÚÐº ÃØ ÑÓÖÞÑÙ ÞÓÖÞØ Ø ÓÑÔÓÞÙÒØ Ò ÐÙ Þ ÙØ ÐØÞ ÝÒØ 0¹ÒØ Ñ ÞØ ÞÓÖÞ ÑòÚÐØØ k¹ðò Ö Ò ØÖ ÞØº Þ Ý ÐÑ Þ ÒØØ ÑÓÖÞÑÙ Ó Þ C Þ ÓØÙÑ Öº Ó ÞÓÖÞ Ø ÑÓÖÞÑÙ ÓÓÒ Þ f f f ÓÒ Ð ÐÔÞ ÒØ Ò ÐÙ Ñ ÞØ ÑòÚÐØØ k¹ðò Ö Ò ØÖ ÞØº ÓÝ ÖØ ÑÓÖÞÑÙ ÓÓÒ 1 Ñ ÞØ ÑòÚÐØØ k¹ðò Ö Ò ØÖ ÞØº Ð¹ Ó Ö ÞÐÖ Ø Þ ÒØØ ÑÓÖÞÑÙ Ó ÞØ º À C ÖÙÔÓ ÞÞ ÑÒÒ ÑÓÖÞÑÙ ÞÓÑÓÖÞÑÙ µ ÓÖ ÒØ ÝÒ ÐÖ ÝÒ ÀÓÔ¹ÐÖº Þ ÒØÔÓØ ÑÓÖÞÑÙ ÓÓÒ Þ ÒÚÖÞÒØ Ò ÐÙ Ñ ÞØ ÑòÚÐØØ k¹ðò Ö Ò ØÖ ÞØº ÒØ ÑÓÒ ÖÑÐÝ n ÔÓÞØÚ Þ Þ Ñ ØÒ Þ n n¹ k¹ð ÐÑò Ñ ØÖÜÓ ÐÖ ÐÐ ØØ ÝÒ ÀÓÔ¹ÐÖ ØÖÙØÖ ÚÐº Ñ ØÖÜ Ý Ò Ó ÞÓÖÞ ÓÒ Ð ÐÔÞ Þ ÒØÔ Ñ ØÖÜ ØÖÒ ÞÔÓÒ Ð º Ð¹ Ó Ö ÞÐÖ Ø ÓÒ Ð Ñ ØÖÜÓ ÐÓØ º ½ºº ÈÐ ½ ÔÔÒÜµº ÖÑÐÝ R ½ ÒÜò ÖÓÒÙ ¹ÐÖ ØÒ B B op ÐÖ ÝÒ ÀÓÔ¹ÐÖº Ó ÞÓÖÞ Þ i e i f i R R Ù Ð Þ Ø ÚÐ Ò ÐØ x y i (x e i) (f i y) ÑÓÒº ÓÝ ψ ÖÓÒÙ ¹ÙÒÓÒ ÐØ ÞÒ ÐÚ x y ψ(xy) Ðº Þ ÒØÔ x y i y e iψ(xf i )º ½ºº¾º ÝÒ ÓÀÓÔ¹ÑÓÙÐÙ Ó ÅÒØ Þ ½º½º ÞØÒ Ð ØØÙ Ý ÐÖ ÐÒÞ ÀÓÔ¹ ØÔÙ ÑÓÙÐÙ Ò Ý ÐÖ Ø Ó ÐÐØÚ ÃÓÔÔÒÒ ÐØÐ ÚÞØØØ Òº ÓÀÓÔ¹ÑÓÙÐÙ Ó Ø ÚÐº ÞÒ ÞØ Þ ÐÔ ÙÐ ÞÓÐ Ð ÔÙÐ Ð Ý ÒÐ ÝÒ ÐÖ ÀÓÔ¹ØÔÙ ÑÓÙÐÙ Ø Ý Ø ÓÐÓÑ ÓÐÓÞ ÚÞ ÐØº Þ ØØ Ø ÖÝÐØ ÝÒ ÓÀÓÔ¹ÑÓÙÐÙ Ó Ò ÔÖ ÐØ ÚÝ ÞØÖÙØÚ Þ ÐÝÓ ÚÝ ÞÙÐ ÓÐ ØÖÙ¹ ØÖ µ ÒÔÐ ÖÓÓØ ÒÚÞ òþ ÝÒ ÐØÐ ÒÓ Ø Ø ÑÒ ÞØÖØ ØÖÐÑÐØ ÑÐÔÓÞ ½¾ ¹Ò ØÐ ÐØº Ý ÝÒ ÐÖ Ó ÓÑÓÙÐÙ ÐÖ ÑÓÒÓ Ó ÓÑÓÙÐÙ Ó ÑÓÒÓ Ð ØÖ Òº ÜÔÐØÒ Þ ÚØÞØ ÐÒØº ½ºº Ò ÒØÓÒ ¾º½µº Ý B ÝÒ ÐÖ Ó ÓÑÓÙÐÙ ÐÖ Ý ÐÖ A ÐÐ ØÚ Ý a a 0 a 1 Ó B ÓÑÓÙÐÙ ØÖÙØÖ ÚÐ Ý ÓÝ ÑÒÒ a,a A ÐÑÖ (aa ) 0 (aa ) 1 = a 0 a 0 a 1 a a 1 1 = a 0 L (a 1 ). Ý ÝÒ ÐÖ Ó ÑÓÙÐÙ ÓÐÖ ÓÑÓÒÓ Ó ÑÓÙÐÙ Ó ÑÓÒÓ Ð ØÖ Òº ÜÔÐØÒ Þ ÚØÞØ ÐÒØº ½ºº Ò ÒØÓÒ ¾º½µº Ý B ÝÒ ÐÖ Ó ÑÓÙÐÙ ÓÐÖ Ý ÓÐÖ C ÐÐ ØÚ Ý Ó B ÑÓÙÐÙ ØÖÙØÖ ÚÐ Ý ÓÝ ÑÒÒ c C b B ÐÑÖ (c.b) 1 (c.b) 2 = c 1.b 1 c 2.b 2 c. L (b) = ε(c 1.b)c 2. ½ºº Ò ÒØÓÒ ¾º¾µº ÂÓ¹Óµ ÝÒ ÓÀÓÔ¹ØÓ ÐØØ ÓÐÝÒ (A,B, C) ÖÑ ÓØ ÖØÒ ÓÐ B ÝÒ ÐÖ A Ó B¹ÓÑÓÙÐÙ ÐÖ C Ó B¹ÑÓÙÐÙ ÓÐÖº ÐÒ Ó¹Ó ÓÀÓÔ¹ØÓØ ÚÞØÒÒ ÞÒ Ø ÐØ Ø ÑÔ¹ Ù ÝÒ ÐÖ Óµ ÞÓÖÞ Ø Þ ÐÐÒØØØÖ ÖÐ Þ Ý ÝÒ ÐÖ Ó¹Ó ÓÀÓÔ¹ØØ ØÒØº ÝÒ ÓÀÓÔ¹ØÓÓÞ ÓÞÞ ÖÒÐØ ÚØÞ ÑÓÙÐÙ ÓÐÑØº

36 ½º ÂÌº ÎÌË ½ºº Ò ÒØÓÒ º½µº ÌÒØ Ò Ý (A,B,C) ÝÒ ÓÀÓÔ¹ØÓØº ÞÒ ØÓ ÝÒ ÓÀÓÔ¹ÑÓÙÐÙ Ò ÓÐÝÒ M k¹ñóùðù ÓØ ÖØÒ ÑÐÝ Ý¹ ÞÖÖ Ó A ÑÓÙÐÙ Ó Ó C¹ÓÑÓÙÐÙ Ó ÑÒÒ m M a A ÐÑÖ Þ (m.a) 0 (m.a) 1 = m 0.a 0 m 1.a 1 ÝÒÐ ØÐ Ðº ÝÒ ÓÀÓÔ¹ÑÓÙÐÙ Ó ÑÓÖÞÑÙ¹ Ó A¹ÑÓÙÐÙ Ó C¹ÓÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Óº ÝÒ ÓÀÓÔ¹ÑÓÙÐÙ Ó ÑÓÖÞÑÙ ØÖ Ø M C A ¹ÚÐ ÐÐº ½ºº ÌØÐ ËØÓÒ ÜÑÔÐ µº ÄÝÒ B Ý ÝÒ ÐÖº À A = B ÖÙÐ Ö B¹ÓÑÓÙÐÙ ÐÖ C = B R ØÖÚ Ð B¹ÑÓÙÐÙ ÓÐÖ r r1 1 S(1 2 ) 1 1 S(1 2 )r Ó ÞÓÖÞ Ð B ÓÝ Ò Ñ ÞÓÖØ ÚÐ Þ r.b := R (rb) Ó B¹Ø Ðµ ÓÖ M C A ÞÓÑÓÖ Ó B¹ÑÓÙÐÙ Ó ØÖ ÚÐ À C = B ÖÙÐ Ö B¹ÑÓÙÐÙ ÓÐÖ A = B R ØÖÚ Ð B¹ÓÑÓÙÐÙ ÐÖ Þ r 1 1 r1 2 ÓØ Ðµ ÓÖ M C A ÞÓÑÓÖ Ó B¹ÓÑÓÙÐÙ Ó ØÖ ÚÐ À C = B ÖÙÐ Ö B¹ÑÓÙÐÙ ÓÐÖ A = B ÖÙÐ Ö B¹ÓÑÓÙÐÙ ÐÖ ÓÖ M C A ÞÓÑÓÖ B ÀÓÔ¹ÑÓÙÐÙ Ò ØÖ ÚÐ À H Ý ÝÒ ÀÓÔ¹ÐÖ B = H op H C = H A = H c.(h h ) := hch a a 2 (S(a 1 ) a 3 ) Ø Ð ÐÐØÚ ÓØ Ð ÓÖ M C A ¾ ØÖ ÚÐº ÞÓÑÓÖ H ØØÖÖÒÐ³¹ÑÓÙÐÙ Ò ÈÖÓÔÓ ØÓÒ º µ ÞÖÒØ Þ M C A M A ÐØ ÙÒØÓÖ Ó ÙÒ ÐØ Þ M C A M C ÐØ ÙÒØÓÖ Ô Ð ÙÒ ÐØº ½º¼º ÌØÐ ÈÖÓÔÓ ØÓÒ º¾µº À Ý (A,B,C) ÝÒ ÓÀÓÔ¹ØÒ C Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÓÖ ØÐ ÐØ Ý ÐÐÑ ÐÖ ÑÐÝÒ ÑÓÙÐÙ Ø¹ Ö ÞÓÑÓÖ M C A ¹ÚÐº Þ ÝÒ ÐÖØ ÞÓÖÞØÒ ÒÚÞØØ ÐÖ ÚØÞÔÔÒ ÓÒ ØÖÙ ÐØ Ñº C ÓÐÖ ØÖÙØÖ Ò ØÖÒ ÞÔÓÒ Ð ÚÐ C := Hom(C,k) ÐÖ C B C Ø ØÖÒ ÞÔÓÒ Ð ÚÐ C Ð B¹ÑÓÙÐÙ º ÞÓ Ó (a φ)(b ψ) := a 0 b (a 1.ψ)φ ÐÖØ ÞÓÖÞ ÓÖÑÙÐ A C ¹ÓÒ ÞÓØÚ ÞÓÖÞ Ø Ò Ðº ÒÒ ÞÓÖÞ Ò ÞÓÒÒ ÒÒ Ý ÐÑ Þ 1 ε ÐÑ ÒØÖ Ð ÑÔÓØÒ º Þ ÐØÐ ÒÖ ÐØ Ð Ö ØØ Ý ÐÑ ÞÓØÚ ÐÖ C A Òº ÝÒ ÐÖØ ÞÓÖÞØº Þ A C k¹ ÑÓÙÐÙ ÖØÖØÙÑ ÐÑ 1 0 a 1 1.φ Ð ÓÐ a A φ C º ½º½º ÌØÐ ËØÓÒ ÜÑÔÐ Ò ÔÔÒÜµº ÄÝÒ B Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÝÒ ÐÖº À A = B ÖÙÐ Ö B¹ÓÑÓÙÐÙ ÐÖ C = B R ØÖÚ Ð B¹ÑÓÙÐÙ ÓÐÖ Ðº ½º ÌØÐµ ÓÖ C A ÝÒ ÐÖØ ÞÓÖÞØ ÞÓÑÓÖ B¹ÚÐ À C = B ÖÙÐ Ö B¹ÑÓÙÐÙ ÓÐÖ A = B R ØÖÚ Ð B¹ÓÑÓÙÐÙ ÐÖ Ðº ½º ÌØÐµ ÓÖ C A ÝÒ ÐÖØ ÞÓÖÞØ ÞÓÑÓÖ B¹Ù Ð ÐÖ ÚÐ À C = B ÖÙÐ Ö B¹ÓÑÓÙÐÙ ÐÖ A = B ÖÙÐ Ö B¹ÓÑÓÙÐÙ ÐÖ ÓÖ C A ÝÒ ÐÖØ ÞÓÖÞØ ÞÓÑÓÖ B À ÒÖ¹ÙÔÐÙÑ ÚÐ

37 ½ºº ÅÇÆýÇÃ Æ ÄÅÄÌ À H Ý Ú Ò ÒÖ ÐØ ÔÖÓØÚ ÝÒ ÀÓÔ¹ÐÖ B = H op H C = H A = H Þ ½º ÌØÐÒ Ð ØÓØØ Ø Ð ÐÐØÚ ÓØ Ð ÓÖ C A ÝÒ ÐÖØ ÞÓÖÞØ ÞÓÑÓÖ H ÖÒÐ³¹ÙÔÐÙÑ ÚÐº ÅÒØ ÞØ Þ ½º½º ÞØÒ Ð ØØÙ ÐÖ Ö ÚÓÒØÓÞ ÓÒÐ ÓÒ ØÖÙ ÐÐØÚ ÐÐØ Ó ÞÔÒ ÐÐ ÞÒ ÚÝ ÞØÖÙØÚ Þ ÐÝÓ ÐØÐ ÒÓ ÖØº ÌÖÑ¹ ÞØ Ò ÑÖÐ Ð Ö ÐÐÐ ÞØØ¹ ÒØ ÐØÐ ÒÓ Ø Ó ÚÐÑÐÝÒ ÝÒ ÚÝ ÞØÖÙØÚ Þ ÐÝÓ ÚÝ ÝÒ ÞÙÐ ÓÐ ØÖÙØÖ ÐÑÐØº ÞÒ ÑÓØÚ ÐÔ Ò ÚØÞ ÒØ ÒÔÐ ÖÓÓØ Ú ÓÐØ ¾ ¹Òº ½º¾º Ò ÒÔÐ Ò ÖÓÓØ ¾ µº Ý ÝÒ ÚÝ ÞØÖÙØÚ Þ ÐÝ ÚÝ ÝÒ ÞÙÐ ÓÐ ØÖÙØÖ ÐÐ Ý A ÐÖ Ð Ý C ÓÐÖ Ð Ý ψ : C A A C k¹ñóùðù ÓÑÓÑÓÖÞÑÙ Ð ÑÐÝÖ Þ Ð ÖÑÓ ÓÑÑÙØ ÐÒº C A A Id µ C A ψ Id A C A Id ψ ψ A A C µ Id A C C A δ Id C C A ψ Id ψ C A C ψ Id A C Id δ A C C δ C C C Id η Id C A C Id η ψ Id A C C Id ε Id C A ψ A C C A ψ A C Id η Id C A A ψ Id A C A Id ε Id A A µ Id ε A Þ Ð Ø ÖÑ ÑÐÝ ψ Óµ ÞÓÖÞ Ð ÚÐ ÓÑÔØÐØ Ø Þ ÙÝÒÞ ÑÒØ Ð ÚÝ ÞØÖÙØÚ Þ ÐÝ ØÒº Ñ Ó Ø ÖÑ ÝÒØØØØØ ÑÐÝ ψ ÓµÝ Ð ÚÐ ÓÑÔØÐØ Ø Þ º ÅÒÒ (A,B,C) ÝÒ ÓÀÓÔ¹Ø ÑØ ÖÓÞ Ý (A,C,ψ) ÝÒ ÞÔ ÓÐ ØÖÙØÖ Ø ÓÐ ψ(c a) = a 0 c.a 1 º ÝÒ ÓÀÓÔ¹ÑÓÙÐÙ Ó ØÖ Ø ÐØÐ ÒÓ ØÚ ÝÒ ÞÙÐ ÓÐ ØÖÙ¹ ØÖ ÓÞ ÓÞÞ ÖÒÐØ ÞÙÐ ÓÐØ ÑÓÙÐÙ M C A ØÖ º ÁÞÓÐØ ÓÝ M C A M A ÐØ ÙÒØÓÖ Ó ÙÒ ÐØ M C A MC ÐØ ÙÒØÓÖ Ô Ð ¹ ÙÒ ÐØº À Ý ÝÒ ÞÙÐ ÓÐ ØÖÙØÖ Ò C Ú Ò ÒÖ ÐØ ÔÖÓØÚ k¹ñóùðù ÓÖ M C A ÞÓÑÓÖ Ý Þ A C := Hom(C,k) ÐÖ ÝÒ ÐÖØ ÞÓÖÞØÒØ ÐÖ ÑÓÙÐÙ ØÖ ÚÐº ÅÒÒÒ Þ ½º½º ÞØÒ Ð ØÓØØÓÞ ÓÒÐ ÞØÖØ ØÖÐÑÐØ ÐÖ ÑÒØÝ ØÞ ÚÚÐ ½¾ ¹Ò ØÖØÒØ Ñ Ðº ½º ÞØº ½ºº ÝÒ ÀÓÔ¹ÐÖ Ö ÔÐ ÓÒ ØÖÙ ÑÓÒ Ó ÝÒ ÐÑÐØ ÞØ Þ ÐÔ ÙÐ ÞÓÐ Ð ½¾ ÑÙÒ Ð ÓÝ ÝÒ ÐÖ Ö ÔÐ ÓÒ ØÖÙÒ Ý Þ ½º ÞØ ÒÒ ØÖÐÑÐØ ÑÐÔÓÞ Ø º ÃÓÐÓÞÞÙ ÑÓÒ Ó ½º½º ÞØÒ Ñ ÑÖØ ÓÖÑ Ð ÐÑÐØÒ ÐÐÑ ÐØÐ ¹ ÒÓ Ø Ø ÑÐÝ ÓÐÝÒÓÖÑ Ò Ô ÐÖÒ ÞØ ÓÒ ØÖÙØ ÓÝ Ð ÞÙ ÐÑÐØ ÐÖØ ÐÖ ÑÐÐØ Ðº ½º½º ÞØº ½ºº½º EM w (K) ÝÒ Ó ÞÓÖ ÞÓÖÞØ Þ ½º½º ÞØÒ ÐÖ ÑÓÙÐÙ ÐÖ Ò ÑÓÙÐÙ ÓÐÖ Ù Ð Ò Ð¹ ÖØ ÞÓÖÞØ Ø ÑÒØ Ó ÞÓÖ ÞÓÖÞØÓØ ÖØÙ Ð ÞÞ ÑÑÙØØØÙ ÓÝ Ý EM(Bim)¹

38 ½º ÂÌº ÎÌË ÑÐ ÐÐØ ØÖ ÑÓÒ Ð Þ ÖÑÞØØØº ÚØÞÒ ÐÖØ ÞÓÖÞØ ½º ÞØÒ ÞÖÔÐ ÝÒ ÐØÐ ÒÓ Ø Ò ÓÒÐ ÞÐÐÑò ÐÖ Ø Ùº ÄÝÒ K Ý ØØ ÞÐ ØÖº ÊÒÐ ÓÞÞ ÚØÞ EM w (K)¹ÚÐ ÐÐØ ØÖ Øº ¼¹ÐÐ ÐÝÒ ÑÓÒ Ó K¹Ò Ðº ½º¾µº Þ (A,t) (A,t ) ½¹ÐÐ ÐÝÒ x : A A ½¹ÐÐ K¹Ò ÐÐ ØÚ Ý ψ : t x xt K¹Ð ¾¹ÐÐ ÚÐ ÑÐÝÖ ½º µ Ð ÖÑ ÓÑÑÙØØÚº Ñ Ó ÖÑÓØ Ý ÞÖòÒ Ú Ðº ÀÓÝ Ñ ØÖ Ø ÔÙÒ ÞØ ¾¹ÐÐ Ö ÖØØ ØÓÚ ÐØØÐÐÐ ÓÑÔÒÞ ÐÙº Þ (x,ψ) (y,φ) ¾¹ÐÐ K¹Ð : x yt ¾¹ÐÐ ÑÐÝÖ Þ ½ºµ ÖÑ x yt η Id t yt φid yt ytt Idµ ÓÑÑÙØØÚº Þ EM w (K) ØÖ ÐÓ Ð Ò ØÐ Ö Þ¹ØÖÒØ ØÖØÐÑÞÞ EM(K)¹ Øº ÎÝ ÞÖ ÓÝ K¹Ð Idη : x xt ¾¹ÐÐ ÒÑ ¾¹ÐÐ EM w (K)¹Òº Þ (x,ψ) (x,ψ) ÒØØ ¾¹ÐÐ ψ.η Id : x xtº Þ ½º µ Ñ Ó ÖÑ Ò ÐÚØ ÑØØ xt η Id t xt ψid xtt Idµ xt ÒÑ ÒØØ ¾¹ÐÐ K¹Ò ÑÔÓØÒ º Þ EM w (K) ØÖ ÑÓÒ Ø ÚØÙ Ö ÝÒ Ó ÞÓÖÒ º ÑÓÒ ÓÐÓÑ ÐØÐ ÒÓ Ø ÒØ ØÒØ Þ Ð ÒØº ½º º Ò ½¾ ÒØÓÒ ¾º½µº Ý ÔÖ¹ÑÓÒ Ý K ØÖ Ò Ý t : A A ½¹ÐÐ ÐÐ ØÚ η : Id A t ÔÖ¹Ý µ µ : tt t ÞÓÖÞ µ ¾¹ÐÐ Ð ÑÐÝÖ Þ Ð ÖÑÓ ÓÑÑÙØØÚº Idµ ttt tt µ µid tt t µ Idη t ηid tt µ tt t µ ηη 1 A tt η t µ ηid tt ttt µ t µ 2 ÖÑÐÝ ÔÖ¹ÑÓÒ Ö µ.tη : t t ¾¹ÐÐ ÑÔÓØÒ º ÌÝ Ð ÓÝ Þ Þ Ñ¹ ÔÓØÒ ¾¹ÐÐ Ð ÞÞ ÐØÞ Ý t : A A ½¹ÐÐ ÚÐÑÒØ i : t t p : t t ¾¹ÐÐ ÑÐÝÖ p.i = Id i.p ÝÒÐ Ö ÑÔÓØÒ ¾¹ÐÐ ÚÐº ÓÖ t ÑÓÒ K¹Ò η p Id A t ii µ p t t t tt t t Ý Ð ÐÐØÚ ÞÓÖÞ Ðº ½ºº ÌØÐ ½¾ ÌÓÖÑ ¾º µº ÌÒØ Ò Ý K ØÖ Ø Ý (A,t) ÑÓÒ ÓØ Ý s : A A ½¹ÐÐ Ø K¹Òº ØÚ Ô ÓÐØ ÚÒ Þ Ð ØÖÙØÖ ÞØØº ((A,t),(s,ψ)) Ð ÑÓÒ Ó EM w (K)¹Ò (A,st) Ð ÔÖ¹ÑÓÒ Ó K¹Ò ÑÐÝ Θ : stst st ÞÓÖÞ Ö Θ.stsµ = sµ.θtº À ÒØ ØØÐÒ Þ (A,st) ÔÖ¹ÑÓÒ ÑÔÓØÒ st st ¾¹ÐÐ Ð ÓÖ Þ Þ ÐØÐ ÑØ ÖÓÞÓØØ (A,ŝt) ÖØÖØÙÑ ÑÓÒ ÓØ K¹Ò ÚØÙ t s ÝÒ Ó ÞÓÖ ÞÓÖÞØ Òº

39 ½ºº ÅÇÆýÇÃ Æ ÄÅÄÌ ½ºº ÈÐº ÄÝÒ (A,B,C) Ý Ó¹Ó ÝÒ ÓÀÓÔ¹Øº Þ ÓÞÞ ÖÒÐ¹ ØÒ Ý ((k,a),(c,ψ)) ÑÓÒ ÓØ EM w (Bim)¹Ò ÓÐ ψ : A C C A a φ φ( a 1 ) a 0. ÓÞÞ ØÖØÓÞ ÔÖ¹ÑÓÒ C A (φ a)(φ a ) = φ ( a 1 )φ a 0 a ÞÓÖÞ Ð Þ ε( 1 1 ) 1 0 ÔÖ¹Ý ÐÑÑÐº C A C A φ a φ( 1 1 ) 1 0 a ÑÔÓØÒ ÐÔÞ Ð Ý Ô Ý ÐÑ ÞÓØÚ ÐÖ ÑÐÝ ØÒÞÓÖ ØÓÖÓ Ð ÖÐ ÖÚÒµ ÞÓÑÓÖ C A ÝÒ ÐÖØ ÞÓÖÞØ ÚÐ Ðº ½º ÞØº ÌÓÚ ÔÐÒØ ÝÒ ÀÓÔ¹ÐÖ Ð ¹Ò ÚØØ Ö ÞØ ÞÓÖÞØÓ ÝÒ¹ Ó ÞÓÖ ÞÓÖÞØÓº ÝÒ Ó ÞÓÖ ÞÓÖÞØ ÒØ ÓÒ ØÖÙ Ý ¼ ¹Ò ÖÑÐ ØÖ ÐÑÐØ ÑÓÒØÓÐ Ö ÚÐ ÚØÓÞ ÒÐÐµ Ú ÓÐØ ÝÒ Ö ÞØ ÞÓÖÞØØÐº ÌÒØ Ò Ý K ØÖ Ø ÑÐÝÒ ÑÒÒ ÑÔÓØÒ ¾¹ÐÐ Ð ÑÐÝ¹ Ò ÐØÞÒ ÑÓÒ Ó ÐÒÖÅÓÓÖ¹ÓØÙÑ ÞÞ K Mnd(K) ÓÒ Ð ¹ ÙÒØÓÖÒ ÚÒ Ý R Ó ÙÒ ÐØ Ðº ½º½º ÞØµº ÞÒ ÐØÚ ÑÐÐØØ ÓÒ ØÖÙ Ð¹ Ø Ý J w : EM w (K) K Ô ÞÙÓ¹ÙÒØÓÖ Ðº ½¾ ÌÓÖÑ ºµº È ÞÙÓ ÚÓÐØ ÒÒÝØ Ø Þ ÓÝ ÓÖÞÓÒØ Ð ÓÑÔÓÞØ ÒÑ ÞÓÖÒ ÓÖÒ ÞÓÑÓÖÞÑÙ Ó Ö ÖÞµº ¼¹ÐÐ ÓÒ ÑÓÒ ÓÓÒµ ÐÝÒ J w (A,t) := R(A,t) = A t ÞÓ Ó ÐÒÖ ÅÓÓÖ¹ÓØÙÑº Ý (x,ψ) : (A,t) (A,t ) ½¹ÐÐ ØÒ Þ ½º¾ ÌØÐÒ ÚÞØØØ f v : A t A ÙÒ ÒÒ ǫ ÓÝ Ø ÚÐ ØÒØ Þ xv η Id t xv ψ xtv Idǫ xv ½º½ µ ÑÔÓØÒ ¾¹ÐÐ Ø K¹Òº ÐØÚ Ò ÞÖÒØ Þ Ð ÞÞ ÐØÞ Ý p : xv x Ô ¾¹ÐÐ i : x xv ÞÐ Ð Ý ÓÝ i.p ÝÒÐ Þ ½º½ µ ÑÔÓØÒ ¾¹ÐÐ ÚÐº ÆÑ ÒÞ Ð ØÒ ÓÝ ( x, ψ := p.idǫ.ψid.idi) : (A t,id) (A,t ) ½¹ÐÐ Mnd(K)¹Òº Ý ÖØÐÑÞØ J w (x,ψ) := R( x, ψ)º À ÓÒÐÒ Ý : (x,ψ) (y,φ) ¾¹ÐÐ Ö := ( x i xv Id ytv Idǫ yv p ŷ ) : ( x, ψ) (ŷ, φ) ¾¹ÐÐ Mnd(K)¹Ò Ý ÖØÐÑÞØ J w ( ) := R( )º ÂÝÞÞ Ñ ÓÝ J w ÓÒ ØÖÙ¹ ÓÞ ÑÔÓØÒ ¾¹ÐÐ Ð Ø ÞÒ ÐØÙº Ð ÞÓÒÒ ÞÓÑÓÖÞÑÙ Ö ÝÖØÐÑò Ý J w Ô ÞÙÓ¹ØÖÑ ÞØ ÞÓÑÓÖÞÑÙ Ö ÝÖØÐÑòº ÍÝÒ¹ Ð ÒÑ ÝÖØÐÑò Ð ÚØÞ ÓÝ ÐØÐ Ò J w Ô ÞÙÓ¹ÙÒØÓÖ ÒÑ ÙÒØÓÖµº Þ Ð ÖÑ ÐÖÐ ÓÖ ÑÙØØ ÒÝÐ ÖÒÖ ÓÒ Ð ÙÒØÓÖØ ¼¹ ½¹ÐÐ ÓÒ ÒØØ ÒØ Ø Ý ω ¾¹ÐÐ Ø Idη.ω¹ ÚÚ ÙÒØÓÖØ ÐÐØÚ ÒÝÐÚ ÒÚÐ ÝÞ Ø ÐÐº J ÙÒØÓÖ J w Ñ ÞÓÖØ º Þ R J J w ÙÒØÓÖÓ ÑÒÝ Ó ÙÒ ÐØ ÓÝ Þ Ö ÑÙØØº K Mnd(K) EM(K) EM w (K) R J J w

40 ¼ ½º ÂÌº ÎÌË ½ºº¾º ÝÒ ÐÞ ÅÚÐ (x,ψ) : (A,t) (A,t ) EM w (K)¹Ð ½¹ÐÐ ØÒ Idη : x xt ÒÑ ¾¹ÐÐ EM w (K)¹Ò ÒÑ ØÖÑ ÞØ Ö ÐÚØ ÓÝ ÑÐÝ ω K¹Ð ¾¹ÐÐ Ö Ð Þ Idη.ω ¾¹ÐÐ EM w (K)¹Òº ÐÝØØ ÑÖÞØ ÓÝ ÑÐÝ ω ØÒ Ð Þ x η Id t x ψ xt ωid yt illetve x ω y η Id t y φ yt (x,ψ) (y,φ) ¾¹ÐÐ EM w (K)¹Òº Ú Ð Þ Þ ÓÝ ÔÓÒØÓ Ò ÓÖ Idη Id t x ψ xt t t x Idψ t xt IdωId t yt ωid yt φid ytt Idµ illetve t x Idω t y ψ xt ωid yt η Id t yt φ φid yt ytt Idµ ½º½µ ÖÑ ÓÑÑÙØØÚº À (x,ψ) ½¹ÐÐ EM(K)¹Ò ÞÞ ÓÑÔØÐ ÑÓÒ Ó Ý ÚÐ ½º µ Ñ Ó ÖÑ Ò ÖØÐÑÒµ ÓÖ Þ ½º½µ¹Ð Ð ÖÑ ÖÙ Ð ½ºµ¹ Ø (y,φ) ½¹ÐÐ EM(K)¹Ò ÓÖ Þ ½º½µ¹Ð Ñ Ó ÖÑ ÖÙ Ð ½ºµ¹Øº ÌØ ÞÐ (x,ψ) (y,φ) EM w (K)¹Ð ½¹ÐÐ ØÒ Ø ½º½µ¹Ð ÖÑ ÞÑÙÐØ Ò ÓÑÑÙØØÚØ ÚÚÐÒ ½ºµ¹ØÐº ÎÞ ÚØÞ Ø Mnd i (K)¹ÚÐ ÐÐØÚ Mnd p (K)¹ÚÐ ÐÐØ ØÖ Øº ¼¹ ÐÐ ½¹ÐÐ ÑÒØØÒ ÐÝÒ ÙÝÒÞÓ ÑÒØ EM w (K)¹Òº Þ (x,ψ) (y,φ) ¾¹ÐÐ Mnd i (K)¹Ò ÐÐØÚ Mnd p (K)¹Ò ÐÝÒ ω : x y K¹Ð ¾¹ÐÐ ÑÖ ½º½µ Ð ÐÐØÚ Ñ Ó ÖÑ ÓÑÑÙØØÚº ÐÐÒÖÞØ ÓÝ Þ ØÖ ÚÒÒ G i : Mnd i (K) EM w (K) Mnd p (K) : G p ÙÒØÓÖÓ ÑÐÝ ÒØØ ÐÔÞ ÒØ ØÒ ¼¹ÐÐ ÓÒ Þ ½¹ÐÐ ÓÒ Ý ω ¾¹ÐÐ Ø Ô ωid.ψ.η Id¹ ÐÐØÚ φ.η Id.ω¹ Ú ÞÒº Þ ØÖ ÓÒØÓ ÞÖÔØ Ø ÞÒ Þ Ð ÝÒ ÐÞ ÔÖÓÐÑ ¹ Òº ½ºº Ò ½¾ ÒØÓÒ º½µº ÄÝÒ K Ý ØÖ ÑÐÝÒ ÐØÞÒ ÑÓÒ ¹ Ó ÐÒÖÅÓÓÖ¹ÓØÙÑº ÄÝÒ ÓØØ K¹Ò Ý (A,t) Ý (B,s) ÑÓÒ º Ý x : A t B s ½¹ÐÐ Ø Þ x : A B ½¹ÐÐ ÝÒ ÐÞ Ò ÑÓÒÙ ÐØÞÒ ¾¹ÐÐ ÑÐÝÖ p.i = Idº A t x v i B s v A x B A t x v p B s v A x B Ý ¾¹ÐÐ ÝÒ ÐÞ Ø ØÐ ÖØÐÑÒ Ò ÐØÙº ½ºº Ò ½¾ ÒØÓÒ º¾µº ÄÝÒ K Ý ØÖ ÑÐÝÒ ÐØÞÒ ÑÓ¹ Ò Ó ÐÒÖÅÓÓÖ¹ÓØÙÑº ÄÝÒ ÓØØ K¹Ò Ý (A,t) Ý (B,s) ÑÓÒ ØÓÚ x,y : A B ½¹ÐÐ x,y : A t B s ÝÒ ÐÞ ÓÐº Ý ω : x y ¾¹ÐÐ Ø Þ ω : x y ¾¹ÐÐ ¹ÐÞ Ò ÑÓÒÙÒ Þ Ð Ð K¹Ð ¾¹ÐÐ Ö ÚÓÒØÓÞµ ÖÑ ÓÑÑÙØØÚº ω Þ ω Ô¹ÐÞ Þ Ð Ñ Ó ÖÑ ÓÑÑÙØØÚº vx i xv vx p xv Id ω vy i yv ω Id Id ω vy p yv ω Id

41 ½ºº ÅÇÆýÇÃ Æ ÄÅÄÌ ½ À Ý K ØÖ Ò ÐØÞÒ ÑÓÒ Ó ÐÒÖÅÓÓÖ ÓØÙÑ ÓÖ ÓØØ (A,t) (B,s) K¹Ð ÑÓÒ ÓÖ ØÒØØ Þ Ð Lift i ((A,t),(B,s))¹ÚÐ ÐÐØÚ Lift p ((A,t),(B,s))¹ÚÐ ÐÐØ ØÖ Øº ÅÒØ ØÖ ÓØÙÑ ÐÝÒ (x : A B,x : A t B s,i : vx xv,p : xv vx) ÒÝ ÓÐ p.i = Id Ý x Þ x ÝÒ ÐÞ µº Þ (x,x,i,p) (x,x,i,p ) ÑÓÖÞÑÙ Ó ÐÝÒ (ω : X x,ω : x x ) Ô ÖÓ ÓÐ ω Þ ω ¹ÐÞ ÐÐØÚ Ô¹ÐÞ º ½ºº ÌØÐ ½¾ ÌÓÖÑ ºµº ÄÝÒ K Ý ØÖ ÑÐÝÒ ÐØÞÒ ÑÓÒ Ó ÐÒÖÅÓÓÖ¹ÓØÙÑ ÑÐÝÒ ÑÒÒ ÑÔÓØÒ ¾¹ÐÐ Ð º ÖÑÐÝ (A,t) (B,s) ÑÓÒ ÓÖ K¹Ò Þ Ð ØÖ ÚÚÐÒ º Lift i ((A,t),(B,s)) Mnd i (K)((A,t),(B,s)) Lift p ((A,t),(B,s)) Mnd p (K)((A,t),(B,s))º À ω : (x,ψ) (x,ψ ) ÑÓÖÞÑÙ ÑÓÒÙµ Mnd i (K)((A,t),(B,s))¹Ò ÓÖ ÒØ ÚÚÐÒ ÐØÐ Ô Þ (ω,j w G i (ω)) : (x,j w G i (x,ψ),i,p) (x,j w G i (x,ψ ),i,p ) ÑÓÖ¹ ÞÑÙ Lift i ((A,t),(B,s))¹Ò ÓÐ i p i p µ J w ÓÒ ØÖÙ ÓÞ ÞÒ ÐØ ¾¹ÐÐ K¹Òº ÒØ ØØÐÒ Ð ØØ ÚÚÐÒ ÐØÐ Ò ÒÑ ÞÓÑÓÖÞÑÙ Ó ÞÑÒ Þ ½º¾ ÌØÐÐÐµ ÑÚÐ ÑÔÓØÒ ÑÓÖÞÑÙ Ó Ð ÞÓÑÓÖÞÑÙ Ö ÝÖ¹ ØÐÑòº ½ºº ÈÐº ÖÑÐÝ B ÝÒ ÐÖ Ö M B ÑÓÒÓ Ð ØÖÙØÖ ÝÒ ÐÞ Ð º Å ÑÓÒÓ Ð ÞÓÖÞ M k ÑÓÒÓ Ð ÞÓÖÞ Ò ÑÒØ : M k M k M k ÙÒØÓÖÒµ ÝÒ ÐÞ ÑÓÒÓ Ð Ý ÒÑ M k ÑÓÒÓ Ð Ý Ò ÒÑ B : 1 M k ÙÒØÓÖÒ ÝÒ ÐÞ ÓÐ 1 ØÖÑÒ Ð ØÖ Ø ÐÐ ÑÐÝÒ ÝØÐÒ ÓØÙÑ ÚÒ ÝØÐÒ ÑÓÖÞÑÙ ÒÒ ÒØØ º ÊÞÒ Ø Ø ÐØÞÒ M B M B M B v 1 B M B v v v i p M k M k M k 1 i p B M k ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ÑÐÝ Ø ÚÐ Þ M B ÑÓÒÓ Ð ØÖ ÓÖÒ ØÖÑ ÞØ ÞÓÑÓÖÞÑÙ ÖÞ ÞÖÒØ ¹ÐÞ Ð Òº ( Id) M B M B M B = M B M B (B Id) = Id = M B (Id ) v v v M k M k M k ( Id) = v M k v M k (B Id) ε Id Id Id ε (Id B) v M k (Id ) (Id B) ÒØ Ö ÓÒ v : M B M k ÐØ ÙÒØÓÖØ ÐÐ Ðº ½º¾ ÈÐº Þ ÐÐØ ÑÓÖØ ÓÞ ÞÞ ÓÞ ÚØÞØØ Þ ÓÝ ÚÐÑÐÝ ÓØØ B ÐÖ ÖÒÐÞ ÝÒ ÐÖ ØÖÙØÖ ÚÐ ÒÑ Ð ÐØÒÒ ÓÝ M B ÑÓÒÓ Ð ÚÐÑÐÝ ÝÒ ÐÞ Ð ÑÓÒÓ Ð ØÖÙØÖ ÚÐ Ö ÙÐ ÝÒ ÐÞ ÖØ ÐÐ Mnd i (Cat)¹Ð ½¹ÐÐ Ð Ö ÐØÚ Ø ÐÐ ØÒÒ Ðº ½º ½º ÌØÐº

42 ¾ ½º ÂÌº ÎÌË ½º¼º ÈÐº ÌÒØ Ò Ý A ÐÖ Ø Ý C ÓÐÖ Ø ÚÐÑÐÝ k ÓÑÑÙØØÚ ÝòÖò ÐØØ ÞØØ Ý ψ : C A A C ÝÒ ÚÝ ÞØÖÙØÚ Þ ÐÝØ Bim¹Òº Þ ÙÝÒÞ ÑÒØ Ý ( ) ψ ÝÒ ÚÝ ÞØÖÙØÚ Þ ÐÝ Cat¹Ò (M k,( ) A) ÑÓÒ (M k,( ) C) ÓÑÓÒ ÞØØº ÞÒ ÐØÚ ÑÐÐØØ (M k,( ) C) ÓÑÓÒ Ò ÐØÞ ¹ÐÞ M A ¹Öº ÊÞÒ ÐØÞÒ M A ( ) C =( ) A (A C) M A v v M k i p ( ) C M k ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ÚÝ Ñ ÙÝÒÞ A C A C A C A C Ð A¹ ÑÓÙÐÙ ÓÑÓÑÓÖÞÑÙ Óµ ÑÐÝ Ø ÚÐ ÝÒÒ ÐÞÓØØ (M A,( ) A (A C)) ÓÑÓÒ Ó ÞÓÖÞ ÓÝ ÖÞ ÞÖÒØ ¹ÐÞ Ð º M A v M k ( ) (A C) A ( ) A ( ) (A δ) A (A C) (A C) A ( ) C ( ) δ ( ) C C M A v M k M A v M k ( ) (A C) A ( ) (A ε) A ( ) A A ( ) C ( ) ε Id M A v M k Ì Þ ½¾ ÈÖÓÔÓ ØÓÒ ºµ ÞÖÒØ Ý A ÐÖ Ý C ÓÐÖ Ý ψ : C A A C ÐÒ Ö ÐÔÞ ÔÓÒØÓ Ò ÓÖ ÐÓØÒ ÝÒ ÚÝ ÞØÖÙØÚ Þ ÐÝØ Bim¹Ò ψ Þ (M k,( ) C) ÓÑÓÒ ¹ÐÞ Ø ÒÙ Ð M A ¹Ö ÒØ ÖØÐÑÒ ÞÞ ((k,a),(c,ψ)) ÓÑÓÒ Mnd i (Bim)¹Ò Þ (M k,( ) A) ÑÓÒ Ô¹ ÐÞ Ø ÒÙ Ð M C ¹Ö Ù Ð ÖØÐÑÒ ÞÞ ((k,c),(a,ψ)) ÑÓÒ Mnd i (Bim ) ¹ Ò ÓÐ ( ) Þ ÐÐÒØØØ ÚÖØ Ð ÓÑÔÓÞ ØÖ Ø ÐÐº Ò Þ ØÒ ÝÒÒ ÐÞÓØØ ÑÓÒ ÓÑÓÒ ÐÒÖÅÓÓÖ¹ØÖ ÞÓÑÓÖ ½ ÞÖÒØ Ú Þ ÒÔÐ ÖÓÓØ ÐØÐ ÚÞØØØ ÞÙÐ ÓÐØ ÑÓÙÐÙ Ó ØÖ Ø Ðº ½º ÞØº ÎÝ ÞØÖÙØÚ Þ ÐÝÓ Ñ ÐØÐ ÒÓ Ø Ø ÒÔÐ ÂÒ Ò ÐØÐ ¾ ¹ Ò ÚÞØØØ ÔÖ Ð ¹ ÐÐØÚ ÐÜ ÞÙÐ ÓÐ ØÖÙØÖ Ø ½¾ ËØÓÒ µ ÚÞ Ð ÝÒ ÐÞ Ó ÒÞÔÓÒØ Ðº ½ºº ÝÒ ÑÓÒ Ó ÅÒØ Þ ½º½º½ ÞØÒ ÐÞØ ÐÖ ÐÐÑÞØ ÑÒØ ÔÓÒØÓ Ò ÞÓ B k¹ ÐÖ ÑÐÝ ÑÓÙÐÙ Ò ØÖ ÞÞ Þ (M k,( ) B) ÑÓÒ ÐÒÖÅÓÓÖ¹ ØÖ µ ÑÓÒÓ Ð Ý ÓÝ ÐØ ÙÒØÓÖ k¹ñóùðù Ó ØÖ ÞÓÖÒ ÑÓÒÓ Ð ÞÒ ÐÖ ÓÒ ÐÔ Þ ÐÖ ÅÓÖ ÐØÐ ¹Ò Ú ÓÐØ ÐØÐ ÒÓ¹ Ø ÑÐÝØ ÖØÐ ÀÓÔ¹ÑÓÒ Ò ÚÓØØ ÑÐÝÖ ÞØ Ò Þ ÑÓÒ ÒÚ ÞÒ ÐØÓ º ÑÓÒ Ó ÞØÒ ÓÒÖØÒ Cat ¾¹ØÖ Òµ Þ ÐÖ ØÖÑ ÞØ Ð¹ ØÐ ÒÓ Ø º Ò ÞÖÒØ Ý ÑÓÒ Cat¹Òµ Ý ÖÑ ÑÐÝ ÐÐ Ý T : C C

43 ½ºº Æ ÁÅÇÆýÇÃ ÙÒØÓÖÐ Ý m : T 2 T Ý u : Id T ØÖÑ ÞØ ØÖÒ ÞÓÖÑ Ð ÑÐÝ ÞÓ Ó ÞÓØÚØ Ý ÐØØÐÒ Ø ÞÒ ÐØº Ý T ÑÓÒ ÓÞ ÓÞÞ ÖÒ¹ ÐØ Þ Òº ÐÒÖÅÓÓÖ ØÖ Ø Ðº ½º¾½ ÈÐµº ÅÓÖ Ò ÞÖÒØ Ý ÑÓÒ Ý ÑÓÒ ÚÐÑÐÝ ÑÓÒÓ Ð ØÖ Ò ÐÐ ØÚ ÑÒÞÓÒ ØÖÙØÖ Ð ÑÐÝ ÚÚÐÒ ÑÓÒ ÐÒÖÅÓÓÖ¹ØÖ ¹ Ò ÑÓÒÓÐØ ÚÐ Ý ÓÝ Þ ÐÔ ØÖ ÑÒ ÐØ ÙÒØÓÖ ÞÓÖÒ ÑÓÒÓ ¹ Ð º ÞÖÒØ Ý B Ó R¹ÐÖÓ ÑØ ÖÓÞ Ý ( ) R op R B ÑÓÒ ÓØ Þ R¹ ÑÓÙÐÙ Ó ØÖ Òºµ ÅÒØ ÅÖÙÒ ¾ ¹Ò ÑÑÙØØØ Ý ÑÓÒ ÙÝÒÞ ÑÒØ Ý ÑÓÒ ÅÓÒÓ Ð ÃØÖ ÇÔÑÓÒÓ Ð ÙÒØÓÖÓ ÇÔÑÓÒÓ Ð ÌÖÑ¹ ÞØ ÌÖÒ ÞÓÖÑ ¾¹ØÖ Òº ÌÖÑ ÞØ Ò ÐÚØ Ö Ñ ÐÖ Ø ÐÒÞ Ö ÒÝ ÐØÐ ÒÓ Ø Ø ÑÓÒ ÓØ ÝÒ ÐÖ Ø ÝÖ ÒØ ÐØÐ ÒÓ Ø ÓÐÓÑº Ö ÑÚ ¹ Ð ÞÓÐ Ø ØòÞØ Ð ÙÐ ËØÚ ÄÐ ÊÓ ËØÖØØÐ Þ ½ ÑÙÒ Òº ½ºº½º Þ ÜÑ ÝÒ ÑÓÒ Ó ÐÒÖÅÓÓÖ¹ØÖ ËÞÐ ÒÝ ½ ÑÙÒ Ð ÑÖØ ÓÝ ÝÒ ÐÖ ÐÐÑÞØ ÑÒØ ÔÓÒØÓ Ò ÞÓ k¹ðö ÑÐÝ ÑÓÙÐÙ Ò ØÖ ÑÓÒÓ Ð ÐØ ÙÒØÓÖ k¹ ÑÓÙÐÙ Ó ØÖ ÖÒÐÞ ÚØÞ Òº ÞÔÖ Ð ÖÓÒÙ ¹ ÞÖÞØØÐº ½º½º Ò ËÞÐ ÒÝ ½ µº ÌÒØ Ò Ý F ÙÒØÓÖØ Þ (N,,R) ÑÓÒÓ Ð Ø¹ Ö Ð Þ (M,, K) ÑÓÒÓ Ð ØÖ º ÞØ ÑÓÒÙ ÓÝ F ÞÔÖ Ð ÖÓÒÙ ÞÖÞØò Ð ÚÒ Ð ØÚ Ý p X,Y : FX FY F(X Y ) p 0 : K FR ÑÓÒÓ Ð ØÖÙØÖ ÚÐ Ý i X,Y : F(X Y ) FX FY i 0 : FR K ÓÔÑÓÒÓ Ð ØÖÙØÖ ÚÐ Ý ÓÝ N ÑÒÒ X,Y,Z ÓØÙÑ Ö ÚØÞ ÖÑÓ ÓÑÑÙØØÚº FX F(Y Z) Id iy,z FX FY FZ p X,Y Z p X,Y Id F(X Y Z) ix Y,Z F(X Y ) FZ F(X Y ) FZ i X,Y Id FX FY FZ p X Y,Z Id p Y,Z F(X Y Z) ix,y Z FX F(Y Z) FX FY i X,Y px,y F(X Y ) F(X Y ) ËÞÓÖÒ ÑÓÒÓ Ð ÙÒØÓÖÓ ÒÝÐÚ ÒÚÐÒ ÖÒÐÞÒ ÞÔÖ Ð ÖÓÒÙ ¹ ÞÖÞØØÐº ÚØÞ Ò Ø Ø ÐØÐ ÒÓ Ø ÑÒ ÑÓÒ ÓØ ÑÒ ÝÒ ÀÓÔ¹ÐÖ Øº ½º¾º Ò ½ ÒØÓÒ ½º µº Ý ÝÒ ÑÓÒ ÐØØ Ý T ÑÓÒ ÓØ ÖØÒ ÚÐÑÐÝ (M,,K) ÑÓÒÓ Ð ØÖ Ò ÐÐ ØÚ ÑÒÞÓÒ ØÖÙØÖ Ð ÑÐÝ ÚÚ¹ ÐÒ Þ M T ÐÒÖÅÓÓÖ¹ØÖ ÑÓÒÓÐØ ÚÐ Ý ÓÝ Þ M T M ÐØ ÙÒØÓÖ ÖÒÐÞ ÞÔÖ Ð ÖÓÒÙ ¹ ÞÖÞØØÐº ÒØ ÚÐ Ó ÐÒØ ò Ñ ÑÐØ Ò ÑÔÐØ Ò ÔÖÔÒÞÖ Ú ÐØØ Ö ÙÐ Þ M ÑÓÒÓ Ð ØÖ ÙÝ¹ØÐ ÞÞ ÑÔÓØÒ ÑÓÖÞÑÙ Ð ¹ Òº Þ ÞØ ÐÒØ ÓÝ ÑÒÒ ÓÐÝÒ M¹Ð e : A A ÑÓÖÞÑÙ Ö ÑÐÝÖ e 2 = e ÐØÞ Ý Q ÓØÙÑ ÚÐÑÒØ p : A Q i : Q A ÑÓÖÞÑÙ Ó M¹Ò ÑÐÝÖ pi = Id Q ip = eº

44 ½º ÂÌº ÎÌË ½º º ÌØÐ ½ ÌÓÖÑ ½ºµº ÌÒØ Ò Ý (T,m,u) ÑÓÒ ÓØ Ý (M,,K) ÙÝ¹ ØÐ ÑÓÒÓ Ð ØÖ Òº Ý ÝÒ ÑÓÒ ØÖÙØÖ T ¹Ò ÔÓÒØÓ Ò ÙÝÒÞ ÑÒØ Ý (T,τ,τ 0 ) ÓÔÑÓÒÓ Ð ØÖÙØÖ ÑÐÝÖ Þ Ð ÖÑÓ ÓÑÑÙØØÚº T 2 (X TK) TτX,TK T(TX T 2 K) T(Id m K) T(TX TK) T(Id τ 0) T 2 X Tu X TK T(X TK) τ X,TK TX T 2 K Id mk TX TK Id τ0 TX T 2 (TK X) TτTK,X T(T 2 K TX) T(m K Id) T(TK TX) T(τ 0 Id) T 2 X Tu TK X T(TK X) τ TK,X T 2 K TX mk Id TK TX τ0 Id X T(Y Z) Id τy,z X TY TZ u X TY Id T(X TY ) TZ τ X,TY Id TX T 2 Y TZ Id u Y Z X Y Z TX m X m X Id m Y Id u X Y Z T(X Y Z) τ X Y,Z T(X Y ) TZ τx,y TX TY TZ Id T(X Y ) Z τ X,Y Id TX TY Z Id u TY Z TX T(TY Z) Id τ TY,Z TX T 2 Y TZ u X Y Id X Y Z Id m Y Id u X Y Z T(X Y Z) τ X Y,Z T(X Y ) TZ τx,y TX TY TZ Id T 2 (X Y ) TτX,Y T(TX TY ) τtx,ty T 2 X T 2 Y m X Y m X m Y T(X Y ) τ X,Y TX TY ÒØ ØØÐÒ ÐÐÔ ÓÔÑÓÒÓ Ð ØÖÙØÖ ÝÒ ÐÖ ÓÐÖ ØÖÙØÖ Ø ÐØÐ ÒÓ Ø Þ Ø ÞÖÔÐ ÖÑ Ô ÖÒÖ ÑÐÐØØØ Þ ½º ÒÒ Ð ØÓØØ Ø ÖÑÒ Úº ½º ÌØÐµº Ý ÝÒ ÑÓÒ ÐÒÖÅÓÓÖ¹ØÖ Ò ÐÔ ØÖ Ò ÑÓÒÓ Ð ØÖÙØÖ ÞØØ Ô ÓÐØ ÑÖØ Þ Ð ØØÐÒ ÐÔ Þº ½ºº ÌØÐ ½ ÌÓÖÑ ½º½¼µº ÄÝÒ (T,m,u) Ý ÑÓÒ ÚÐÑÐÝ (M,,K) ÑÓÒÓ ¹ Ð ØÖ Ò ÑÐÝÒ (T,τ,τ 0 ) : (M,,K) (M,,K) Ý ÓÔÑÓÒÓ Ð ÙÒØÓÖº ÞÒ ÐØÚ ÑÐÐØØ Þ ½º ÌØÐÒ ÞÖÔÐ ÖÑÓ ÔÓÒØÓ Ò ÓÖ ÓÑÑÙØØÚ Þ Ð ØÐ ÐÒº K ÑÒØ ØÖÑÒ Ð ØÖ Ð M¹ ÑÒ ÙÒØÓÖ Ø ÚÐ Ò ÐØ 1 K M T M ÙÒØÓÖ := ( TK u TK T 2 K τk,tk TK T 2 K Id m K TK TK Id τ 0 TK ) ØÖÑ ÞØ ØÖÒ ÞÓÖÑ Ð ÐÔÐ T 2 K mk TK TK ØÖÑ ÞØ ØÖÒ ÞÓÖÑ (1,Id) (M,T) ½¹ÐÐ Ø ÐÓØÒ Mnd i (Cat)¹Ò

45 ½ºº Æ ÁÅÇÆýÇÃ Þ M M M ÙÒØÓÖ T( ) τ T( ) T( ) ØÖÑ ÞØ ØÖÒ ÞÓÖÑ (M,T) (M,T) (M,T) ½¹ÐÐ Ø ÐÓØÒ Mnd i (Cat)¹Ò Þ M M Id T M M Id τ 0 Id K M Id M M M T Id M M K Id τ 0 Id M Id M ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ¾¹ÐÐ Mnd i (Cat)¹Ò Þ Ð ÓÖÑÙÐ Ð Ò ÐØ E TX,TY E (3) TX,TY,TZ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ TX TY u TX TY T(TX TY )τ TX,TY T 2 X T 2 Ym X m Y TX TY u TX TY TZ TX TY TZ T(TX TY TZ) ÓÑÑÙØØÚÚ Ø Þ Þ Ð ÖÑÓØ τ (3) TX,TY,TZ T 2 X T 2 Y T 2 Z m X m Y m Z TX TY TZ, TX TY TZ E TX,TY Id TX TY TZ E (3) TX,TY,TZ TX TY TZ ETX,TY TX TY TZ Id Id E TY,TZ Id E TY,TZ M ÑÒÒ X,Y,Z ÓØÙÑ Öº ÞØ ÝØØ ÐÐÑÞÚ Þ ½º ÌØÐÐÐ ÚØÞ ÞÓÐØº ½ºº ÌØÐ ½ ÈÖÓÓ Ó ÈÖÓÔÓ ØÓÒ ½º½½µº Ý T ÝÒ ÑÓÒ Ö ÚÐÑÐÝ (M,,K) ÙÝ¹ØÐ ÑÓÒÓ Ð ØÖ Ò ÚØÞ ÐÐØ Ó ØÐ ÐÒº M T ÑÓÒÓ Ð Ý Þ 1 K M T M ÙÒØÓÖ ÝÒ ÐÞ M T ÑÓÒÓ Ð ÞÓÖÞ M ÑÓÒÓ Ð ÞÓÖÞ Ò ÝÒ ÐÞ M T ÞÓ ØÓÖ ÞÓÑÓÖÞÑÙ M ÞÓ ØÓÖ ÞÓÑÓÖÞÑÙ Ò ¹ÐÞ M T ÑÓÒÓ Ð Ý Ð Ô ÓÐØÓ ÓÖÒ ÞÓÑÓÖÞÑÙ Þ ½º ÌØÐ ÖÑ¹ ÔÓÒØ Ò ÞÖÔÐ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ¹ÐÞ º ½ºº¾º ÝÒ ÑÓÒ Ó ÑÒØ ÑÓÒ Ó ÓÝ ÝÒ ÐÖ Ô Ð ÐÖÓÓ ØØ ÓÝ ÝÒ ÑÓÒ Ó ÑÔØ ÑÒØ Ô Ð ÑÓÒ Óº Ò ÞØÒ ½ ËØÓÒ ¾µ ÐÔ Ò ØØ¹ ÒØ ÓÝ ÙÝÒ ÝÒ ÑÓÒ Ó ØÖ ÚÚÐÒ ÑÓÒ Ó Ý ÐÐ¹ Ñ ØÖ ÚÐº

46 ½º ÂÌº ÎÌË ½ºº Ò ½ ÒØÓÒ ¾ºµº Ý M ÑÓÒÓ Ð ØÖ Ò ÖØÐÑÞØØ ÝÒ ÑÓÒ Ó ÞØØ ÑÓÖÞÑÙ ÓØ Ý Ò ÐÙ ÑÒØ ÓÔÑÓÒÓ Ð ÑÓÒ ÑÓÖÞÑÙ Ó¹ Øº ÞÞ ÑÒØ g : T T ØÖÑ ÞØ ØÖÒ ÞÓÖÑ Ø ÑÐÝ ÓÔÑÓÒÓ Ð Ò Þ ÖØÐÑÒ ÓÝ T(X Y ) g X Y T (X Y ) τ X,Y τ X,Y TX TY gx g Y T X T Y TK g K τ 0 K T K K τ 0 ÖÑÓ ÓÑÑÙØØÚ M ÑÒÒ X Y ÓØÙÑ Ö ÑÐÝ ÑÓÒ ÑÓÖÞÑÙ Ó ÚØÞ ÓÑÑÙØØÚ ÖÑÓ ÖØÐÑÒ M ÑÒÒ X ÓØÙÑ Öº T 2 X Tg X TT X g T X T 2 X X X m X m X TX g X T X u X u X TX gx T X. ÝÒ ÑÓÒ Ó M¹Ò ÑÒØ ÓØÙÑÓ ÑÓÖÞÑÙ ÑÒØ ÒÝÐ Ý Wbm(M)¹ ÑÐ ÐÐØ ØÖ Ø ÐÓØÒ ÑÒ ÑÓÒ Ó ØÖ ØÐ Ö ÞØÖº ÞØ ÑÓÒÙ ÓÝ Ý (R,µ,η) ÑÓÒÓ ÚÐÑÐÝ (M,,K) ÑÓÒÓ Ð ØÖ Ò ÖÒÐÞ ÖÓÒÙ ¹ØÙÐÓÒ Ð ÐØ ÙÒØÓÖ Þ R¹ÑÓÙÐÙ Ó ØÖ Ð M¹ ÓÐÝÒ ÓÝ Ó Ð ÙÒ ÐØ ÑÝÞº Þ ÚÚÐÒ ÞÞÐ ÓÝ ÐØÞ Ý (R,δ,ε) ÓÑÓÒÓ M¹Ò ÚØÞ ÖÑ ÓÑÑÙØØÚº δ Id R R µ R R R Id δ R R R R Id µ δ R R µ Id ËÞÔÖ Ð ÖÓÒÙ ¹ÑÓÒÓÖÐ ÞÐÒ ØÓÚ δ µ ÞÓÖÞ ÞÐ ÞÞ µδ = Id R º Ð Ø ÞÐ Ø ØÖÑÒÓÐ ÑÝÖ ÞØ Þ ÓÝ Ý R ÑÓÒÓ ÔÓÒØÓ Ò ÓÖ ÞÔÖ Ð µ ÖÓÒÙ ¹ØÙÐÓÒ ÐØ ÙÒØÓÖ Þ R¹ÑÓÙÐÙ Ó ØÖ Ð M¹ ÞÔÖ Ð µ ÖÓÒÙ ¹ ÞÖÞØòºµ ÌÒØ ÚØÞ Sfbm(M)¹ÑÐ ÐÐØ ØÖ Øº ÇØÙÑ ÐÝÒ (R, T) Ô ¹ ÖÓ ÓÐ R ÞÔÖ Ð ÖÓÒÙ ¹ÑÓÒÓ M¹Ò T ÑÓÒ Þ R ÑÓÙÐÙ Ó Ø¹ Ö Òº Þ (R, T) (R, T ) ÑÓÖÞÑÙ Ó ÐÝÒ (γ,γ) Ô ÖÓ ÓÐ γ : R R ÑÓÒÓ ÓÑÓÒÓ ÞÓÑÓÖÞÑÙ Ý Ý γ ÞÓÑÓÖÞÑÙ Ø ÒÙ Ð Þ R¹ÑÓÙÐÙ Ó Þ R ¹ ÑÓÙÐÙ Ó ÑÓÒÓ Ð ØÖ ÞØØµ Γ Ô T γ Tγ 1 ÑÓÒ ÑÓÖÞÑÙ Þ ½º Ò ÖØÐÑÒµº ½ºº ÌØÐ ½ ÌÓÖÑ ¾º½½µº ÖÑÐÝ M ÙÝ¹ØÐ ÑÓÒÓ Ð ØÖ ØÒ Wbm(M) Sfbm(M) ØÖ ÚÚÐÒ º

47 ½ºº Æ ÁÅÇÆýÇÃ ½ºº º ÝÒ ÑÓÒ Ó ÝÒ ÐÖ ÝÒ ÐÖ ½º Ò ÐÐØÚ ÝÒ ÀÓÔ¹ÐÖ ½º Ò ÑÒÒ ÒÞ ÒÐÐ Ñ ÑØÐØ k¹ñóùðù Ó ØÖ ÐÝØØ ØØ ÞÐ ÞÑÑØÖÙ ÚÝ Ö ÓÒÓØØ ÑÓÒÓ Ð ØÖ Ò Ðº ¾ º ½ºº ÌØÐ ½ ÌÓÖÑ º½µº ÌÒØ Ò Ý (B,µ,η) ÑÓÒÓÓØ ÚÐÑÐÝ (M,,K,c) ÙÝ¹ØÐ ÓÒÓØØ ÑÓÒÓ Ð ØÖ Òº ØÚ Ô ÓÐØ ÐÐ ÒØ ÚØÞ ØÖÙ¹ ØÖ ÞØØº (B,µ,η,δ,ε) Ð ÝÒ ÐÖ M¹Ò (( ) B,( ) µ,( ) η,τ,τ 0 ) Ð ÝÒ ÑÓÒ Ó M¹Ò ÑÐÝÖ Þ Ð ÖÑ ÓÑÑÙØØÚ M ÖÑÐÝ X,Y ÓØÙÑ Öº X Y B Id Id τk,k X Y B B τ X,Y X B Y B Id c Y,B Id ½º½µ À ÚÐÑÐÝ B ÑÓÒÓÖ Ý (M,,K,c) ÙÝ¹ØÐ ÓÒÓØØ ÑÓÒÓ Ð ØÖ Ò ( ) B ÙÒØÓÖ ÝÒ ÑÓÒ M¹Ò ÓÖ ØÖÑ ÞØ ÑØØ ÑÒÒ f : K X g : K Y h : K B ÑÓÖÞÑÙ Ö Þ Ð ÖÑ ÓÑÑÙØØÚº f g h K X Y B f g h Id Id τk,k X Y B B h B X Y B τ X,Y τ K,K B B X B Y B f Id g Id Id c Y,B Id Ý ½º½µ ØÐ Ð K ÑÓÒÓ Ð Ý ÒÖ ØÓÖ ÚØÞ ÖØÐÑÒ À ÚÐÑÐÝ p,q : X Y Z W M¹Ð ÑÓÖÞÑÙ ÓÖ p (f g h) = q (f g h) ÑÒÒ f : K X g : K Y h : K Z ÑÓÖÞÑÙ ØÒ ÓÖ p = qº ÑÓÒÓ Ð Ý ÒÖ ØÓÖ ÔÐ ÙÐ k¹ñóùðù Ó ÞÑÑØÖÙ ÑÓÒÓ Ð ØÖ Òº Ý Ø Ø ÒØ ØØÐ Ñ Ò ÓÐÐ k ÐØØ ÝÒ ÐÖ ÑÒØ M k ¹Ò ÖØÐÑÞØØ ÝÒ ÑÓÒ Ó ÐÖ Ø Ðº ËÞÐ ÒÝ ½ µº ½ººº ÝÒ ÀÓÔ¹ÑÓÒ Ó Ý (M,,K) ÑÓÒÓ Ð ØÖ Ø ÓÖÐ Þ ÖØÒ ÑÓÒÙÒ M ÑÒÒ X ÓØÙÑ ¹ Ö ( ) X : M M ÙÒØÓÖ Ð ÙÒ ÐØº ÈÐ ÙÐ k¹ñóùðù Ó ØÖ ÓÖÐ Þ ÖØº ÂÓÖÐ Þ ÖØ ØÓÚ Ý B k¹ðö ÑÓÙÐÙ Ò ØÖ ËÙÒÙÖ ÞÖÚØÐ ÞÖÒØ Þ M B M k ÐØ ÙÒØÓÖ ÔÓÒØÓ Ò ÓÖ ÓÑÔØÐ Þ ÖØ ØÖÙØÖ Ð B ÀÓÔ¹ÐÖº ÞÒ Þ ÐÔÓÒ ÞÒÚ Ý ÑÓÒ ÓØ Ý ÓÖÐ Þ ÖØ ÑÓÒÓ Ð ØÖ Ò Ó ÀÓÔ¹ ÑÓÒ Ò ÑÓÒÒ ÐÒÖÅÓÓÖ¹ØÖ ÓÖÐ Þ ÖØ Ý ÓÝ ÐØ ÙÒØÓÖ ÓÑÔØÐ Þ ÖØ ØÖÙØÖ Ðº ÄÛÚÖ ËÙÒÙÖ ÒØ ÞÖÚØÐÒÐ ÐØÐ ÒÓ µ ØØÐ Ðº ¼ µ ÞÖÒØ Ý Ó ÙÒ ÐØ ÙÒØÓÖ ÔÐº ÐØ ÙÒØÓÖµ ÔÓÒØÓ Ò ÓÖ ÓÑÔØÐ Þ ÖØ ØÖÙØÖ Ð Þ Òº ÖÓÒÙ ¹ÖÔÖÓØ ØÐ Ðº Ý (T,m,u,τ,τ 0 ) ÑÓÒ ØÒ Þ ÞØ ÐÒØ

48 ½º ÂÌº ÎÌË ÓÝ Þ M T M ÐØ ÙÒØÓÖ ÔÓÒØÓ Ò ÓÖ ÓÑÔØÐ ÓÖÐ Þ ÖØ ØÖÙØÖ Ð can X,Y := ( T(TX Y ) τ TX,Y T 2 X TY m X Id TX TY ) ½º½µ ÒÓÒÙ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ÞÓÑÓÖÞÑÙ º ÖÙÙÖ Ø Ö ¾¾ ¹Ò ÞØ ¹ ÒØ Ú ÓÐØ ÓÝ Ý ÑÓÒ ÓØ Ý ØØ ÞÐ ÑÓÒÓ Ð ØÖ Ò ÒÚÞÞÒ Ó ÀÓÔ¹ÑÓÒ Ò ½º½µ ØÖÑ ÞØ ÞÓÑÓÖÞÑÙ º ËÞÑÑØÖÙ Ò Ò ÐØ Ð ÀÓÔ¹ÑÓÒ Ó T(X TY ) TX TY ÒÓÒÙ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ÞÓÑÓÖ¹ ÞÑÙ ÚÓÐØ ÚÐº Þ ½º½µ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ Ø ØÒØØ Ý ÝÒ ÑÓÒ ØÒ º Ý ÝÒ ÀÓÔ¹ÐÖ ÐØÐ ÒÙ ÐØ ÝÒ ÑÓÒ Ö ÞÓÒÒ ½º½µ ÒÑ ØÖÑ ÞØ ÞÓ¹ ÑÓÖÞÑÙ ØÖÑ ÞØ ÞÓÑÓÖÞÑÙ Ø ÒÙ Ð T(TX Y ) TX TY Ý¹Ý ÐÐÑ ÖØÖØÙÑ ÞØØº ÝÒ ÀÓÔ¹ÑÓÒ Ò ÓÞ Þ ½º ÌØÐÒ Ð ØÓØØ E TX,TY : TX TY TX TY Ý T ¹Þ ÞÒØÒ ÒÓÒÙ Ò ÖÒÐØ Ðº ½ º µµ F X,Y : T(TX Y ) T(TX Y ) ÑÔÓØÒ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ Ø ÞÒ ÐÙÒº ½ºº Ò ½ ÒØÓÒ º½µº Ý T ÝÒ ÑÓÒ ÓØ ÚÐÑÐÝ (M,,K) ÑÓÒÓ¹ Ð ØÖ Ò ÝÒ Ó ÀÓÔ¹ÑÓÒ Ò ÑÓÒÙÒ ÐØÞ Ý χ X,Y : TX TY T(TX Y ) ØÖÑ ÞØ ØÖÒ ÞÓÖÑ ÑÐÝÖ χ X,Y E TX,TY = χ X,Y = F X,Y χ X,Y, χ X,Y can X,Y = F X,Y, can X,Y χ X,Y = E TX,TY, M ÖÑÐÝ X,Y ÓØÙÑ ØÒº ËÞÑÑØÖÙ Ò Ò ÐÙ ÝÒ Ð ÀÓÔ¹ÑÓÒ ÓØ T(X TY ) TX TY ÒÓÒÙ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ Ø ÚÐº Ò ÓÓ Ø ÞÓÐ ÚØÞº ½º¼º ÌØÐ ½ ÌÓÖÑ º¾µº ÌÒØ Ò Ý T ÝÒ ÑÓÒ ÓØ ÚÐÑÐÝ (M,,K) ÙÝ¹ØÐ ÑÓÒÓ Ð ØÖ Ò Þ ½º ÌØÐ ÞÖÒØ Ò ÑÐÐ T ÑÓÒ ÓØ Ý ÞÔÖ Ð ÖÓÒÙ ¹ÑÓÒÓ ÑÓÙÐÙ ØÖ Òµº ÚØÞ ÐÐØ Ó ÚÚÐÒ º T Ó ÐÐº Ðµ ÀÓÔ¹ÑÓÒ T ÝÒ Ó ÐÐº Ðµ ÀÓÔ¹ÑÓÒ º ÄÝÒ T Ý ÝÒ Ó ÀÓÔ¹ÑÓÒ ÚÐÑÐÝ (M,,K) ÙÝ¹ØÐ ÑÓÒÓ Ð ØÖ Ò ÐÝÒ R Þ ½º ÌØÐ ÞÖÒØ ÓÞÞ ÖÒÐØ ÞÔÖ Ð ÖÓÒÙ ¹ÑÓÒÓº À Þ R¹ÑÓÙÐÙ Ó ØÖ ÓÖÐ Þ ÖØ ÓÖ ÒØ ØØÐ ÞÖÒØ M T ÓÖÐ Þ ÖØ Ý ÓÝ ÐØ ÙÒØÓÖ Þ R¹ÑÓÙÐÙ Ó ØÖ ÓÑÔØÐ Þ ÖØ ØÖÙØÖ Ðº À M ÓÖÐ Þ ÖØ ÓÖ Þ R¹ÑÓÙÐÙ Ó ØÖ ÓÖÐ Þ ÖØ ÑÚ ÒØ Ó ÙÒ ÐØ ÐÐÑ ÑÔÓØÒ ØÖÑ ÞØ ØÖÒ ÞÓÖÑ Ð ÚÐ Òµº ÎÐ ÖÑÐØ Ô ÓÐØ ÞÓÒÝØØ ÝÒ ÀÓÔ¹ÑÓÒ Ó ÝÒ ÀÓÔ¹ÐÖ ÞØØº ½º½º ÌØÐ ¾¾ ÌÓÖÑ º Ò ºµº ÄÝÒ B Ý ÝÒ ÐÖ ÚÐÑÐÝ (M,,K,c) ÙÝ¹ØÐ ÓÒÓØØ ÑÓÒÓ Ð ØÖ Òº Þ ÒÙ ÐØ ( ) B ÙÒØÓÖ ÔÓÒØÓ Ò ÓÖ ÝÒ Ó ÀÓÔ¹ÑÓÒ B ÝÒ ÀÓÔ¹ÐÖº ÌÓÚ ( ) B ÔÓÒØÓ Ò ÓÖ ÝÒ Ð ÀÓÔ¹ÑÓÒ B ÝÒ ÀÓÔ¹ÐÖ ÒØÔ ÒÚÖØ ÐØº

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71 3. fejezet Hopf algebroids with bijective antipodes: axioms, integrals, and duals Böhm, G.; Szlachányi, K. J. Algebra 274 (2004), no. 2,

72 72 3. FEJEZET. HOPF ALGEBROIDS WITH BIJECTIVE ANTIPODES

73 Journal of Algebra 274 (2004) Hopf algebroids with bijective antipodes: axioms, integrals, and duals Gabriella Böhm 1 and Kornél Szlachányi Research Institute for Particle and Nuclear Physics, Budapest, PO Box 49, H-1525 Budapest 114, Hungary Received 6 March 2003 Communicated by Eva Bayer-Fluckiger Abstract Motivated by the study of depth 2 Frobenius extensions, we introduce a new notion of Hopf algebroid. It is a 2-sided bialgebroid with a bijective antipode which connects the two, left and right handed, structures. While all the interesting examples of the Hopf algebroid of J.H. Lu turn out to be Hopf algebroids in the sense of this paper, there exist simple examples showing that our definition is not a special case of Lu s. Our Hopf algebroids, however, belong to the class of L -Hopf algebras proposed by P. Schauenburg. After discussing the axioms and some examples, we study the theory of non-degenerate integrals in order to obtain duals of Hopf algebroids Elsevier Inc. All rights reserved. 1. Introduction There is a consensus in the literature that bialgebroids, invented by Takeuchi [24] as R -bialgebras, are the proper generalizations of bialgebras to non-commutative base rings [5,13,17,20,21,25]. The situation of Hopf algebroids, i.e., bialgebroids with some sort of antipode, is less understood. The antipode proposed by J.H. Lu [13] is burdened by the need of a section for the canonical epimorphism A A A R A the precise role of which remained unclear. The R -Hopf algebras proposed by P. Schauenburg in [18] have a clearcut categorical meaning. They are the bialgebroids A over R such that the * Corresponding author. addresses: bgabr@rmki.kfki.hu (G. Böhm), szlach@rmki.kfki.hu (K. Szlachányi). 1 Supported by the Hungarian Scientific Research Fund, OTKA T , FKFP 0043/2001, and the Bolyai János Fellowship /$ see front matter 2004 Elsevier Inc. All rights reserved. doi: /j.jalgebra

74 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) forgetful functor A M R M R is not only strict monoidal, which is the condition for A to be a bialgebroid over R, but preserves the closed structure as well. In this very general quantum groupoid, however, antipode, as a map A A, does not exist. Our proposal of an antipode, announced in [2], is based on the following simple observation. The antipode of a Hopf algebra H is a bialgebra map S : H Hcop. The opposite of a bialgebroid A, however, is not a bialgebroid in the same sense. In the terminology of [8] there are left bialgebroids and right bialgebroids; corresponding to whether A M or M A is given a monoidal structure. This suggests that the existence of antipode on a bialgebroid should be accompanied with a two-sided bialgebroid structure and the antipode should swap the left and right handed structures. More explicit guesses for what to take as a definition of the antipode can be obtained by studying depth 2 Frobenius extensions. In [8] it has been shown that for a depth 2 ring extension N M the endomorphism ring A = End N M N has a canonical left bialgebroid structure over the centralizer R = C M (N). IfN M is also Frobenius and a Frobenius homomorphism ψ : M N is given, then A has a right bialgebroid structure, too. There is a candidate for the antipode S : A A as transposition w.r.t. the bilinear form m, m M ψ(mm ), see (3.6). In fact, this definition of S does not require the depth 2 property, so in this example antipode exists prior to comultiplication. Assuming the extension M/N is either H-separable or Hopf Galois, L. Kadison has shown [9,10] that the bialgebroid A or its dual B = (M N M) N has an antipode in the sense of [13]. In a recent paper [7] B. Day and R. Street give a new characterization of bialgebroids in the framework of symmetric monoidal autonomous bicategories. They also introduce a new notion called Hopf bialgebroid. It is more restrictive than Schauenburg s R -Hopf algebra since it requires star autonomy [1] rather than to be closed. We will show in Section 4.2 that the categorical definition [7] of the Hopf bialgebroid is equivalent to our purely algebraic Definition 4.1 of Hopf algebroid, apart from the tiny difference that we allow S 2 to be nontrivial on the base ring. This freedom can be adjusted to the Nakayama automorphism of ψ in case of the Frobenius depth 2 extensions. It is a new feature of our Hopf algebroids, compared to (weak) Hopf algebras, that the antipode is not unique. The various antipodes on a given bialgebroid were shown to be in one-to-one correspondence with the generalized characters (called twists) in [2]. It is encouraging that one can find quantum groupoids in the literature that satisfy our axioms. Such are the weak Hopf algebras with bijective antipode, the examples of Lu Hopf algebroids in [5], and the extended Hopf algebras in [11]. In particular, the Connes Moscivici algebra [6] is a Hopf algebroid it this sense. We also present an example which is a Hopf algebroid in the sense of this paper but does not satisfy the axioms of [13]. This proves that the two notions of Hopf algebroid the one in the sense of this paper and the one in the sense of [13] are not equivalent. Until now we could neither prove nor exclude by examples the possibility that the latter was a special case of the former. The left and right integrals in Hopf algebra theory are introduced as the invariants of the left and right regular module, respectively. In this analogy one can define left integrals in a left bialgebroid and right integrals in a right bialgebroid. Since a Hopf algebroid has both left and right bialgebroid structures both left and right integrals can be defined. The properties of the integrals in a (weak) Hopf algebra over a field k carry information about its algebraic structure. For example, the Maschke s theorem [3,12] states that it is

75 710 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) a semi-simple algebra if and only if it has a normalized integral. The Larson Sweedler theorem [12,26] implies that it is finite dimensional over k if and only if it has a nondegenerate left (hence also a right) integral. In this case the k-dual also has a (weak) Hopf algebra structure. Therefore, in Section 5, we analyze the consequences of the existence of a nondegenerate integral in a Hopf algebroid. We show that if there exists a non-degenerate integral in a Hopf algebroid A over the base L then the ring extension L A is a Frobenius extension hence also finitely generated projective. We do not investigate, however, the opposite implication, i.e., we do not study the question under what conditions on the Hopf algebroid the existence of a non-degenerate integral follows. If some of the L-module structures of a Hopf algebroid is finitely generated projective then the corresponding dual can be equipped with a bialgebroid structure [8]. There is no obvious way, however, how to equip it with an antipode in general. We show that in the case of Hopf algebroids possessing a non-degenerate integral the dual bialgebroids are all (anti-) isomorphic and they combine into a Hopf algebroid depending on the choice of the nondegenerate integral. Therefore, we do not associate a dual Hopf algebroid to a given Hopf algebroid rather a dual isomorphism class to an isomorphism class of Hopf algebroids. The well-known k-dual of a finite (weak) Hopf algebra H over the commutative ring k turns out to be the unique (distinguished) (weak) Hopf algebra in the dual isomorphism class of the isomorphism class of the Hopf algebroid H. The paper is organized as follows. In Section 2 we introduce some technical conventions about bialgebroids that are used in this paper. Our motivating example, the Hopf algebroid corresponding to a depth 2 Frobenius extension of rings is discussed in Section 3. In Section 4 we give some equivalent definitions of Hopf algebroids. We prove that our definition is equivalent to the one in [7] hence gives a special case of the one in [18]. In the final subsection of Section 4 we present a collection of examples. In Section 5 we propose a theory of non-degenerate integrals as a tool for the definition of the dual Hopf algebroid. 2. Preliminaries on bialgebroids In this technical section we summarize our notations and the basic definitions of bialgebroids that will be used later on. For more about bialgebroids we refer to the literature [5,8,18,19,21,22,24]. Definition 2.1. A left bialgebroid (or Takeuchi L -bialgebra) A L consists of the data (A,L,s L,t L,γ L,π L ).TheA and L are associative unital rings, the total and base rings, respectively. The s L : L A and t L : L op A are ring homomorphisms such that the images of L in A commute making A an L-L-bimodule via l a l := s L (l)t L (l )a. (2.1)

76 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) The bimodule (2.1) is denoted by L A L. The triple ( L A L,γ L,π L ) is a comonoid in L M L, the category of L-L-bimodules. Introducing the Sweedler s notation γ L (a) a (1) a (2) A L A the identities a (1) t L (l) a (2) = a (1) a (2) s L (l), (2.2) γ L (1 A ) = 1 A 1 A, (2.3) γ L (ab) = γ L (a)γ L (b), (2.4) π L (1 A ) = 1 L, (2.5) π L ( asl π L (b) ) = π L (ab) = π L ( atl π L (b) ) (2.6) are required for all l L and a,b A. The requirement (2.4) makes sense in the view of (2.2). The L actions of the bimodule L A L in (2.1) are given by left multiplication. Using right multiplication there exists another L-L-bimodule structure on the total ring A of a left bialgebroid A L : l a l := at L (l)s L (l ). (2.7) This L-L-bimodule is called L A L.ThiswayA carries four commuting actions of L. If A L = (A,L,s L,t L,γ L,π L ) is a left bialgebroid then so is its co-opposite: A L cop = (A, L op,t L,s L,γ op L,π L). The opposite A op L = (Aop,L,t L,s L,γ L,π L ) has a different structure that was introduced under the name right bialgebroid in [8]. Definition 2.2. A right bialgebroid A R consists of the data (A,R,s R,t R,γ R,π R ).TheA and R are associative unital rings, the total and base rings, respectively. The s R : R A and t R : R op A are ring homomorphisms such that the images of R in A commute making A an R-R-bimodule: r a r := as R (r )t R (r). (2.8) The bimodule (2.8) is denoted by R A R. The triple ( R A R,γ R,π R ) is a comonoid in R M R. Introducing the Sweedler s notation γ R (a) a (1) a (2) A R A the identities are required for all r R and a,b A. s R (r)a (1) a (2) = a (1) t R (r)a (2), γ R (1 A ) = 1 A 1 A, γ R (ab) = γ R (a)γ R (b), π R (1 A ) = 1 R, π R ( sr π R (a)b ) = π R (ab) = π R ( tr π R (a)b )

77 712 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) For the right bialgebroid A R we introduce the R-R-bimodule R A R via r a r := s R (r)t R (r )a. (2.9) This way A caries four commuting actions of R. Left (right) bialgebroids can be characterized by the property that the forgetful functor AM L M L (M A R M R ) is strong monoidal [17,20]. It is natural to consider the homomorphisms of bialgebroids to be ring homomorphisms preserving the comonoid structure. We do not want to make difference however between bialgebroids over isomorphic base rings. This leads to the following. Definition 2.3 [21]. A left bialgebroid homomorphism A L A L homomorphisms (Φ : A A,φ: L L ) such that is a pair of ring s L φ = Φ s L, t L φ = Φ t L, π L Φ = φ π L, γ L Φ = (Φ Φ) γ L. The last condition makes sense since by the first two conditions Φ Φ is a well-defined map A L A A L A. The pair (Φ, φ) is an isomorphism of left bialgebroids if it is a bialgebroid homomorphism such that both Φ and φ are bijective. A right bialgebroid homomorphism (isomorphism) A R A R is a left bialgebroid homomorphism (isomorphism) (A R ) op (A R )op. Let A L be a left bialgebroid. The equation (2.1) describes two L-modules A L and L A. Their L-duals are the additive groups of L-module maps A := {φ : A L L L } and A := { φ : L A L L}, where L L stands for the left regular and L L for the right regular L-module. Both A and A carry left A module structures via the transpose of the right regular action of A. For φ A, φ A,anda,b A we have (a φ )(b) = φ (ba) and (a φ)(b) = φ(ba). Similarly, in the case of a right bialgebroid A R denoting the left and right regular R-modules by R R and R R, respectively the two R-dual additive groups A := { φ : A R R R} and A := { φ : R A R R } carry right A-module structures: (φ a)(b) = φ (ab) and ( φ a)(b) = φ(ab).

78 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) The comonoid structures can be transposed to give monoid (i.e., ring) structures to the duals. In the case of a left bialgebroid A L ( ) (φ ψ )(a) = ψ sl φ (a (1) )a (2) ( φ ψ)(a) = ψ ( ) t L φ(a (2) )a (1) for φ, ψ A, φ,ψ A,anda A. Similarly, in the case of a right bialgebroid A R, and (φ ψ )(a) = φ ( a (2) t R ψ ( a (1))) and (2.10) ( φ ψ)(a) = φ ( a (1) s R ψ ( a (2))) (2.11) for φ,ψ A, φ, ψ A,anda A. In the case of a left bialgebroid A L also the ring A has right A - and right A-module structures: a φ = s L φ (a (1) )a (2) and a φ = t L φ(a (2) )a (1) (2.12) for φ A, φ A and a A. Similarly, in the case of a right bialgebroid A R the ring A has left A -andleft A- structures: φ a= a (2) t R φ ( a (1)) and φ a= a (1) s R φ ( a (2)) (2.13) for φ A, φ A,anda A. In the case when the L (R) module structure on A is finitely generated projective then the corresponding dual has also a bialgebroid structure: if A L is a left bialgebroid such that the L-module A L is finitely generated projective then A is a right bialgebroid over the base L as follows: ( s R (l) ) (a) = π L ( asl (l) ), ( t R (l) ) (a) = lπ L (a), γ R (φ ) = b i φ β i, π R(φ ) = φ (1 A ), where {b i } is an L-basis in A L and {β i } is the dual basis in A. Similarly, if A L is a left bialgebroid such that the L-module L A is finitely generated projective then A is a right bialgebroid over the base L as follows: ( s R (l) ) (a) = π L (a)l, ( t R (l) ) (a) = π L ( atl (l) ), γ R ( φ) = β i b i φ, π R ( φ) = φ(1 A ), where {b i } is an L-basis in L A and { β i } is the dual basis in A. If A R is a right bialgebroid such that the R-module A R is finitely generated projective then A is a left bialgebroid over the base R as follows:

79 714 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) ( s L (r) ) (a) = rπ R (a), ( t L (r) ) (a) = π R ( sr (r)a ), γ L (φ ) = φ b i β i, π L (φ ) = φ (1 A ), where {b i } is an R-basis in A R and {β i } is the dual basis in A. If A R is a right bialgebroid such that the R-module R A is finitely generated projective then A is a left bialgebroid over the base R as follows: ( s L (r) ) (a) = π R ( tr (r)a ), γ L ( φ) = β i φ b i, ( t L (r) ) (a) = π R (a)r, π L ( φ) = φ(1 A ), where {b i } is an R-basis in R A and { β i } is the dual basis in A. 3. The motivating example: D2 Frobenius extensions 3.1. The forefather of antipodes In this subsection N M denotes a Frobenius extension of rings. This means the existence of N-N-bimodule maps ψ : M N possessing quasibases. An element i u i v i M N M is called the quasibasis of ψ [27] if ψ(mu i ) v i = m = u i ψ(v i m), m M. (3.1) i i As we shall see, already in this general situation there exist anti-automorphisms S on the ring A := End N M N, one for each Frobenius homomorphism ψ. TheS will become an antipode if the extension N M is also of depth 2, so A also has coproduct(s). The idea of writing the antipode as the difference of two Fourier transforms goes back to Radford s paper [16] but plays important role in Szymanski s treatment of finite index subfactors [23], too. If l is a non-degenerate left integral in a finite dimensional Hopf algebra H and λ H is the dual left integral, i.e., λ l= 1, then the antipode can be written as S(h) = (λ h) l. This formula generalizes also to weak Hopf algebras [3]. Here we will give the analogous formula for the two-step centralizer A of any Frobenius extension N M. Instead of a dual algebra of A we have the second two-step centralizer B in the Jones tower of N M which is the center of the N-N-bimodule M N M B := (M N M) N {X M N M n X = X n n N}. It is a ring with multiplication (b 1 b 2 )(b 1 b 2 ) = b 1 b 1 b 2 b 2 and unit 1 B = 1 M 1 M. Note that the ring structures of neither A nor B depend on the Frobenius structure. But if there is a Frobenius homomorphism ψ then Fourier transformation makes A and B isomorphic as additive groups.

80 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Fixing a Frobenius homomorphism ψ with quasibasis i u i v i, we can introduce convolution products on both A and B as follows. From now on we omit the summation symbol for summing over the quasibasis α, β A α β := α(u i )β(v i _ ) A, a,b B a b := a 1 ψ ( a 2 b 1) b 2 B. The convolution product lends A and B new ring structures. The unit of A is ψ and the unit of B is u i v i. The Fourier transformation is to relate these new algebra structures to the old ones. There are two natural candidates for a Fourier transformation: F : A B, F(α) := u i α(v i ), Ḟ : A B, Ḟ(α) := α(u i ) v i, F 1 : B A, F 1 (b) = ψ ( _b 1) b 2, F 1 : B A, F 1 (b) = b 1 ψ ( b 2 _ ). They relate the convolution and ordinary products or their opposites as follows: F(α β) = F(α)F(β), F(αβ) = F(β) F(α), Ḟ(α β) = F(β)Ḟ(α), F(αβ) = Ḟ(α) Ḟ(β). The difference between F and F is therefore an anti-automorphism on both A and B.This leads to the antipodes S A : A A op, S B : B B op, S A := F 1 F, S A (α) = u i ψ ( α(v i )_ ), (3.2) S B := F F 1, S B (b) = ψ ( u i b 1) b 2 v i (3.3) with inverses SA 1 _α(u i ) ) v i, (3.4) SB 1 i b 1 ψ ( b 2 ) v i. (3.5) Notice that S A is just transposition w.r.t. the bi-n-linear form (m, m ) = ψ(mm ) since ψ ( ms A (α)(m ) ) = ψ ( α(m)m ). (3.6) These antipodes behave well also relative to the bimodule structures over the centralizer C M (N) := {c M cn = nc, n N}. Let us consider S A. The centralizer is embedded into A twice: via left multiplications and right multiplications, λ ρ L A R,

81 716 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) where L stands for C M (N) and R for C M (N) op. Clearly, λ(l) C M (ρ(r)). Introducing the Nakayama automorphism ν : C M (N) C M (N), ν(c) := ψ(u i c)v i of ψ and using its basic identities ψ(mc) = ψ ( ν(c)m ), u i c v i = u i ν(c)v i, m M, c C M (N), c C M (N), we obtain S A λ = ρ ν 1, S A ρ = λ and therefore S A ( λ(l)ρ(r)α ) = SA (α)λ(r)ρ ( ν 1 (l) ), l L, r R, α A. (3.7) In order to interpret the latter relation as the statement that S A is a bimodule map we define L-L-andR-R-bimodule structures on A by where we introduced the ring homomorphisms l 1 α l 2 := s L (l 1 )t L (l 2 )α, (3.8) r 1 α r 2 := αt R (r 1 )s R (r 2 ), (3.9) s L := L λ A, s R := R ρ A, t L := L op id R ρ A, t R := R op ν L λ A. (3.10) Also using the notation θ for the inverse of the Nakayama automorphism when considered as a map Eq. (3.7) can be read as θ : L ν 1 R op, S A (l 1 α l 2 ) = θ(l 2 ) S A (α) θ(l 1 ). (3.11) Remark 3.1. The apparent asymmetry between t L and t R in (3.10) disappears if one repeats the above construction for the more general situation of a Frobenius N-M-bimodule X instead of the N M M arising from a Frobenius extension of rings. As a matter of fact,

82 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) denoting by X the (two-sided) dual of X and setting A = End X M X, L = EndX, and R = EndX, we find the obvious ring homomorphisms s L (l) = l X, s R (r) = X r; but there is no distinguished map L R op like the identity is in the case of X = N M M. Instead we have two distinguished maps given by the left and right dual functors (transpositions). It is easy to check that in case of X = N M M they are the identity and the ν 1, respectively, as we used in (3.10). In order to restore the symmetry, let us introduce the counterpart of θ which is the identity as a homomorphism ι : R id L op. Then, in addition to (3.11) the antipode satisfies also S A (r 1 α r 2 ) = ι(r 2 ) S A (α) ι(r 1 ). (3.12) The most important consequence of (3.11) and (3.12) is the existence of a tensor square of S A. In the case of Hopf algebras, one often uses expressions like (S S) Σ,whereΣ is the symmetry A A A A, x y y x in the category of k-modules. Now we have bimodule categories L M L and R M R without braiding, so Σ does not exist and neither do S A S A nor SA 1 S 1 A because S A is not a bimodule map. Instead we have the twisted bimodule properties (3.11) and (3.12) which guarantee the existence of the composite of S A S A and Σ although individually they do not exist. More precisely, there exist twisted bimodule maps S A L A : A L A A R A, S A R A : A R A A L A, α L β S A (β) R S A (α), α R β S A (β) L S A (α). For later convenience let us record some useful formulas following directly from (3.2) and (3.10): S A s L = t L ν 1, S A t L = s L, t L ι = s R, S A s R = t R ν 1, S A t R = s R, t R θ = s L. (3.13) Notice also that s L (L) and t R (R) are the same subrings of A and similarly t L (L) = s R (R) Two-sided bialgebroids Recall from [8] that for depth 2 extensions N M the A has a canonical left bialgebroid structure over L in which the coproduct γ L : A A L A is an L-L-bimodule map with respect to the bimodule structure (3.8). If N M is also Frobenius then there is another right bialgebroid structure on A, canonically associated to a choice of ψ, in which R is the base and A is an R-R-bimodule via (3.9). Moreover, these two structures

83 718 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) are related by the antipode. This two-sided structure is our motivating example of a Hopf algebroid. We start with a technical lemma on the left and right quasibases. A and B denotes the rings as before. Lemma 3.2. Let N M be a Frobenius extension and ψ, u i v i be a fixed Frobenius structure. Let n be a positive integer and β i, γ i A and b i, c i B, fori = 1,...,n. Assume they are related via b i = F(γ i ) and c i = F(β i ). Then the following conditions are equivalent (summation symbols over i suppressed ): (i) bi 1 N bi 2β i(m) = m N 1 M, m M; (ii) γ i (m)ci 1 N ci 2 = 1 M N m, m M; (iii) γ i (m)β i (m ) = ψ(mm ), m, m M; (iv) bi 1 N bi 2c1 i N ci 2 = u k N 1 M N v k. If such elements exist the extension is called D2, i.e., of depth 2. The first two conditions are meaningful also in the non-frobenius case and therefore {b i,β i } was called in [8] a left D2 quasibasis and {c i,γ i } a right D2 quasibasis. The equivalence of conditions (i) and (ii) was shown in [8, Proposition 6.4]. The rest of the proof is left to the reader. For D2 extensions the map α β {m m α(m)β(m )} is an isomorphism A L A Hom N N (M N M,M), see [8, Proposition 3.11]. Then the coproduct γ L : A A L A is the unique map α α (1) α (1) which satisfies α(mm ) = α (1) (m)α (1) (m ). We can dualize this construction for D2 Frobenius extensions. We have the isomorphism A R A Hom N N (M, M N M), α R β α(_u i ) N β(v i ). Then γ R : A A R A is defined as the unique map α α (1) α (2) for which α(m)u i N v i = α (1) (mu i ) N α (2) (v i ). (3.14) Explicit formulas for both coproducts, as well as their counits, are given in the corollary below. But even without these formulas we can find out how the two coproducts are related by the antipode. Theorem 3.3. For a D2 Frobenius extension N M of rings the endomorphism ring A = End N M N is a left bialgebroid over L = C M (N) and a right bialgebroid over R = L op such that the antipode defined in (3.2) gives rise to isomorphisms

84 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) (A,R,s R,t R,γ R,π R ) op (S A,ι) (A,L,s L,t L,γ L,π L ) cop, of left bialgebroids. That is to say, (A,L,s L,t L,γ L,π L ) (S A,θ) (A,R,s R,t R,γ R,π R ) op cop (3.15) S A : A A op is a ring isomorphism, (3.16) S A s R = s L ι, S A t R = t L ι, S A s L = s R θ, S A t L = t R θ, (3.17) γ L S A = S A R A γ R, γ R S A = S A L A γ L, (3.18) π L S A = ι π R, and π R S A = θ π L. (3.19) Proof. The left bialgebroid structure of A has been constructed in [8, Theorem 4.1]. The right bialgebroid structure will follow automatically after establishing the four properties of the antipode. The first two have already been discussed before. In order to prove (3.18) recall the definition (3.14) of γ R. Thus, (3.18) is equivalent to S 1 A ( ) SA (α) (1) (mui ) N S 1 ( ) SA (α) (1) (vi ) = α(m)u i N v i, (3.20) S A ( S 1 A (α) (1) A ) ( (mui ) N S A S 1 A (α) (1)) (vi ) = u i N v i α(m) (3.21) for all m M. Expanding the left-hand side of (3.20) then using (3.6), then (3.4), then the definition of γ L, and finally (3.6) again we obtain S 1 A ( ) SA (α) (1) (mui )ψ ( S 1 ( ) ) SA (α) (1) (vi )u k N v k = S 1 A = S 1 A A ( SA (α) (1) ) (mui )ψ ( v i S A (α) (1) (u k ) ) N v k ( SA (α) (1) )( msa (α) (1) (u k ) ) N v k = ψ ( ms A (α) (1) (u k )S A (α) (1) (u j ) ) v j N v k = ψ ( ms A (α)(u k u j ) ) v j N v k = ψ ( α(m)u k u j ) vj N v k = α(m)u k N v k and analogously (3.21). Now it is easy to see that both π L S A and θ π L SA 1 are counits for γ R. Therefore, both are equal to the counit π R. This finishes the proof of the isomorphisms (3.15). Corollary 3.4. Explicit formulas for the left and right bialgebroid structures can be given using the quasibases of Lemma 3.2 as follows:

85 720 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) γ L (α) = γ i L ci 1 α( ci 2 _) = α ( _bi 1 ) b 2 i L β i = γ i L β i α = α γ i L β i, π L (α) = α(1 M ), γ R (α) = α ( _ci 1 ) c 2 i R ψ ( _γ i (u k ) ) v k = u k ψ ( β i (v k )_ ) R bi 1 α( bi 2 _) = α S A (β i ) R S A (γ i ) = S A (β i ) R S A (γ i ) α, π R (α) = u i ψ α(v i ). 4. Hopf algebroids 4.1. The definition The total ring of a Hopf algebroid carries eight canonical module structures over the base ring modules of the kind (2.1), (2.7) (2.9). In this situation the standard notation for the tensor product of modules, e.g., A R A, would be ambiguous. In order to avoid any misunderstandings, we therefore put marks on both modules, as in A R R A for example, that indicate the module structures taking part in the tensor product. Other module structures (commuting with those taking part in the tensor product) are usually unadorned and should be clear from the context. For coproduts of left bialgebroids we use the Sweedler s notation in the form γ L (a) = a (1) a (2) and of right bialgebroids γ R (a) = a (1) a (2). Definition 4.1. The Hopf algebroid is a pair (A L,S) consisting of a left bialgebroid A L = (A,L,s L,t L,γ L,π L ) and an anti-automorphism S of the total ring A satisfying (i) S t L = s L and (4.1) (ii) S 1 (a (2) ) (1 ) S 1 (a (2) ) (2 )a (1) = S 1 (a) 1 A, (4.2) S(a (1) ) (1 )a (2) S(a (1) ) (2 ) = 1 A S(a) (4.3) as elements of A L L A,foralla A. The axiom (4.3) implies that S(a (1) )a (2) = t L π L S(a) (4.4) for all a A. Introduce the map θ L := π L S s L : L L. Owing to (4.4), it satisfies t L θ L (l) = t L π L S s L (l) = S s L (l), θ L (l)θ L (l ) = π L S s L (l)π L S s L (l ) = π L ( tl π L S s L (l )S s L (l) ) = π L ( S sl (l )S s L (l) ) = θ L (ll ). (4.5)

86 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) In view of (4.5) S is a twisted bimodule map R A R L A L where R is a ring isomorphic to L op and the R-R-bimodule structure of A is given by fixing an isomorphism µ : L op R: r a r := as L θ 1 L µ 1 (r)t L µ 1 (r ). (4.6) The usage of the same notation R A R as in (2.8) is not accidental. It will turn out from the next Proposition 4.2 that there exists a right bialgebroid structure on the total ring A over the base R for which the R-R-bimodule (2.8) is (4.6). It makes sense to introduce the maps S A L A : A L L A A R R A, a b S(b) S(a) and S A R A : A R R A A L L A, a b S(b) S(a). (4.7) It is useful to give some alternative forms of the Definition 4.1. Proposition 4.2. The following are equivalent: (i) (A L,S) is a Hopf algebroid; (ii) A L = (A,L,s L,t L,γ L,π L ) is a left bialgebroid and S is an anti-automorphism of the total ring A satisfying (4.1), (4.4), and S A L A γ L S 1 = SA 1 R A γ L S, (4.8) (γ L id A ) γ R = (id A γ R ) γ L, (γ R id A ) γ L = (id A γ L ) γ R, (4.9) where we introduced the ring R and the R-R-bimodule R A R as in (4.6) and the map γ R := S A L A γ L S 1 S 1 A R A γ L S : A A R R A. The equations in (4.9) are equalities of maps A A L L A R R A and A A R R A L L A, respectively. (iii) A L = (A,L,s L,t L,γ L,π L ) is a left bialgebroid and A R = (A,R,s R,t R,γ R,π R ) is a right bialgebroid such that the base rings are related to each other via R L op. S is a bijection of additive groups and s L (L) = t R (R), t L (L) = s R (R) as subrings of A, (4.10) (γ L id A ) γ R = (id A γ R ) γ L, (γ R id A ) γ L = (id A γ L ) γ R, (4.11) S ( t L (l)at L (l ) ) = s L (l )S(a)s L (l), S ( t R (r )at R (r) ) = s R (r)s(a)s R (r ), (4.12) S(a (1) )a (2) = s R π R (a), a (1) S ( a (2)) = s L π L (a) (4.13) hold true for all l,l L, r, r R, and a A.

87 722 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) (iv) A L is a left bialgebroid over L and A R is a right bialgebroid over R such that the base rings are related to each other via R L op and Eqs. (4.10) and (4.11) hold true. Furthermore, the maps of additive groups α : A R R A A L L A, a b a (1) a (2) b and β : A R R A A L L A, a b b (1) a b (2) (4.14) are bijective. (All modules appearing in (4.14) are the canonical modules introduced in Section 2.) Each characterization of Hopf algebroids in Proposition 4.2 will be relevant in what follows. The one in (ii) is as similar to [13] as possible which will be useful in Section 4.3 both in checking that concrete examples of Lu Hopf algebroids satisfy the axioms in Definition 4.1 and also in constructing Hopf algebroids in the sense of Definition 4.1 which do not satisfy the Lu axioms. As it will turn out from the following proof of Proposition 4.2, the Definition 4.1 implies the existence of a right bialgebroid structure on the total ring of the Hopf algebroid. The characterization in (iii) uses the left and right bialgebroids underlying a Hopf algebroid in a perfectly symmetric way. This characterization will be appropriate in developing the theory of integrals in Section 5. The characterization in (iv) is formulated in the spirit of [18], that is the bijectivity of certain Galois maps is required. The relevance of this form of the definition is that it shows that the Hopf algebroid in the sense of Definition 4.1 is a special case of Schauenburg s L -Hopf algebra. Proof. (ii) (iii). We construct a right bialgebroid A R such that (A L, A R,S) satisfies the requirements in (iii). Let R be a ring isomorphic to L op and µ : L op R afixed isomorphism. Set A R = ( A,R, s R := t L µ 1,t R := S 1 t L µ 1,γ R := SA 1 R A γ L S, π R := µ π L S ). (4.15) (iii) (iv). We construct the inverses of the maps (4.14): α 1 : A L L A A R R A, a b a (1) S ( a (2)) b and β 1 : A L L A A R R A, a b S 1( b (1)) a b (2). (4.16) (iv) (i). Recall that the requirements in (iv) imply that both left bialgebroids A L and A L cop are L -Hopf algebras in the sense of [18]. In particular, [18, Proposition 3.7] holds true for both. That is, denoting α 1 (a 1 A ) := a + a and β 1 (1 A a) := a [ ] a [+], we have

88 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) (i) a +(1) a +(2) a = a 1 A, a [+](1) a [ ] a [+](2) = 1 A a, (ii) a (1)+ a (1) a (2) = a 1 A, a (2)[ ] a (1) a (2)[+] = 1 A a, (iii) (ab) + (ab) = a + b + b a, (ab) [ ] (ab) [+] = b [ ] a [ ] a [+] b [+], (iv) (1 A ) + (1 A ) = 1 A 1 A, (1 A ) [ ] (1 A ) [+] = 1 A 1 A, (v) a +(1) a +(2) a = a (1) a (2)+ a (2), a [ ] a [+](1) a [+](2) = a (1)[ ] a (1)[+] a (2), (vi) a + a (1) a (2) = a ++ a a +, a [ ](2) a [ ](1) a [+] = a [ ] a [+][ ] a [+][+], (vii) a = a + t L π L (a ), a = a [+] s L π L (a [ ] ), (viii) a + a = s L π L (a), a [+] a [ ] = t L π L (a). (4.17) We define the antipode as and what is going to be its inverse as S(a) := s R π R (a + )a (4.18) S (a) := t R π R (a [+] )a [ ]. (4.19) The maps (4.18) and (4.19) are well-defined due to the R-module map property of π R. Since α(1 A s L (l)) = 1 A s L (l) t L (l) 1 A, the requirement (4.1) holds true. By making use of (vi) and (i) of (4.17), one verifies and similarly S(a (1) ) (1) a (2) S(a (1) ) (2) = a (1) (1) a (2) s R π R (a (1)+ )a (1) (2) = a (1) a (2) S(a (1)+ ) = 1 S(a) S (a (2) ) (1) S (a (2) ) (2) a (1) = S (a) 1 A, which becomes the requirement (4.2) once we proved S = S 1. As a matter of fact by (vi) of (4.17) hence, using (ii) of (4.17), S(a) (1) S(a) (2) = a (1) s R π R (a + )a (2) = a S(a + ) so, by (viii) of (4.17) and (4.10), β ( a (2) S(a (1) ) ) = a (1) a (2) S(a (1)+ ) = 1 A S(a),

89 724 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) S S(a) = t R π R ( sr π R (a (1)+ )a (1) ) a(2) = t R π R (a (1)+ a (1) )a (2) = s L π L (a (1) )a (2) = a. In a similar way one checks that S S = id A. The anti-multiplicativity of S is proven as follows. We have α(t R (r) 1 A ) = t R (r) 1 A and by β(1 A s R (r)) = 1 A s R (r) and S = S 1 also S(t R (r)a) = S(a)s R (r) hence, S(ab) = s R π R ( (ab)+ ) (ab) = s R π R (a + b + )b a = s R π R ( tr π R (a + )b + ) b a = s R π R ([ tr π R (a + )b ] +)[ tr π R (a + )b ] a = S ( t R π R (a + )b ) a = S(b)s R π R (a + )a = S(b)S(a). (i) (ii). The requirements (4.1) and (4.4) hold obviously true. One easily checks that the maps α and β in (4.14) are bijective with inverses α 1 (a b) = S 1( S(a) (2) ) S(a)(1) b, β 1 (a b) = S 1 (b) (2) a S ( S 1 (b) (1) ). This implies that the [18, Proposition 3.7] holds true both in A L and A L cop. In particular, introducing the maps the part (v) of (4.17) reads as γ R : A A R R A, a ( id A S 1) α 1 (a 1 A ), γ R : A AR R A, a (S id A ) β 1 (1 A a) (γ L id A ) γ R = (id A γ R ) γ L, (4.20) (id A γ L ) γ R = (γ R id A) γ L. (4.21) This means that both (4.8) and (4.9) follow provided γ R = γ R. Using the Sweedler s notation γ R (a) = a (1) a (2) and γ R (a) = a 1 a 2 by the repeated use of (4.20) and (4.21), we obtain (id A γ R ) γ R (a) = a 1 s L π L ( a 2 (1) ) a 2 (2) (1) a 2 (2) (2) = a (1) 1 s L π L ( a(1) 2 ) a (2) (1) a (2) (2) = a (1) (1) 1 s L π L ( a (1) (1) 2 ) a (1) (2) a (2) = a (1) 1 s L π L ( a (1) 2 (1) ) a (1) 2 (2) a (2) = ( id A γ R) γr (a).

90 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Since by (viii) of (4.17) both a (1) S(a (2) ) = s L π L (a) and a 1 S(a 2 ) = s L π L (a), we have and ε(a) := S s L π L S 1 (a) = S(a (1) )a (2) = S 1 s L π L S(a) (m A id A ) (id A ε id A ) (id A γ R ) γ R (a) = a 1 ε ( a 2 (1)) a 2 (2) = a 1 a 2 (2) S 2 s L π L S ( a 2 (1)) = a 1 S 1( S ( a 2 ) (1)) S 1 ( ( t L π L S a 2 ) (2)) = γ R (a), (m A id A ) (id A ε id A ) ( γ R id A) γr (a) = a (1) 1 ε ( a (1) 2 ) a (2) = a (1) 1 S s L π L S 1( a (1) 2 ) a (2) = S ( S 1( a (1)) (2)) S sl π L ( S 1 ( a (1)) (1)) a (2) = γ R (a) hence by the equality of the left-hand sides γ R = γ R. The following is a consequence of the proof of Proposition 4.2. Proposition 4.3. Let (A L,S) be a Hopf algebroid and A R a right bialgebroid such that (A L, A R,S) satisfies the requirements of Proposition 4.2(iii). Then both (S : A A op,ν := π R s L : L R op ) and (S 1 : A A op,µ := π R t L : L R op ) are left bialgebroid isomorphisms A L (A R ) op cop. In particular, A R is unique up to an isomorphism of the form (id A,φ). One easily checks that µ 1 ν = θ L. For the sake of symmetry we introduce also θ R := ν µ 1 with the help of which the right analogue of (4.5) holds true: S s R = t R θ R. Proposition 4.3 has an interpretation in terms of the forgetful functors Φ R : M A RM R and Φ L : A M R M R as follows. The antipode map defines two functors S and S : M A A M. They have object maps (M, ) (M, S) and (M, ) (M, S 1 ), respectively, and the identity maps on the morphisms. It is clear that S and S are strict antimonoidal equivalence functors. The ring automorphisms µ and ν define endo-functors µ and ν of R M R. The object maps are (M,, ) (M, µ, µ) and (M,, ) (M, ν, ν), respectively, and the identity map on the morphisms. The µ and ν are also strict antimonoidal equivalence functors. We have then equalities of strong monoidal functors Φ L S = ν Φ R and Φ L S = µ Φ R. Finally, we define the morphisms of Hopf algebroids. Definition 4.4. A Hopf algebroid homomorphism (isomorphism) (A L,S) (A L,S ) is a left bialgebroid homomorphism (isomorphism) A L A L. A Hopf algebroid homomorphism (Φ, φ) is strict if S Φ = Φ S.

91 726 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) The existence of non-strict isomorphisms of Hopf algebroids that is the nonuniqueness of the antipode in a Hopf algebroid is a new feature compared to (weak) Hopf algebras. The antipodes making a given left bialgebroid into a Hopf algebroid are characterized in [2]. In the following (in particular in Section 5) we are going to call a triple (A L, A R,S) satisfying Proposition 4.2(iii) a symmetrized form of the Hopf algebroid (A L,S).The A R is called the right bialgebroid underlying (A L,S). In the view of Proposition 4.3 the symmetrized form is unique up to the choice of the base ring R of A R within the isomorphism class of L op the opposite of the base ring of A L and the isomorphism µ : L op R in (4.15). Let us define the homomorphisms of symmetrized Hopf algebroids A = (A L, A R,S) A = (A L, A R,S ) as pairs of bialgebroid homomorphisms (Φ L,φ L ) : A L A L, (Φ R,φ R ) : A R A R. Then by Proposition 4.3 the Hopf algebroid homomorphisms (Φ, φ) : (A L,S) (A L,S ) are injected into the homomorphisms of symmetrized Hopf algebroids (A L, A R,S) (A L, A R,S ) via (Φ, φ) ((Φ, φ), (S 1 Φ S,µ φ µ 1 )) where µ = π R t L and µ = π R t L are the ring isomorphisms introduced in Proposition 4.3. This implies that two symmetrized Hopf algebroids (A L, A R,S) and (A L, A R,S ) are isomorphic if and only if the Hopf algebroids (A L,S) and (A L,S ) are isomorphic. A homomorphism ((Φ L,φ L ), (Φ R,φ R )) of symmetrized Hopf algebroids A A is strict if Φ L = Φ R as homomorphisms of rings A A. We leave it to the reader to check that this is equivalent to the requirement that (Φ L,φ L ) is a strict homomorphism of Hopf algebroids (A L,S) (A L,S ), that is two symmetrized Hopf algebroids (A L, A R,S) and (A L, A R,S ) are strictly isomorphic if and only if the Hopf algebroids (A L,S) and (A L,S ) are strictly isomorphic. The usage of symmetrized Hopf algebroids allows for the definition of the opposite and co-opposite structures A op = (A op R, Aop L,S 1 ) and A cop = (A L cop, A R cop,s 1 ), respectively Relation to the Hopf bialgebroid of Day and Street In [22] left bialgebroids over the base L have been characterized as opmonoidal monads T on the category L M L such that their underlying functors T have right adjoints. The endofunctor T is given by tensoring over L e = L L op with the L e -L e -bimodule A. The monad structure makes A into an L e -ring via an algebra map η : L e A, while the opmonoidal structure comprises the coproduct and counit of the bialgebroid A. In a recent preprint [7] Day and Street put this into the more general context of pseudomonoids in monoidal bicategories. Using the notion of -autonomy [1] they propose a definition of Hopf bialgebroid as a -autonomous structure on a bialgebroid. We are going to show in this section that their definition coincides with our Definition 4.1. Working with k-algebras the base category V is the category M k of k-modules. Then the monoidal bicategory in question is Mod(V) having as objects the k-algebras and as hom-categories Mod(V)(A, B) the category B M A of B-A-bimodules. The tensor product = k of k-modules makes Mod(V) monoidal. A pseudomonoid in Mod(V) is a triple A,M,J where A is an object, M : A A A and J : k A are 1-cells satisfying

92 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) associativity and unitality up to invertible 2-cells that in turn satisfy the pentagon and the triangle constraints. If A,L,s,t,γ,π is a left bialgebroid in V then we have a strong monoidal morphism of pseudomonoids where η : A,M,J L e,m,j (4.22) M = A L A with actions a (x L y) (a 1 a 2 ) = a (1) xa 1 L a (2) ya 2, (4.23) J = L with action a x = π ( as(x) ), (4.24) m = L e L L e with actions (l l ) ((x x ) L (y y ) ) (( l 1 l 1) ( l2 l 2 )) = ( lxl 1 l 1 x ) ( L yl2 l 2 y l ), (4.25) j = L with action (l l ) x = lxl (4.26) and the bimodule η = L ea A, induced by the algebra map η, has a left adjoint η = A A L e with the counit ɛ : η η A induced by multiplication of A. The connection of this bimodule picture with module categories can be seen by applying the monoidal pseudofunctor Mod(V)(k, ) : Mod(V) Cat. Then pseudomonoids become monoidal categories, η becomes the strong monoidal forgetful functor η A : A M L M L, and η its opmonoidal left adjoint. An important piece of the structure is the bimodule morphism ψ : η m M (η η ), a (l l ) a (1) s(l) a (2) t(l ), (4.27) which makes η opmonoidal and encodes the comultiplication γ : A A L A of the bialgebroid. In [7] a Hopf bialgebroid is defined to be a bialgebroid A over L together with a strong -autonomous structure on the opmonoidal morphism η : L e A. The latter means: (1) A -autonomous structure on the pseudomonoid A,i.e., (a) a right A A-module σ defined on the k-module A in terms of an algebra isomorphism ξ : A A op by (b) an isomorphism x (a b) = ξ 1 (b)xa, x, a, b A, and (4.28) Γ : σ (M A) σ (A M). (2) A canonical -autonomous structure on the pseudomonoid L e which consists of the following:

93 728 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) (a) A right L e L e -module σ 0 defined on L e using the isomorphism ξ 0 : L e (L e ) op, l l θ 1 (l ) l for some algebra automorphism θ : L L. Thus (x x ) (( l 1 l 1) ( l2 l 2)) = θ 1 ( l 2) xl1 l 1 x l 2 for x,x,l 1,l 1,l 2,l 2 L where juxtaposition is always multiplication in L and never in L op. (Note that allowing a non-trivial θ in the definition of σ 0 is a slight deviation from [7] which is, however, well motivated by Section 3.) (b) The isomorphism Γ 0 : σ 0 ( m L e) σ 0 ( L e m ). (3) The arrow η is strongly -autonomous in the following sense: There exists a 2-cell τ : σ 0 (η η ) σ such that [ Γ0 (η η η ) ] [ σ (ψ η ) ] [ τ ( m L e)] and such that the map = [ σ (η ψ) ] [ τ ( L e m )] Γ 0 (4.29) τ l : σ 0 ( η L e) σ (A η ), τ l = [ σ (ɛ η ) ] [ τ ( η L e)] (4.30) is an isomorphism. (We used and to denote horizontal and vertical compositions in the bicategory Mod(V).) Now we are going to translate this categorical definition into simple algebraic expressions. Lemma autonomous structures on the pseudomonoid A,M,J are in one-to-one correspondence with the data ξ, k e k f k where ξ is an anti-automorphism of the ring A and k e k f k A L A is such that for all a A ξ(a (1) ) (1 )e k a (2) ξ(a (1) ) (2 )f k = e k f k ξ(a) (4.31) k k and such that there exists j g j h j A L A satisfying, for all a A: ξ 1 (a (2) ) (1 )g j ξ 1 (a (2) ) (2 )h j a (1) = g j ξ 1 (a) h j, (4.32) j j ξ(g j ) (1) e k h j ξ(g j ) (2) f k = 1 A 1 A, (4.33) j,k ξ 1 (f k ) (1) g j ξ 1 (f k ) (2) h j e k = 1 A 1 A (4.34) j,k as elements of A L A.

94 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Proof. In order to find explicit formulas for Γ and its inverse, we introduce the isomorphisms with inverses ϕ + : σ (M A) A L A, (4.35) x A A (a L b c) ( ξ 1 (c)x ) (1) a ( L ξ 1 (c)x ) (2)b, (4.36) ϕ : σ (A M) A L A, (4.37) x A A (a b L c) ξ(xa) (1) b L ξ(xa) (2) c, ϕ 1 + (a L b) = 1 A A (a L b 1), ϕ 1 (b L c) = 1 A A (1 b L c). Then Γ := ϕ Γ ϕ 1 + is a twisted A 3 -automorphism of A L A in the sense of Γ ( ξ 1 (c) (x L y) (a b) ) = ξ(a) Γ (x L y) (b c). (4.38) Hence, Γ is uniquely determined by i e i f i = Γ (1 1) as Γ ( x A A (a L b c) ) = 1 A A ( x (1) a k ) e k x (2) b L f k c (4.39) and k e k f k satisfies (4.31). Invertibility then implies that Γ 1 (1 1) = j g j h j satisfies the remaining equations. We need also the expression for Γ 0. We leave it to the reader to check that ( Γ 0 (x x ( ) L e L e l1 l 1) ( L l2 l 2) ( l3 l 3 )) ( = (1 1) L e L e xl1 l 1) ( l2 l 2 x ) ( L l3 l 3 ) (4.40) is well-defined and is invertible. Lemma 4.6. The -autonomous property of η is equivalent to the existence of an element i A satisfying iη ( θ 1 (l ) l ) = ξ 1( η(l l ) ) i, l,l L, (4.41) e k f k ξ(i) = ξ(i) (1) ξ(i) (2), (4.42) k while strong -autonomy adds the requirement that i be invertible.

95 730 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Proof. Since 1 1 is a cyclic vector in the L e L e -module σ 0, the module map τ is uniquely determined by i := τ(1 1) as τ(x x ) = iη(x x ). (4.43) Tensoring with η = A A L e from the right being the restriction via η : L e A the element i is subject to condition (4.41) due to (4.28). Using the expressions (4.43), (4.39), (4.40), and (4.27) the -autonomy condition (4.29) becomes equation (4.42). The mate of τ in (4.30) can now be written as τ l( (x x ( ) L e L e a (l l ) )) = iη(x x ( )a A L e 1 (l l ) ). Hence τ l is invertible iff the map a ia is, i.e., iff i is invertible. Theorem 4.7. A strong -autonomous structure on the bialgebroid A over L in the sense of [7] is equivalent to a Hopf algebroid structure (A L,S) in the sense of Definition 4.1. Proof. Using invertibility of i A condition (4.41) has the equivalent form ξ ( iη(l l)i 1) = η ( l θ(l ) ) (4.44) and (4.42) can be used to express the element k e k f k in terms of i, e i f i = ξ(i) (1) ξ(i) (2) ξ(i) 1. The conditions (4.31) (4.34) are then equivalent to g j h j = i (1) i 1 i (2), (4.45) [ i 1 ξ 1 (a (2) )i ] (1 ) [ i 1 ξ 1 (a (2) )i ] (2 ) a (1) = i 1 ξ 1 (a)i 1 A, (4.46) ξ ( ia (1) i 1) (1 ) a (2) ξ ( ia (1) i 1) (2 ) = 1 A ξ ( iai 1). (4.47) This means that introducing S : A A op, a ξ(iai 1 ), the conditions (4.44), (4.46), and (4.47) are identical to the axioms (4.1) (4.3), respectively Examples In addition to our motivating example in Section 3, let us collect some more examples of Hopf algebroids. Example 4.8. Weak Hopf algebras with bijective antipode.let(h,,ε,s)be a weak Hopf algebra [3,4,14] (WHA) over the commutative ring k with bijective antipode. This means that H is an associative unital k algebra, : H H k H is a coassociative coproduct.

96 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) It is an algebra map (i.e., multiplicative) but not unit preserving in general. In its stead we have weak comultiplicativity of the unit 1 [1] 1 [2] 1 [1 ] 1 [2 ] = 1 [1] 1 [2] 1 [3] = 1 [1] 1 [1 ]1 [2] 1 [2 ], where 1 [1] 1 [2] = (1).Themapε : H k is the counit of the coproduct. Instead of being multiplicative, it is weakly multiplicative: ε ( ab [1] ) ε ( b[2] c ) = ε(abc) = ε ( ab [2] ) ε ( b[1] c ) for a,b,c H. The bijective map S : H H is the antipode, subject to the axioms h [1] S ( h [2] ) = ε ( 1[1] h ) 1 [2], S ( h [1] ) h[2] = 1 [1] ε ( h1 [2] ), S ( h[1] ) h[2] S ( h [3] ) = S(h) for all h H. (If H is finite over k then the assumption made about the bijectivity of S is redundant.) The WHA (H,,ε,S) is a Hopf algebra if and only if is unit preserving. The algebra H contains two commuting subalgebras: R is the image of H under the projection R : h 1 [1] ε(h1 [2] ) and L under L : h ε(1 [1] h)1 [2] generalizing the subalgebra of the scalars in a Hopf algebra. Both maps S and S 1 restrict to algebra antiisomorphisms R L. We have four commuting actions of L and R on H : H R : h r := hr, R H : r h := hs 1 (r), H L : h l := S 1 (l)h, LH : l h := lh. Introduce the canonical projections p R : H k H H R R H and p L : H k H H L L H. There exists a left and a right bialgebroid structure corresponding to the weak Hopf algebra: H R := ( H,R,id R,S 1 R, p R, R), H L := ( H,L,id L,S 1 L, p L, L). We leave it as an exercise to the reader to check that (H L, H R,S)satisfies the requirements of Proposition 4.2(iii). Notice that the examples of the above class are not necessarily finite dimensional and not even finitely generated over R. To have a trivial counterexample, think of the group Hopf algebra kg of an infinite group. Example 4.9. An example that does not satisfy the Lu-axioms [13]. Let k be a field the characteristic of which is different from 2. Consider the group bialgebra kz 2 with presentation kz 2 = bialg- t t 2 = 1, (t)= t t, ε(t) = 1

97 732 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) as a left bialgebroid over the base k.thatistosay,weseta L = (kz 2,k,η,η,,ε)where η is the unit map k kz 2, λ λ1. Introduce the would-be-antipode S : kz 2 kz 2, t t. Proposition The pair (A L,S) in Example 4.9 satisfies the axioms in Definition 4.1 but not the Lu-axioms. Proof. One easily checks the conditions of Proposition 4.2(ii) on the single algebraic generator t, proving that (A L,S) is a Hopf algebroid in the sense of Definition 4.1. Now,thebaseringLbeing k itself, the canonical projection kz 2 k kz 2 kz 2 L kz 2 is the identity map leaving us with the only section kz 2 L kz 2 kz 2 k kz 2, the identity map. Since t (1) S(t (2) ) = 1andη ε(t) = 1, this contradicts to that (A L,S)is a Lu Hopf algebroid. In the Example 4.8 the left and right coproducts γ L and γ R are compositions of a coproduct : A A k A with the canonical projections p L and p R, respectively. Actually, many other examples can be found this way by allowing for not to be counital. Proposition Let A L be a left bialgebroid such that A and L are k-algebras over some commutative ring k, S an anti-automorphism of A such that the axioms (4.1) and (4.4) hold true. Suppose that γ L = p L,wherep L is the canonical projection A k A A L L A and : A A k A is a coassociative ( possibly non-counital) coproduct satisfying p L (S S) op = p L S, p L ( S 1 S 1) op = p L S 1. (4.48) Then (A L,S) is a Hopf algebroid in the sense of Definition 4.1. Proof. We leave it to the reader to check that all the requirements of Proposition 4.2(ii) are satisfied. Example The groupoid Hopf algebroid. Let G be a groupoid that is a small category with all morphisms invertible. Denote the object set by G 0 and the set of morphisms by G 1. For a commutative ring k the groupoid algebra is the k-module spanned by the elements of G 1 with the multiplication given by the composition of the morphisms if the latter makes sense and 0 otherwise. It is an associative algebra and if G 0 is finite it has a unit 1 = a G0 a. The groupoid algebra admits a left bialgebroid structure over the base subalgebra kg 0.Themaps L = t L is the canonical embedding, γ L is the diagonal map g g L g,andπ L (g) := target(g). This left bialgebroid together with the antipode S(g) := g 1 is a Hopf algebroid in the sense of Definition 4.1. Actually this example is of the kind described in Proposition 4.11 with (g) := g k g. Example The algebraic quantum torus. Let k be a field and T q the unital associative k algebra generated by two invertible elements U and V subject to the relation UV = qvu where q is an invertible element in k. As it is explained in [11], the algebra T q admits a

98 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Lu Hopf algebroid structure over the base subalgebra L generated by U: themaps L = t L is the canonical embedding, γ L ( U n V m) := U n V m L V m V m L U n V m, π L ( U n V m) := U n, and the antipode S(U n V m ) := V m U n. The section ξ of the canonical projection p L : T q k T q T q L T q appearing in the Lu axioms is of the form ξ ( U n V m L U k V l) := U (n+k) V m k V l. The reader may check that these maps satisfy the Definition 4.1 as well. This example is also of the type considered in Proposition 4.11 with (U n V m ) := U n V m k V m. Example Examples by Brzezinski and Militaru [5]. In the paper [5] a wide class of examples of Lu Hopf algebroids is described. Some other examples [13,15] turn out to belong also to this class. The examples of [5] are Lu Hopf algebroids of the type considered in Proposition Let (H, H,ε H,τ) be a Hopf algebra over the field k with bijective antipode τ and the triple (L,,ρ) a braided commutative algebra in the category H D H of Yetter Drinfel d modules over H. Then the crossed product algebra L#H carries a left bialgebroid structure over the base algebra L: s L (l) = l #1 H, t L (l) = ρ(l) l 0 # l 1, γ L (l # h) = (l # h (1) ) L (1 L # h (2) ), π L (l # h) = ε H (h)l, (4.49) where h (1) k h (2) H (h). It is proven in [5] that the left bialgebroid (4.49) and the bijective antipode S(l # h) := ( τ(h (2) )τ 2 (l 1 ) ) l 0 # τ(h (1) )τ 2 (l 2 ) (4.50) form a Lu Hopf algebroid. It is obvious that γ L is of the form p L with (l # h) := (l # h (1) ) k (1 L # h (2) ).Themap is well-defined since L # H is L k H as a k-space and H maps H into H k H. We leave it to the reader to check that satisfies (4.48) hence the left bialgebroid (4.49) and the antipode (4.50) form a Hopf algebroid in the sense of Definition 4.1. The Example 4.9 is not of the type considered in Proposition Although γ L is of the form p L,the does not satisfy (4.48). In [11] data (A L,S, S) satisfying compatibility conditions somewhat analogous to (4.48) were introduced under the name extended Hopf algebra. The next proposition states that extended Hopf algebras with S bijective (such as Example 4.9) provide examples of Hopf algebroids. Proposition Let (A L,S, S) be an extended Hopf algebra. This means that A L is a left bialgebroid such that A and L are k-algebras over some commutative ring k. The maps S and S are anti-automorphisms of the algebra A, S 2 = id A and both pairs (A L,S) and

99 734 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) (A L, S) satisfy (4.1) and (4.4).Themapγ L is a composition of a coassociative coproduct : A A k A and the canonical projection p L : A k A A L A. The compatibility relations S = (S S) op and S = ( S S ) op hold true. Then the pair (A L, S) is a Hopf algebroid in the sense of Definition 4.1. Proof. We leave to the reader to check that the condition (4.9) hence all requirements of Proposition 4.2(ii) hold true. 5. Integral theory and the dual Hopf algebroid In this section we generalize the notion of non-degenerate integrals in (weak) bialgebras to bialgebroids. We examine the consequences of the existence of a non-degenerate integral in a Hopf algebroid. We do not address the question, however, under what conditions on the Hopf algebroid does the existence of a non-degenerate integral follow. That is we do not give a generalization of the Larson Sweedler theorem on bialgebroids and neither of the (weaker) [3, Theorem 3.16] stating that a weak Hopf algebra possesses a non-degenerate integral if and only if it is a Frobenius algebra. (About the implications in one direction see however [2, Theorem 6.3] and Theorem 5.5 below, respectively.) Assuming the existence of a non-degenerate integral in a Hopf algebroid we show that the underlying bialgebroids are finite. The duals of finite bialgebroids w.r.t. the base rings were shown to have bialgebroid structures [8] but there is no obvious way how to transpose the antipode to (either of the four) duals. As the main result of this section we show that if there exists a non-degenerate integral in a Hopf algebroid then the four dual bialgebroids are all (anti-) isomorphic and they can be made Hopf algebroids. This dual Hopf algebroid structure is unique up to isomorphism (in the sense of Definition 4.4). For the considerations of this section the symmetric definition of Hopf algebroids, i.e., the characterization of Proposition 4.2(iii) is the most appropriate. Throughout the section we use the symmetrized form of the Hopf algebroid introduced at the end of Section 4.1. It is important to emphasize that although the Definitions 5.1 and 5.3 are formulated in terms of a particular symmetrized Hopf algebroid A = (A L, A R,S), actually they depend only on the Hopf algebroid (A L,S).Thatistosay,ifl is a (non-degenerate) left integral in a symmetrized Hopf algebroid then it is one in any other symmetrized form of the same Hopf algebroid. Therefore, l can be called a (non-degenerate) left integral of the Hopf algebroid. Analogously, although the anti-automorphism ξ in Lemma 5.9 is defined in terms of a particular symmetrized Hopf algebroid, it is invariant under the change of the underlying right bialgebroid. For a symmetrized Hopf algebroid A = (A L, A R,S)we use the notations of Section 2: the A and A are the L-duals of A L,theA and A the R-duals of A R.Alsofor the coproducts of A L and A R we write γ L (a) = a (1) a (2) and γ R (a) = a (1) a (2), respectively.

100 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Non-degenerate integrals Definition 5.1. The left integrals in a left bialgebroid A L = (A,L,s L,t L,γ L,π L ) are the invariants of the left regular A module I L (A) := { l A al = sl π L (a)l a A }. The right integrals in a right bialgebroid A R = (A,R,s R,t R,γ R,π R ) are the invariants of the right regular A module I R (A) := { Υ A Υa= ΥsR π R (a) a A }. The left/right integrals in a symmetrized Hopf algebroid (A L, A R,S) are the left/right integrals in A L /A R. Lemma 5.2. For a symmetrized Hopf algebroid A the following properties of the element l of A are equivalent: (i) l I L (A); (ii) al = t L π L (a)l for all a A; (iii) S(l) I R (A); (iv) S 1 (l) I R (A); (v) S(a)l (1) l (2) = l (1) al (2) as elements of A R R A, for all a A. Proof. Left to the reader. Definition 5.3. The left integral l in the symmetrized Hopf algebroid A is non-degenerate if the maps l R : A A, φ φ l, and Rl : A A, φ φ l (5.1) are bijective. The right integral Υ in the symmetrized Hopf algebroid A is non-degenerate if S(Υ ) is a non-degenerate left integral, i.e., if the maps are bijective. Υ L : A A, φ Υ φ, and LΥ : A A, φ Υ φ (5.2) Remark 5.4. If l is a non-degenerate left integral in the symmetrized Hopf algebroid A then so is in A cop and when replacing A with A cop the roles of l R and R l become interchanged. Hence, any statement proven in a symmetrized Hopf algebroid possessing a non-degenerate left integral on l R implies that the co-opposite statement holds true on R l.

101 736 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Theorem 5.5. Let A be a symmetrized Hopf algebroid possessing a non-degenerate left integral. Then the ring extensions s R : R A, t R : R op A, s L : L A, and t L : L op A are all Frobenius extensions. Proof. Let l be a non-degenerate left integral in A. With its help we construct the Frobenius system for the extension s R : R A. It consists of a Frobenius map and a quasi-basis (in the sense of (3.1)) for it λ := l 1 R (1 A) : R A R R (5.3) l (1) S ( l (2)) A R R A. As a matter of fact the λ is a right R-module map A R R by construction. We claim that it is also a left R-module map R A R. Since ( λ S(a) ) l= l (2) t R λ ( S(a)l (1)) = a(λ l)= a, (5.4) the inverse l 1 R maps a A to λ S(a).Thisimpliesthatλ s R (r) = l 1 R t R(r). Now for a given element r R the map χ(r) : A R R, a rλ (a) is also equal to t R(r) as l 1 R χ(r) l= l (2) t R ( rλ ( l (1))) = (λ l)t R (r) = t R (r). Applying the two equal maps λ s R (r) and χ(r) to an element a A, we obtain λ ( s R (r)a ) = rλ (a). (5.5) This proves that λ is an R-R-bimodule map R A R A.Also s R λ ( al (1)) S ( l (2)) = S(λ l)a= a and (5.6) l (1) s R λ ( S ( l (2)) a ) = al (1) s R λ S ( l (2)). (5.7) Now we claim that l (1) s R λ S(l (2) ) λ S lis equal to 1 A, which proves the claim. (Recall that by (5.5) λ S is an element of A.) Since l 1 R (a) = λ S(a), φ (a) = [ λ S(φ l) ] (a) = λ ( s R φ ( l (1)) S ( l (2)) a ) (5.8) for all φ A and a A. By the bijectivity of R l, we can introduce the element λ := R l 1 (1 A ) A. Analogously to (5.5) and (5.4), we have λ ( t R (r)a ) = λ(a)r and (5.9) Rl 1 (a) = λ S 1 (a). (5.10)

102 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Since both R l and S 1 are bijective, so is the map A A, a λ a.usingthe identities (5.9), (5.5), and (5.8), compute ( λ S 1 (λ S l) ) (a) = λ ( t R λ S ( l (2)) S 1( l (1)) a ) = λ ( s R λ S 1( l (1)) S ( l (2)) S(a) ) = λ(a) for all a A. This is equivalent to λ S l= 1 A that is λ = λ S proving that (λ,l (1) S(l (2) )) is a Frobenius system for the extension s R : R A. By repeating the same proof in A cop, we obtain the Frobenius system ( λ,l (2) S 1 (l (1) )) for the extension t R : R A. It is straightforward to check that (µ 1 λ,l (1) S(l (2) )) is a Frobenius system for the extension t L : L A, and(ν 1 λ,l (2) S 1 (l (1) )) is a Frobenius system for the extension s L : L A. From now on, let l be a non-degenerate left integral in the symmetrized Hopf algebroid A, setλ = l 1 R (1 A) and λ = R l 1 (1 A ). Theorem 5.5 implies that for a symmetrized Hopf algebroid A possessing a nondegenerate integral the modules A R, R A, A L,and L A are finitely generated projective. Hence, by the result of [8], their duals A and A carry left bialgebroid structures over the base R, anda and A carry right bialgebroid structures over the base L s L (r)(a) = rπ R(a), tl (r)(a) = π R( sr (r)a ), γl (φ ) = φ l (1) l 1 ( R l (2) ), πl (φ ) = φ (1 A ), ( s L (r)(a) = π R tr (r)a ), t L (r)(a) = π R (a)r, γ L ( φ) = R l 1( l (1)) φ l (2), π L ( φ) = φ(1 A ), ( s R (l)(a) = π L asl (l) ), s R (l)(a) = π L (a)l, t R (l)(a) = lπ L (a), ( t R (l)(a) = π L atl (l) ), γ R (φ ) = l (1) φ l 1 L (l (2)), γ R ( φ) = L l 1 (l (1) ) l (2) φ, π R (φ ) = φ (1 A ), π R ( φ) = φ(1 A ). (5.11) Lemma 5.6. Let l be a non-degenerate left integral in the symmetrized Hopf algebroid A. Then for λ = l 1 R (1 A), λ = R l 1 (1 A ), and any element a A, the identities λ a= s R λ (a), (5.12) λ a= t R λ(a) (5.13) hold true.

103 738 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Proof. One checks that φ λ = l 1 R (φ 1 A ) = s L φ (1 A )λ for all φ A.Thisimpliesthatφ (λ a)= φ (s R λ (a)) for all φ A.Since A R is finitely generated projective by Theorem 5.5, this proves (5.12). The identity (5.13) follows by Remark 5.4. The left integrals in a Hopf algebroid were defined in Definition 5.1 as the left integrals in the underlying left bialgebroid. The non-degeneracy of the left integral was defined in Definition 5.3 using however the underlying right bialgebroid as well, that is it relies to the whole of the Hopf algebroid structure. Therefore, it is not obvious whether the nonstrict isomorphisms of Hopf algebroids preserve non-degenerate integrals. In the rest of this subsection, we prove that this is the case: Proposition 5.7. Let both (A L,S) and (A L,S ) be Hopf algebroids. Then their nondegenerate left integrals coincide. Proof. Aleftintegrall in (A L,S) is a left integral in (A L,S ) by definition. Let A R = (A,R,s R,t R,γ R,π R ) and A R = (A, R,s R,t R,γ R,π R ) be the right bialgebroids underlying the Hopf algebroids (A L,S) and (A L,S ), respectively. It follows from the uniqueness of the maps α 1 and β 1 in (4.16) that the coproducts γ R (a) = a (1) a (2) of A R and γ R (a) = a{1} a {2} of A R are related as a (1) S 1 S ( a (2)) = a {1} a {2} = S S 1( a (1)) a (2). (5.14) With the help of the maps µ = π R t L, µ = π R t L, ν = π R s L,andν = π R s L,we can introduce the isomorphisms of additive groups: A A, φ µ µ 1 φ, A A, φ ν ν 1 φ. Then the canonical actions (2.13) of A and A and of A and A on A are related as µ µ 1 φ a= S 1 S(φ a), ν ν 1 φ a= S S 1 ( φ a) what implies the non-degeneracy of the left integral l in (A L,S ) provided it is nondegenerate in (A L,S).

104 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Two-sided non-degenerate integrals Proposition 5.7 above proves that the structure of the non-degenerate left integrals is the same within an isomorphism class of Hopf algebroids. In this subsection we prove that for a non-degenerate left integral l in the Hopf algebroid (A L,S) there exists a distinguished representative (A L,S l ) in the isomorphism class of (A L,S)with the property that l is not only a non-degenerate left integral in (A L,S l ) but also a non-degenerate right integral. The Hopf algebroids with two-sided non-degenerate integral are of particular interest. Both the Hopf algebroid structure constructed on the dual of a Hopf algebroid in Section 5.3 and the one associated to a depth 2 Frobenius extension in Section 3 belong to this class. Lemma 5.8. Let l be a non-degenerate left integral in a symmetrized Hopf algebroid A. Set λ := l 1 R (1 A) and λ := R l 1 (1 A ). Then any (not necessarily non-degenerate) left integral l I L (A) satisfies ls R λ (l ) = l = lt R λ(l ). Proof. Observe that for φ A and l I L (A) we have φ S 1 (l ) = t L φ S 1 (l ); hence l = ( λ S 1 (l ) ) l= t L λ S 1 (l ) l= ls R λ (l ). The identity l = lt R λ(l ) follows by Remark 5.4. Lemma 5.9. Let l be a non-degenerate left integral in a symmetrized Hopf algebroid A. Setλ := l 1 R (1 A). Then the map ξ : A A; a S((λ l) a)is a ring antiautomorphism. Proof. Using Lemma 5.8 one checks that [ (λ l) a ][ (λ l) b ] = a (2) t R λ ( la (1)) b (2) t R λ ( lb (1)) = a (2) b (2) t R λ ( ls R λ ( la (1)) b (1)) = a (2) b (2) t R λ ( la (1) b (1)) = (λ l) ab for a,b A, hence the map ξ is anti-multiplicative. By analogous calculations the reader may check that it is bijective with inverse ξ 1 (a) = S 1 (( λ l) a),where λ := Rl 1 (1 A ) = λ S. Proposition Let l be a non-degenerate left integral in a symmetrized Hopf algebroid A. Then the following maps are bijective:

105 740 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) l L : A A, φ l φ, Ll : A A, φ l φ. Proof. We claim that with the help of the ring isomorphism ν (introduced in Proposition 4.3) we have l φ = ξ 1 l R (ν φ S 1 ), which implies the bijectivity of L l. As a matter of fact, [ l ν 1 l 1 R (a) S] = t L ν 1 ( λ S(a) ) S(l (2) )l (1) = S 1 t R λ(l (2) a)l (1) = S 1( λ l (2) a ) l (1) = S 1( S(l (1) )l (1) (2) a (1) s R λ ( l (2) (2) a (2))) = S 1( ( s R π R l (1) ) a (1) s R λ ( l (2) a (2))) = S 1( a (1) s R λ ( la (2))) = S 1( ( λ l) a ) = ξ 1 (a). (5.15) Similarly, by the application of (5.15) to A cop, l φ = ξ R l(µ φ S), (5.16) hence l L is also bijective. Using (5.16) we have an equivalent form of the anti-automorphism ξ introduced in Lemma 5.9: ξ(a) = l ( a l 1 L (1 A) ). (5.17) Lemma Let l be a non-degenerate left integral in a symmetrized Hopf algebroid A. Then for all elements a,b A we have the identities l 1 R (b) a = Rl 1 (a) b, (5.18) a l 1 L (b) = b Ll 1 (a), (5.19) l 1 R (b) a = a l 1 L (b), (5.20) Rl 1 (b) a = a L l 1 (b). (5.21) Proof. We illustrate the proof on (5.18). Use Lemma 5.6 to see that Rl 1 (a) b = b (1)( λ S ( b (2)) a ) = s L π L (b (1) )a (2) t R λ ( S(b (2) )a (1)) = l 1 R (b) a, where λ = l 1 R (1 A). The rest of the proof is analogous.

106 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Lemma Let l be a non-degenerate left integral in a symmetrized Hopf algebroid A. Set λ = l 1 R (1 A). Then the map κ : R R, r λ (lt R (r)) is a ring automorphism. Proof. It follows from Lemma 5.8 that κ is multiplicative: for r, r R κ(r)κ(r ) = λ ( lt R (r) ) λ ( lt R (r ) ) = λ ( ls R λ ( lt R (r ) ) t R (r) ) = λ ( lt R (r )t R (r) ) = κ(rr ). In order to show that κ is bijective, we construct the inverse κ 1 : r λ(ls R (r)) where λ = R l 1 (1 A ) = λ S. Proposition Let l be a non-degenerate left integral in the Hopf algebroid (A L,S). Then there exists a unique Hopf algebroid (A L,S l ) such that l is a two-sided nondegenerate integral in (A L,S l ). Proof. Uniqueness. Suppose that (A L,S l ) is a Hopf algebroid of the required kind. Denote the underlying right bialgebroid by A R = (A, R,s R,t R,γ R,π R ).Defineλ A with the property that λ l= 1 A (where denotes the canonical action (2.13) of A on A). Introducing the notation γ R (a) = a{1} a {2}, one checks that S 1 (l) = ( λ l ) l= l {2} t R λ ( ll {1}) = l {2} t R λ ( ls R π R ( l {1} )) = lt R λ (l) = l. With the help of the element π L S 1 S A,wehave S ( a π L S 1 S ) = S S 1( S (a) {1}) s R π R ( S (a) {2}) = S (a) for all a A, where in the last step the relation (5.14) has been used. Then the condition S (l) = l is equivalent to S (a) = S ( a l 1 L S 1 (l) ). This proves the uniqueness of S. Existence. Let ξ be the anti-automorphism of A introduced in Lemma 5.9. We claim that (A L,ξ) is a Hopf algebroid of the required kind. Introduce the right bialgebroid A R on the total ring A over the base R with structural maps s R = s R, t R = ξ 1 s R, γ R = ξ A 1 L A S A R A γ R S 1 ξ, π R = π R S 1 ξ, where A R = (A,R,s R,t R,γ R,π R ) is the right bialgebroid underlying (A L,S).First,we check that the triple (A L, A R,ξ) satisfies Proposition 4.2(iii). Since ξ 1 S s R = ξ 1 t R θ R = S 1 t R θ R = s R and ξ 1 S t R = ξ 1 s R,

107 742 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) the A R is a right bialgebroid isomorphic to A R via the isomorphism (ξ 1 S,id R ). The requirement s R (R) s R(R) = t L (L) obviously holds true. Since t R (r) = ξ 1 s R (r) = S 1 s R λ ( ls R (r) ) = t R κ 1 (r), also t R (R) t R(R) = s L (L).Since γ R (a) a{1} a {2} = ξa 1 L A γ L ξ(a) = a (1) ξ 1 S ( a (2)) = a (1) S 1( Rl 1 S(l) S ( a (2))) = a (1) S 1( S ( a (2)) L l 1 S(l) ) = a (1) s R κ µ L l 1 S(l) S ( a (2)) a (3), we have (γ L id A ) γ R (a) = a(1) (1) a (1) (2) ξ 1 S ( a (2)) = a (1) a (1) (2) ξ 1 S ( (2) a ) (2) = ( id A γ R) γl (a), (id A γ L ) γ R (a) = a(1) s R κ µ L l 1 S(l) S ( a (2)) a (3) (1) a (3) (2) = a (1) (1) s R κ µ L l 1 S(l) S ( (2) a ) (1) a (3) (1) a (2) = ( γ R id A) γl (a). By Lemma 5.9, the ξ is an anti-automorphism of the ring A. The identity ξ t R = s R is obvious and also ξ t L = ξ s R µ = S s R µ = s L. Finally, ξ(a (1) )a (2) = s R π R ( (λ l) a ) = s R π R S 1 ξ(a) = s R π R (a), a {1} ξ ( a {2}) = a (1) ξ ξ 1 S ( a (2)) = s L π L (a). This proves that A l = (A L, A R,ξ) satisfies the conditions in Proposition 4.2(iii); hence (A L,ξ) is a Hopf algebroid. Since ξ(l) = S ( (λ l) l ) = S S 1 (l) = l, the l is a two-sided non-degenerate integral in A l.

108 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Duality It follows from Theorem 5.5 that for a symmetrized Hopf algebroid A possessing a nondegenerate left integral l the dual rings (with respect to the base ring) carry bialgebroid structures. These bialgebroids (5.11) are independent of the particular choice of the nondegenerate integral. In this subsection we analyze these bialgebroids. We show that the four bialgebroids (5.11) are all (anti-) isomorphic and can be equipped with an l-dependent Hopf algebroid structure. Because of the l-dependence of this Hopf algebroid structure the duality of Hopf algebroids is sensibly defined on the isomorphism classes of Hopf algebroids. Lemma Let l be a non-degenerate left integral in the symmetrized Hopf algebroid A. Then with the help of the anti-automorphism ξ of Lemma 5.9 we have the equalities in A L L A. ξ 1( l (2)) S 1( l (1)) = l (1) l (2) = S ( l (2)) ξ ( l (1)) (5.22) Proof. The element ξ 1 (l (2) ) S 1 (l (1) ) is in A L L A since S 1 s R = t R = s L ν 1 and ξ 1 t R = S 1 t R = t L ν 1. Using (5.15), in A L L A we have ξ 1( l (2)) S 1( l (1)) = l (1) t R λ ( l (2) l (2)) S 1( l (1)) = l (1) l (2). The other equality follows by repeating the proof in A cop. Corollary For a non-degenerate left integral l in the symmetrized Hopf algebroid A the maps l L and L l satisfy the identities l L (a φ ) = l L (φ )ξ(a), (5.23) Ll(a φ) = L l( φ)ξ 1 (a), (5.24) where ξ is the anti-automorphism of A introduced in Lemma 5.9. Theorem Let l be a non-degenerate left integral in the symmetrized Hopf algebroid A = (A L, A R,S). Then the left bialgebroids A L, A L, (A R ) op cop, and ( A R ) op cop in (5.11) are isomorphic via the isomorphisms ( l 1 R l L,ν ) (A R ) op cop A L (Ll 1 ξ 1 l L,id R ) (Rl 1 ξ 1 l R,θ 1 ) R ( A R ) op cop A L ( Rl 1 L l,µ ), where ξ is the anti-automorphism of A introduced in Lemma 5.9 and the maps µ, ν, and θ R are the ring isomorphisms introduced in Proposition 4.3.

109 744 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Proof. By Proposition 4.3, the map ν is a ring isomorphism L op R. By Proposition 5.10, l 1 R l L is bijective. Its anti-multiplicativity follows from (5.20). The comultiplicativity follows by the successive use of the identity l R (φ a)= S 1 (a)l R (φ ), the integral property of l, (5.22), and (5.23): One checks also γ L l 1 R l L(φ ) = l 1 R = l 1 R l L(φ ) l (1) l 1 ( R l (2) ) ( S 1 ( l (1)) l L (φ ) ) l 1 ( l (2) ) ( S 1 ( l (1))) l 1 ( ll (φ )l (2)) = l 1 R R = l 1 R (l (2)) l 1 ( R ll (φ )ξ(l (1) ) ) = l 1 R l ( L l 1 L (l (2)) ) l 1 R l L(l (1) φ ) = ( l 1 R l L l 1 R l L) op γ R (φ ). ( l 1 R l L s R (l) ) (a) = λ ( S 1 [ (a)s L π L l(1) s L (l) ] ) l (2) = ν(l)πr (a) = ( sl ν(l)) (a), ( l 1 R l L t R (l) ) (a) = λ ( S 1 [ (a)s L lπl (l (1) ) ] ) ( l (2) = πr sr ν(l)a ) = ( tl ν(l)) (a), πl l 1 R l ( L(φ ) = ν φ tl ν 1 ) λ(l (2) )l (1) = ν φ (1 A ) = ν π R (φ ). This proves that (l 1 R l L,ν) is a bialgebroid isomorphism (A R ) op cop A L. By Remark 5.4, ( R l 1 L l, µ) is a bialgebroid isomorphism ( A R ) op cop A L. By (5.16), R l 1 ξ 1 l R = µ l 1 L l R(φ ) S, hence we have to prove that (µ S,µ) is a bialgebroid isomorphism (A R ) op cop A L. The map µ is a ring isomorphism L op R by Proposition 4.3. The map φ µ φ S is bijective. Its anti-multiplicativity is obvious. The anti-comultiplicativity follows from (5.16) and (5.22): γ L (µ φ S) = R l 1( l (1)) µ φ S l (2) Finally, by Proposition 4.3 = µ l 1 L ξ( l (1)) S µ ( S ( l (2)) ) φ S = µ φ (2) S µ φ (1) S. ( µ s R (l) S ) (a) = µ π L ( S(a)sL (l) ) = π R ( sr µ(l)a ) = ( s L µ(l) ) (a), ( µ t R (l) S ) (a) = µ ( lπ L S(a) ) = π R (a)µ(l) = ( t L µ(l) ) (a), π L (µ φ S) = µ φ S(1 A ) = µ π R (φ ). R This proves the theorem.

110 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Theorem Let l be a non-degenerate left integral in the symmetrized Hopf algebroid A. Then the left bialgebroid A l L = (A,R,s L,t L,γ L,π L ) where s L (r)(a) = µ 1 (r)π L (a), t L (r)(a) = π L ( atr κ 1 (r) ), γ L (φ ) = ξ 2 (l (2) ) φ l L 1 (l (1) ), π L (φ ) = λ (l φ ), the right bialgebroid A R in (5.11), and the antipode S l Hopf algebroid denoted by A l. := l 1 L ξ l L form a symmetrized Proof. We show that the triple A l := (Al L, A R,S l ) satisfies the conditions in Proposition 4.2(iii). The A l L is a left bialgebroid isomorphic to ( A R ) op cop via the isomorphism ( L l 1 l L,µ).Also s L (R) = t R (L) and t L (R) = s R (L) hold obviously true. Making use of the identities (5.22) and (5.23), one checks that (id A γ R ) γ L (φ ) = ξ 2 (l (2) ) φ l (1 ) l 1 L (l (1)) l 1 L (l (2 )) = ξ 2 (l (2) ) φ l 1 ( L l(1) ξ(l (1 )) ) l 1 L (l (2 )) = ξ 2( l (2) ξ 2 (l (1 )) ) φ l 1 L (l (1)) l 1 L (l (2 )) = (γ L id A ) γ R (φ ), (γ R id A ) γ L (φ ) = l (1 )ξ 2 (l (2) ) φ l 1 L (l (2 )) l 1 L (l (1)) = l (1 ) φ l 1 ( L l(2 )ξ 1 (l (2) ) ) l 1 L (l (1)) = l (1 ) φ ξ 2 (l (2) ) l 1 L (l (2 )) l 1 L (l (1)) = (id A γ L ) γ R (φ ). The S l = (l 1 L ξ Ll) ( L l 1 l L ) is a composition of ring isomorphisms L l 1 l L : (A ) op A and l 1 L ξ Ll : A A, hence it is an anti-automorphism of the ring A. Also S l t R(l)(a) = µ 1 R l 1 l L t R (l) S 1 (a) = µ 1 λ ( S 1[ s L ( lπl (l (1) ) ) l (2) ] S 1 (a) ) = µ 1 π R S 1( as L (l) ) = π L ( asl (l) ) = s R (l)(a), S l t L(r)(a) = µ 1 R l 1 l L t L (r) S 1 (a) = µ 1 λ ( as L π L ( l(1) t R κ 1 (r) ) l (2) ) = µ 1 λ ( t L π L (a)lt R κ 1 (r) ) = µ 1 (r)π L (a) = s L (r)(a).

111 746 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) Since γ L ξ 1 (a) = ξ 1( a (2)) S 1( a (1)) and γ R ξ 1 (a) = ξ 1 (a (2) ) S 1 (a (1) ), we have γ L ξ 2 (a) = ξ 1 S 1 (a (1) ) S 1 ξ 1 (a (2) ). Then we can compute [ S l 1 ] [ (φ (2) )φ (1) (a) = l 1 L ξ 1 (l (1) ) ( ξ 2 )] (l (2) ) φ (a) ([ = φ a l 1 L ξ 1 (l (1) ) ] ξ 2 (l (2) ) ) ([ = φ ξ 1 (l (1) ) L l 1 (a) ] ξ 2 (l (2) ) ) = φ ξ 2( ξ ( l (2) (1)l (2) (2) )Ll 1 (a) S 1( l (1)) = φ (1 A )π L (l (1) ) L l 1 (a)(l (2) ) = φ (1 A )π L (a); hence S l (φ (1))φ (2) = s R π R (φ ) for all φ A.Also [ φ (1) Sl ( φ (2) )] (a) = [ (l(1) φ )l 1 L ξ(l (2)) ] (a) = µ 1 R l 1 (l (2) ) S 1( ) s L φ (a (1) l (1) )a (2) = µ 1 λ ( ) s L φ (a (1) l (1) )a (2) l (2) = µ 1 λ (al φ ) = µ 1 λ ( ) t L π L (a)l φ = µ 1 λ (l φ )π L (a) = [ s L π L (φ ) ] (a). This proves that A l = (Al L, A R,S l ) is a symmetrized Hopf algebroid. Obviously, the strong isomorphism class of the Hopf algebroid (A l L,Sl ) depends only on the Hopf algebroid (A L,S) and the non-degenerate left integral l of it. It is insensitive to the particular choice of the underlying right bialgebroid A R. The antipode S l has a form analogous to (5.17): S l (φ )(a) = [ (l φ ) l 1 L (1 A) ] (a). (5.25) Using the left bialgebroid isomorphisms of Theorem 5.16, also the dual left bialgebroids ( A R ) op cop, A L,and A L can be made Hopf algebroids, all strictly isomorphic to the above Hopf algebroid (A l L,Sl ). They have the antipodes S l = L l 1 ξ L l, (5.26) Sl = l 1 R ξ l R, (5.27) S l = R l 1 ξ R l. (5.28) Let us turn to the interpretation of the role of the Hopf algebroid (A l L,Sl ).AsAl L is isomorphic to ( A R ) op cop and the right bialgebroid underlying (A l L,Sl ) is A R, onthe

112 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) first sight it seems to be natural to consider it as some kind of a dual of (A L,S).There are however two arguments against this interpretation. First, the Hopf algebroid (A l L,Sl ) depends on l. Second, as it is proven in Proposition 5.19, (A l L,Sl ) belongs to a special kind of Hopf algebroids: it possesses a two-sided non-degenerate integral. Lemma Let A be a symmetrized Hopf algebroid such that the R-module A R in (4.6) is finitely generated projective. Then a left integral l I L (A) is non-degenerate if and only if the map l R is bijective. Proof. The only if part is trivial. In order to prove the if part, recall that by the proof of Lemma 5.6 for λ := l 1 R (1 A) and all a A the identity λ a= s R λ (a) holds true, λ is an R-R-bimodule map RA R R, and the inverse of l R reads as l 1 R (a) = λ S(a).Then φ (a) = l 1 R l R(φ )(a) = λ ( s R φ ( l (1)) S ( l (2)) a ) = φ ( al (1) s R λ S ( l (2))) for all φ A and a A. Using the finitely generated projectivity of A R,wehave l (1) s R λ S(l (2) ) = 1 A.Sinceλ S A,theinverse R l 1 can be defined as R l 1 (a) = λ S S 1 (a). Proposition Let A be a symmetrized Hopf algebroid possessing a non-degenerate left integral l. Then the element ll 1 (1 A) in A is a two-sided non-degenerate integral in the symmetrized Hopf algebroid A l (constructed in Theorem 5.17). Proof. It follows from (5.16) that l 1 L (1 A) = µ 1 λ where λ := l 1 R (1 A) and µ is the ring isomorphism introduced in Proposition 4.3. For all φ A and a A,wehave [( µ 1 λ ) ] φ (a) = φ S ( λ S 1 (a) ) = φ (1 A )µ 1 λ (a), hence [( µ 1 λ ) t R π R (φ ) ] (a) = t R π R (φ )(1 A )µ 1 λ (a) = φ (1 A )µ 1 λ (a), which proves that µ 1 λ is a right integral. Using (5.17), S l ( µ 1 λ ) = l 1 L ξ l ( L µ 1 λ ) = l 1 L ξ 2 (1 A ) = l 1 L (1 A) = µ 1 λ, hence µ 1 λ is also a left integral. As it is proven in [8], since A L is finitely generated projective, so is the left L L- module L (A ). The corresponding dual bialgebroid (A ) L is isomorphic to A L via the isomorphism (ι, id L ) of left bialgebroids where Since ι : A (A ), ι(a)(φ ) := φ (a). (5.29)

113 748 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) ( ) ([ ι(a) φ (b) = πl b (l(1) φ ) ] s L l 1 L (l (2))(a) ) ( = φ bsr λ ( al (1)) S ( l (2))) = (a φ )(b), the map L (µ 1 λ ) : (A ) A, ι(a) ι(a) µ 1 λ a µ 1 λ is bijective with inverse L ( µ 1 λ ) 1 : φ ι l L S l 1 ι R l(µ φ S). The application of Lemma 5.18 finishes the proof. In the view of Proposition 5.7, the following definition makes sense. Definition The dual of the isomorphism class of a Hopf algebroid (A L,S)possessing a non-degenerate left integral l is the isomorphism class of the Hopf algebroid (A l L,Sl ) (constructed in the Theorem 5.17). The next proposition shows that this notion of duality is involutive. Proposition Let l be a non-degenerate left integral in the Hopf algebroid (A L,S). Then the Hopf algebroid ( (A l ) l 1 L (1 A) L, ( S l ) l 1 L (1 A) ) (5.30) is strictly isomorphic to (A L,S l ) the Hopf algebroid constructed in Proposition 5.13.In particular, the Hopf algebroid (5.30) is isomorphic to (A L,S). Proof. Since the Hopf algebroids ( (A ) l l 1, ( S l L (1 A) L ) l 1 L (1 A) ) and ( (A ) L, ( S l ) ) l 1 L (1 A) are strictly isomorphic, it suffices to show that the isomorphism (ι, id L ) of left bialgebroids A L (A ) L in (5.29) extends to a strict isomorphism of Hopf algebroids (A L,S l ) ( (A ) L, (S l ) l 1 L (1 A) ). By (5.28) for a non-degenerate left integral l in the Hopf algebroid (A L,S),the antipode S l of the Hopf algebroid ( A L, S l ) reads as S l ( φ) = λ ( φ S 1 (l) ). Applying it to the non-degenerate integral l 1 L (1 A) in (A l L,Sl ), we obtain ( S l ) l 1 L (1 A) ( ι(a) ) = ι(l) [ ι(a) S l 1 l 1 L (1 A) ] = ι ( l ( a l 1 L (1 A) )) = ι ξ(a).

114 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) The duality of (weak) Hopf algebras is re-obtained from Definition 5.20 as follows. Let H be a finite weak Hopf algebra over a commutative ring k. Let(H L,S) be the corresponding Hopf algebroid (introduced in the Example 4.8) and let l be a nondegenerate left integral in H. Recall that, in order to reconstruct the weak Hopf algebra from the Hopf algebroid in Example 4.8, one needs a distinguished separability structure on the base ring. The dual weak Hopf algebra is the unique weak Hopf algebra in the isomorphism class of (H L l,sl ) corresponding to the same separability structure on L as H corresponds to. If H is a Hopf algebra over k then since the separability structure on k is unique the dual Hopf algebra is the only Hopf algebra in the isomorphism class of (H L l,sl ). References [1] M. Barr, -Autonomous Categories, in: Lecture Notes Math., vol. 752, Springer-Verlag, Berlin, [2] G. Böhm, An alternative notion of Hopf algebroid, math.qa/ [3] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf algebras I: Integral theory and C -structure, J. Algebra 221 (1999) 385. [4] G. Böhm, K. Szlachányi, A coassociative C -quantum group with non-integral dimensions, Lett. Math. Phys. 35 (1996) 137. [5] T. Brzezinski, G. Militaru, Bialgebroids, R -bialgebras and duality, J. Algebra 251 (2002) 279. [6] A. Connes, H. Moscovici, Differential cyclic cohomology and Hopf algebraic structures in transverse geometry, math.dg/ [7] B. Day, R. Street, Quantum categories, star autonomy, and quantum groupoids, math.ct/ [8] L. Kadison, K. Szlachányi, Dual bialgebroids for depth two ring extensions, Adv. Math., in press, math.ra/ [9] L. Kadison, Hopf algebroids and H-separable extensions, Proc. Amer. Math. Soc., in press, MPS [10] L. Kadison, A Hopf algebroid associated to a Galois extension, preprint. [11] M. Khalkhali, B. Rangipour, On cohomology of Hopf algebroids, math.kt/ [12] R.G. Larson, M.E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969) 75. [13] J.H. Lu, Hopf Algebroids and quantum groupoids, Internat. J. Math. 7 (1) (1996) 47. [14] F. Nill, Weak bialgebras, math.qa/ [15] F. Panaite, Doubles of (quasi)hopf algebras and some examples of quantum groupoids and vertex groups related to them, math.qa/ [16] D.E. Radford, The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (1976) 333. [17] P. Schauenburg, Bialgebras over noncommutative rings, and a structure theorem for Hopf bimodules, Appl. Categ. Structures 6 (1998) 193. [18] P. Schauenburg, Duals and doubles of quantum groupoids, in: New Trends in Hopf Algebra Theory, Proc. Colloq. on Quantum Groups and Hopf Algebras, La Falda, Sierra de Cordoba, Argentina, 1999, in: Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, p [19] P. Schauenburg, Weak Hopf algebras and quantum groupoids, preprint. [20] K. Szlachányi, Finite quantum groupoids and inclusions of finite type, Fields Inst. Commun. 30 (2001) 393. [21] K. Szlachányi, Galois actions by finite quantum groupoids, in: L. Vainerman (Ed.), Locally Compact Quantum Groups and Groupoids, Proc. Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 2002, in: V. Turaev (Ed.), IRMA Lectures in Math. Theoret. Phys., vol. 2, de Gruyter, [22] K. Szlachányi, The monoidal Eilenberg Moore construction and bialgebroids, J. Pure Appl. Algebra 182 (2003) 287. [23] W. Szymanski, Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120 (1994) 519.

115 750 G. Böhm, K. Szlachányi / Journal of Algebra 274 (2004) [24] M. Takeuchi, Groups of algebras over A Ā, J. Math. Soc. Japan 29 (1977) 459. [25] P. Xu, Quantum groupoids and deformation quantization, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 289. [26] P. Vecsernyés, Larson Sweedler theorem, group like elements, invertible modules and the order of the antipode in weak Hopf algebras, J. Algebra, in press, math.qa/ [27] Y. Watatani, Index for C -subalgebras, Mem. Amer. Math. Soc. 23 (424) (1990).

116 FEJEZET. HOPF ALGEBROIDS WITH BIJECTIVE ANTIPODES

117 4. fejezet Integral theory for Hopf algebroids Böhm, G. Algebr. Represent. Theory 8 (2005), no. 4, Erratum in press. 117

118 FEJEZET. INTEGRAL THEORY FOR HOPF ALGEBROIDS

119 Algebras and Representation Theory (2005) 8: Springer 2005 DOI: /s Integral Theory for Hopf Algebroids GABRIELLA BÖHM Research Institute for Particle and Nuclear Physics, H-1525 Budapest 114, POB 49, Hungary. (Received: May 2004; accepted in revised form: February 2005) Presented by: S. Montgomery Abstract. The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras. Mathematics Subject Classification (2000): 16W30. Key words: Hopf algebroid, integral, Maschke theorem, (quasi-)frobenius extension. 1. Introduction The notion of integrals in Hopf algebras has been introduced by Sweedler [33]. The integrals in Hopf algebras over principal ideal domains were analyzed in [19, 32] where the following by now classical results have been proven: A free, finite-dimensional bialgebra over a principal ideal domain is a Hopf algebra if and only if it possesses a nondegenerate left integral. (Larson Sweedler Theorem.) The antipode of a free, finite-dimensional Hopf algebra over a principal ideal domain is bijective. A Hopf algebra over a field is finite-dimensional if and only if it possesses a nonzero left integral. The left integrals in a finite-dimensional Hopf algebra over a field form a one dimensional subspace. A Hopf algebra over a field is semi-simple if and only if it possesses a normalized left integral. (Maschke s Theorem.) There are numerous generalizations of these results in the literature. Historically the first is due to Pareigis [27] who proved the following statements on a finitely generated and projective Hopf algebra (H,,ɛ,S)over a commutative ring k:

120 564 GABRIELLA BÖHM H is a Frobenius extension of k if and only if there exists a Frobenius functional ψ: H k satisfying (H ψ) = 1 H ψ( ). The antipode, S, is bijective. The left integrals form a projective rank 1 direct summand of the k-module H. H is a quasi-frobenius extension of k. A finitely generated and projective bialgebra over a commutative ring k, such that pic(k) = 0, is a Hopf algebra if and only if it possesses a nondegenerate left integral. The generalization of the Maschke theorem to Hopf algebras H over commutative rings k states that the existence of a normalized left integral in H is equivalent to the separability of H over k, which is further equivalent to its relative semisimplicity in the sense of [15, 17] that any H -module is (H, k)-projective [12, 20]. This is equivalent to the true semi-simplicity of H (i.e. the true projectivity of any H -module [28]) if and only if k is a semi-simple ring [20]. As a nice review on these results we recommend Section 3.2 in [13]. Similar results are known also for the generalizations of Hopf algebras. Integrals for finite-dimensional quasi-hopf algebras [14] over fields were studied in [16, 25, 26, 11] and for finite-dimensional weak Hopf algebras [4, 3] over fields in [3, 40]. The purpose of the present paper is to investigate which of the above results generalizes to Hopf algebroids. Hopf algebroids with bijective antipode have been introduced in [1, 5]. It is important to emphasize that this notion of Hopf algebroid is not equivalent to the one introduced under the same name by Lu in [21]. Here we generalize the definition of [5, 1] by relaxing the requirement of the bijectivity of the antipode. A Hopf algebroid consists of a compatible pair of a left and a right bialgebroid structure [21, 34, 35, 38] on the common total algebra A. The antipode relates these two left- and right-handed structures. Left/right integrals in a Hopf algebroid are defined as the invariants of the left/right regular A-module in terms of the counit of the left/right bialgebroid. Integrals on a Hopf algebroid are the comodule maps from the total algebra to the base algebra (reproducing the integrals in the dual bialgebroids, provided the duals possess bialgebroid structures). The total algebra of a bialgebroid can be looked at as an extension of the base algebra or its opposite via the source and target maps, respectively. This way there are four algebra extensions associated to a Hopf algebroid. The main results of the paper relate the properties of these extensions to the existence of integrals with special properties: A Maschke-type theorem, proving that the separability, and also the (in two cases left in two cases right) semi-simplicity of any of the four extensions is equivalent to the existence of a normalized integral in the Hopf algebroid (Theorem 3.1).

121 INTEGRAL THEORY FOR HOPF ALGEBROIDS 565 Any of the four extensions is a Frobenius extension if and only if there exists a nondegenerate integral in the Hopf algebroid (Theorem 4.7). Any of the four extensions is (in two cases a left in two cases a right) quasi- Frobenius extension if and only if the total algebra is a finitely generated and projective module, and the (left or right) integrals on the Hopf algebroid form a flat module, over the base algebra (Theorem 5.2). Our main tool in proving the latter two points is the Fundamental Theorem for Hopf modules over Hopf algebroids (Theorem 4.2). The paper is organized as follows: We start Section 2 with reviewing some results on bialgebroids from [9, 18, 21, 30, 31, 34 36, 38], the knowledge of which is needed for the understanding of the paper. Then we give the definition of Hopf algebroids and discuss some of its immediate consequences. Integrals both in and on Hopf algebroids are introduced and some equivalent characterizations are given. In Section 3 we prove two Maschke-type theorems. The first collects some equivalent properties (in particular the separability) of the inclusion of the base algebra in the total algebra of a Hopf algebroid. These equivalent properties are related to the existence of a normalized integral in the Hopf algebroid. The second collects some equivalent properties (in particular the coseparability) of the coring, underlying the Hopf algebroid. These equivalent properties are shown to be equivalent to the existence of a normalized integral on the Hopf algebroid. In Section 4 we prove the Fundamental Theorem for Hopf modules over a Hopf algebroid. This theorem is somewhat stronger than the one that can be obtained by the application of [7, Theorem 5.6], to the present situation. The main result of the section is Theorem 4.7. In proving it we follow an analogous line of reasoning as in [19]. That is, assuming that one of the module structures of the total algebra over the base algebra is finitely generated and projective, we apply the Fundamental Theorem to the Hopf module, constructed on the dual of the Hopf algebroid (w.r.t. the base algebra). Similarly to the case of Hopf algebras, our result implies the existence of nonzero integrals on any finitely generated projective Hopf algebroid. Since the dual of a (finitely generated projective) Hopf algebroid is not known to be a Hopf algebroid in general, we have no dual result, that is, we do not know whether there exist nonzero integrals in any finitely generated projective Hopf algebroid. As a byproduct, also a sufficient and necessary condition on a finitely generated projective Hopf algebroid is obtained, under which the antipode is bijective. We do not know, however, whether this condition follows from the axioms. In Section 5 we use the results of Section 4 to obtain conditions which are equivalent to the (either left or right) quasi-frobenius property of any of the four extensions behind a Hopf algebroid. In order to show that these conditions do not hold true in general, we construct a counterexample. Throughout the paper we work over a commutative ring k. That is, the total and base algebras of our Hopf algebroids are k-algebras. For an (always associative and unital) k-algebra A (A, m A, 1 A ) we denote by A M, M A and A M A the

122 566 GABRIELLA BÖHM categories of left, right, and bimodules over A, respectively. For the k-module of morphisms in A M, M A and A M A we write A Hom(, ),Hom A (, )and A Hom A (, ), respectively. 2. Integrals for Hopf Algebroids Hopf algebroids with bijective antipode have been introduced in [5], where several equivalent reformulations of the definition [5, Definition 4.1] have been given. The definition we give in this section generalizes the form in [5, Proposition 4.2(iii)] by allowing the antipode not to be bijective. Integrals in Hopf algebroids have also been introduced in [5]. As we shall see, the definition [5, Definition 5.1] applies also in our more general setting. In this section we introduce integrals also on Hopf algebroids. In order for the paper to be self-contained we recall some results on bialgebroids from [38, 21, 34, 35, 18]. For more on bialgebroids we refer to the papers [30, 9, 31, 36]. The notions of Takeuchi s R -bialgebra [38], Lu s bialgebroid [21] and Xu s bialgebroid with anchor [41] have been shown to be equivalent in [9]. We are going to use the definition in the following form: DEFINITION 2.1. A left bialgebroid A L = (A,B,s,t,γ,π) consists of two algebras A and B over the commutative ring k, which are called the total and base algebras, respectively. A is a B k Bop -ring (i.e. a monoid in B B opm B B op) via the algebra homomorphisms s: B A and t: B op A, called the source and target maps, respectively. In terms of s and t one equips A with a B B bimodule structure BA B as b a b := s(b)t(b )a for a A, b, b B. The triple ( B A B,γ,π) is a B-coring, that is a comonoid in B M B. Introducing Sweedler s convention γ(a)= a (1) B (2) for a A,theaxioms a (1) t(b) a B (2) = a (1) B (2)s(b), (2.1) γ(1 A ) = 1 A B A, (2.2) γ(aa ) = γ(a)γ(a ), (2.3) π(1 A ) = 1 B, (2.4) π(a s π(a )) = π(aa ), (2.5) π(a t π(a )) = π(aa ) (2.6) are required for all b B and a,a A. Notice that although A B A is not an algebra axiom (2.3) makes sense in view of (2.1). The homomorphisms of left bialgebroids A L = (A,B,s,t,γ,π) A L = (A,B,s,t,γ,π ) are pairs of k-algebra homomorphisms (: A A,

123 INTEGRAL THEORY FOR HOPF ALGEBROIDS 567 φ: B B ) satisfying s φ = s, (2.7) t φ = t, (2.8) γ = ( B ) γ, (2.9) π = φ π. (2.10) The bimodule B A B, appearing in Definition 2.1, is defined in terms of multiplication on the left. Hence following the terminology of [18] we use the name left bialgebroid for this structure. In terms of right multiplication one defines right bialgebroids analogously. For the details we refer to [18]. Once the map γ : A A A is given we can define γ op B : A A A via B op a a (2) a (1). It is straightforward to check that if A L = (A,B,s,t,γ,π)is a left bialgebroid then A L cop = (A, B op,t,s,γ op,π) is also a left bialgebroid and = (Aop,B,t,s,γ,π)is a right bialgebroid. In the case of a left bialgebroid A L = (A,B,s,t,γ,π)the category A M of left A-modules is a monoidal category. As a matter of fact, any left A-module is a B B bimodule via s and t. The monoidal product in A M is defined as the B-module tensor product with A-module structure A op L a (m B m ) := a (1) m B a (2) m for a A, m B m M B M. Just the same way as axiom (2.3), also this definition makes sense in the view of (2.1). The monoidal unit is B with A-module structure a b := π(as(b)) for a A, b B. Analogously, in the case of a right bialgebroid A R the category M A of right A-modules is a monoidal category. The B-coring structure ( B A B,γ,π), underlying the left bialgebroid A L = (A,B,s,t,γ,π),givesrisetoak-algebra structure on any of the B-duals of B A B [10, 17.8]. The multiplication on the k-module A := B Hom(A, B), for example, is given by ( φ ψ)(a) = ψ(t φ(a (2) )a (1) ) for φ, ψ A,a A (2.11) and the unit is π. A is a left A-module and A is a right A-module via a φ := φ( a) and a φ := t φ(a (2) )a (1) (2.12) for φ A, a A. As it is well known [39, 18], A is also a B k B op -ring via the inclusions s: B A, b π( )b, t: B op A, b π( s(b)). Both maps s and t are split injections of B-modules with common left inverse π: A B, φ φ(1 A ).Whatismore,ifAis finitely generated and projective

124 568 GABRIELLA BÖHM as a left B-module, then A has also a right bialgebroid structure (with source and target maps s and t, respectively, and counit π). Notice that the algebra A reduces to the opposite of the usual dual algebra if ( B A B,γ,π) is a coalgebra over a commutative ring B. In the case when A is a finitely generated projective left B-module, also the coproduct specializes to the opposite of the usual one in the case when A is a bialgebra. This convention is responsible for duality to flip the notions of left and right bialgebroids. Applying the above formulae to the left bialgebroid (A L ) cop we obtain a B k B op -ring structure on A := Hom B (A, B). The inclusions B A and B op A will be denoted by s and t, respectively. In particular, A is a left A-module and A is a right A -module via a φ := ( a) and a φ := s φ (a (1) )a (2). (2.13) If the module A is finitely generated and projective as a right B-module then A is also a right bialgebroid. In the case of a right bialgebroid A R = (A,B,s,t,γ,π) the application of the opposite of the multiplication formula (2.11) to (A R ) op cop and to (A R ) op results B k B op -ring structures on A := Hom B (A, B) and A := B Hom(A, B), respectively. We have the inclusions s : B A, t : B op A, s: B A and t: B op A. In particular, A and A are right A-modules and A is a left A -module and a left A-module via the formulae φ a: = φ (a ) and φ a:= a (2) t φ (a (1) ), (2.14) φ a:= φ(a ) and φ a:= a (1) s φ(a (2) ) (2.15) for φ A, φ A and a A. IfA is finitely generated and projective as a right, or as a left B-module then the corresponding dual is also a left bialgebroid. Before defining the structure that is going to be the subject of the paper let us stop here and introduce some notations. Analogous notations were already used in [5]. When dealing with a B k B op -ring A, we have to face the situation that A carries different module structures over the base algebra B. In this situation the usual notation A B A would be ambiguous. Therefore we make the following notational convention. In terms of the maps s: B A and t: B op A we introduce four B-modules BA : b a := s(b)a, A B : a b := t(b)a, A B : a b = as(b), B A : b a = at(b). (2.16) (Our notation can be memorized as left indices stand for left modules and right indices for right modules. Upper indices for modules defined in terms of right multiplication and lower indices for the ones defined in terms of left multiplication.)

125 INTEGRAL THEORY FOR HOPF ALGEBROIDS 569 In writing B-module tensor products we write out explicitly the module structures of the factors that are taking part in the tensor products, and do not put marks under the symbol. For example, we write A B B A. Normally we do not denote the module structures that are not taking part in the tensor product, this should be clear from the context. In writing elements of tensor product modules we do not distinguish between the various module tensor products. That is, we write both a a A B B A and c c A B B A, for example. AleftB-module can be considered as a right B op -module, and sometimes we want to take a module tensor product over B op.inthiscaseweusethenameof the corresponding B-module and the fact that the tensor product is taken over B op should be clear from the order of the factors. For example, B A A B is the B op -module tensor product of the right B op module defined via multiplication by s(b) on the left, and the left B op -module defined via multiplication by t(b) on the left. In writing multiple tensor products we use different types of letters to denote which module structures take part in the same tensor product. For example, the B-module tensor product A B B A can be given a right B module structure via multiplication by t(b) on the left in the second factor. The tensor product of this right B-module with B A is denoted by A B B A B B A. We are ready to introduce the structure that is going to be the subject of the paper: DEFINITION 2.2. A Hopf algebroid A = (A L, A R,S)consists of a left bialgebroid A L = (A,L,s L,t L,γ L,π L ), a right bialgebroid A R = (A,R,s R,t R,γ R,π R ) and a k-module map S: A A, called the antipode, such that the following axioms hold true: (i) s L π L t R = t R, t L π L s R = s R and s R π R t L = t L, t R π R s L = s L, (2.17) (ii) (γ L R A) γ R = (A L γ R ) γ L as maps A A L L A R R A and (γ R L A) γ L = (A R γ L ) γ R as maps A A R R A L L A, (2.18) (iii) S is both an L L bimodule map L A L L A L and an R R bimodule map R A R R A R, (2.19) (iv) m A (S L A) γ L = s R π R and m A (A R S) γ R = s L π L. (2.20) If A = (A L, A R,S)is a Hopf algebroid then so is A op cop = ((A R ) op cop,(a L ) op cop,s) and if S is bijective then also A cop = ((A L ) cop,(a R ) cop,s 1 ) and A op = ((A R ) op, (A L ) op,s 1 ). The following modification of Sweedler s convention will turn out to be useful. For a Hopf algebroid A = (A L, A R,S) we use the notation γ L (a) = a (1) a (2)

126 570 GABRIELLA BÖHM with lower indices, and γ R (a) = a (1) a (2) with upper indices for a A in the case of the coproducts of A L and of A R, respectively. The axioms (2.18) read in this notation as a (1) (1) a (1) (2) a (2) = a (1) a (1) (2) a (2) (2), a (1) (1) a (2) (1) a (2) = a (1) a (2) (1) a (2) (2) for a A. Examples of Hopf algebroids (with bijective antipode) are collected in [5]. PROPOSITION 2.3. (1) The base algebras L and R of the left and right bialgebroids in a Hopf algebroid are anti-isomorphic. (2) For a Hopf algebroid A = (A L, A R,S) the pair (S, π L s R ) is a left bialgebroid homomorphism (A R ) op cop A L and (S, π R s L ) is a left bialgebroid homomorphism A L (A R ) op cop. Proof. (1) Both π R s L and π R t L are anti-isomorphisms L R with inverses π L t R and π L s R, respectively. (2) We have seen that the map π L s R : R op L is an algebra homomorphism. It follows from (2.19), (2.20) and some bialgebroid identities that S: A op A is an algebra homomorphism, as for a,b A we have S(1 A ) = 1 A S(1 A ) = s L π L (1 A ) = 1 A and S(ab) = S[t L π L (a (2) )a (1) b] = S[a (1) t L π L (b (2) )b (1) ] a (1) (2) S(a (2) (2) ) = S[a (1) (1)b (1) (1)] a (1) (2)b (1) (2)S(b (2) )S(a (2) ) = s R π R (a (1) b (1) )S(b (2) )S(a (2) ) = S[b (2) t R π R (t R π R (a (1) )b (1) )]S(a (2) ) = S(b) s R π R (a (1) )S(a (2) ) = S(b)S(a). The properties (2.7) (2.8) follow from (2.19) and (2.17) as s L π L s R = S t L π L s R = S s R, t L π L s R = s R = S t R. The properties (2.9) (2.10) are checked on an element a A as γ L S(a) = S(a (1) ) (1) s L π L (a (2) ) S(a (1) ) (2) = S(a (1) (1)) (1) a (1) (2)S(a (2) ) S(a (1) (1)) (2) = S(a (1) (1)) (1) t L π L (a (1) (2)(2))a (1) (2)(1) S(a (2) ) S(a (1) (1)) (2) = S(a (1)(1) (1)) (1) a (1)(1) (2)(1)S(a (2) ) S(a (1)(1) (1)) (2) a (1)(1) (2)(2)S(a (1)(2) ) = S(a (2) ) s R π R (a (1)(1) )S(a (1)(2) ) = (S S) γ op R (a) and π L S(a) = π L [S(a (1) )s L π L (a (2) )]=π L s R π R (a).

127 INTEGRAL THEORY FOR HOPF ALGEBROIDS 571 The proof is completed by the observation that in passing from the Hopf algebroid A to A op cop the roles of (S, π L s R ) and (S, π R s L ) become interchanged. PROPOSITION 2.4. The left bialgebroid A L in a Hopf algebroid A is a L -Hopf algebra in the sense of [30]. That is, the map α: L A A L A L L A, a b a (1) a (2) b is bijective. Proof. The inverse of α is given by α 1 : A L L A L A A L, a b a (1) S(a (2) )b. The relation between the left and the right bialgebroids in a Hopf algebroid A implies relations between the dual algebras A Hom R (A R,R) and A Hom L (A L,L)and also between A R Hom( R A, R) and A L Hom( L A, L): LEMMA 2.5. For a Hopf algebroid A there exist algebra anti-isomorphisms σ : A A and χ: A A satisfying and a φ = σ( φ) a (2.21) φ a= a χ(φ ) (2.22) for all φ A, φ A and a A. Proof. We leave it to the reader to check that the maps and σ : A A, φ π R ( φ) χ: A A, φ π L (φ ) are algebra anti-homomorphisms satisfying (2.21) (2.22). The inverses are given by σ 1 : A A, φ π L ( φ ) and χ 1 : A A, φ π R ( φ ). LEMMA 2.6. The following properties of a Hopf algebroid A = (A L, A R,S)are equivalent: (1.a) The module A L is finitely generated and projective. (1.b) The module A R is finitely generated and projective.

128 572 GABRIELLA BÖHM The following are also equivalent: (2.a) The module L A is finitely generated and projective. (2.b) The module R A is finitely generated and projective. If furthermore S is bijective then all the four properties (1.a), (1.b), (2.a) and (2.b) are equivalent. Proof. (1.a) (1.b) In terms of the dual bases, {b i } Aand {β i} A for the module A L, the dual bases, {k j } Aand {κj } A for the module A R, can be constructed by the requirement that k j κj = b (1) i [χ 1 (β i ) s R π R s L π L (b (2) i )] j i as elements of A R R A,whereχ is the isomorphism (2.22). The expression on the right-hand side is well defined since though the map A L L A A, a φ χ 1 (φ ) s R π R s L π L (a) is not a left R-module map R A L L A R A its restriction to the R-submodule { k a k φ k A L L A k a kt L (l) φ k = k a k φ ks (l) l L } is so. (2.a) (2.b) Similarly, in terms of the dual bases, {b i } Aand { β i } A for the module L A, the dual bases, {k j } Aand { κ j } A for the module R A, can be constructed by the requirement that κ j k j = [σ( β i ) t R π R t L π L (b (1) i )] b (2) i j i as elements of A R R A,whereσ is the isomorphism (2.21). (1.b) (1.a) follows by applying (2.a) (2.b) to the Hopf algebroid A op cop. (2.b) (2.a) follows by applying (1.a) (1.b) to the Hopf algebroid A op cop. Now suppose that S is bijective. (1.a) (2.b) In terms of the dual bases, {b i } Aand {β i} A for the module A L, the dual bases, {k j } Aand { κ j } A for the module R A, can be constructed by the requirement that κ j k j = π R t L β i S S 1 (b i ) as elements of A R R A. j i (2.b) (1.a) follows by applying (1.a) (2.b) to the Hopf algebroid A op cop. Now we turn to the study of the notion of integrals in Hopf algebroids. For a left bialgebroid A L = (A,L,s L,t L,γ L,π L ) and a left A-module M the invariants of M with respect to A L are the elements of Inv(M) := {n M a n = s L π L (a) n a A}.

129 INTEGRAL THEORY FOR HOPF ALGEBROIDS 573 Clearly, the invariants of M with respect to (A L ) cop coincide with its invariants with respect to A L. The invariants of a right A-module M with respect to a right bialgebroid A R are defined as the invariants of M (viewed as a left A op -module) with respect to (A R ) op. DEFINITION 2.7. The left integrals in a left bialgebroid A L are the invariants of the left regular A-module with respect to A L. The right integrals in a right bialgebroid A R are the invariants of the right regular A-module with respect to A R. The left/right integrals in a Hopf algebroid A = (A L, A R,S)are the left/right integrals in A L /A R, that is the elements of and L(A) ={l A al = s L π L (a) l a A} R(A) ={ A a = s R π R (a) a A}. For any Hopf algebroid A = (A L, A R,S)we have L(A) = R(A op cop) and if S is bijective then also L(A) = L(A cop ) = R(A op ).Sinceforl L(A) and a A, S(l)a = S[t L π L (a (1) )l]a (2) = S(a (1) l)a (2) = S(l) s R π R (a), we have S(L(A)) R(A) and, similarly, S(R(A)) L(A). SCHOLIUM 2.8. The following properties of an element l A are equivalent: (1.a) l L(A), (1.b) S(a)l (1) l (2) = l (1) al (2) a A, (1.c) al (1) S(l (2) ) = l (1) S(l (2) )a a A. The following properties of the element A are also equivalent: (2.a) R(A), (2.b) (1) (2) S(a) = (1) a (2) a A, (2.c) S( (1) ) (2) a = as( (1) ) (2) a A. By comodules over a left bialgebroid A L = (A,L,s L,t L,γ L,π L ) we mean comodules over the L-coring ( L A L,γ L,π L ), and by comodules over a right bialgebroid A R = (A,R,s R,t R,γ R,π R ) comodules over the R-coring ( R A R,γ R,π R ). The pair ( L A, γ L ) is a left comodule, and (A L,γ L ) is a right comodule over the left bialgebroid A L. Since the L-coring ( L A L,γ L,π L ) possesses a grouplike element 1 A,also(L, s L ) is a left comodule and (L, t L ) is a right comodule over A L (see [10, 28.2]). Similarly, (A R,γ R ) and (R, s R ) are right comodules, and ( R A, γ R ) and (R, t R ) are left comodules over A R. DEFINITION 2.9. An s-integral on a left bialgebroid A L = (A,L,s L,t L,γ L,π L ) is a left A L -comodule map ρ: ( L A, γ L ) (L, s L ). That is, an element of R( A) := { ρ A (A L ρ) γ L = s L ρ}.

130 574 GABRIELLA BÖHM A t-integral on A L is a right A L -comodule map (A L,γ L ) (L, t L ).Thatis,an element of R(A ) := {ρ A (ρ L A) γ L = t L ρ }. An s-integral on a right bialgebroid A R = (A,R,s R,t R,γ R,π R ) is a right A R -comodule map (A R,γ R ) (R, s R ). That is, an element of L(A ) := {λ A (λ R A) γ R = s R λ }. A t-integral on A R is a left A R -comodule map ( R A, γ R ) (R, t R ).Thatis,an element of L( A) := { λ A (A R λ) γ R = t R λ}. The right/left s- and t-integrals on a Hopf algebroid A = (A L, A R,S) are the s-andt-integrals on A L /A R. The integrals on a left/right bialgebroid are checked to be invariants of the appropriate right/left regular module justifying our usage of the terms right and left integrals for them (cf. the remark in Section 2 about using the opposite co-opposite of the convention, usual in the case of bialgebras, when defining the dual bialgebroids A and A ). As a matter of fact, for example, if ρ R( A) then [ ρ φ](a) = φ(a ρ) = φ(s L ρ(a)) = ρ(a) φ(1 A ) = [ ρ s π( φ)](a) (2.23) for all φ A and a A. If the module L A is finitely generated and projective (hence A is a right bialgebroid) then also the converse is true, so in this case the s-integrals on A L are the same as the right integrals in A. Similar statements hold true on the elements of R(A ), L(A ) and L( A). The reader should be warned that integrals on Hopf algebras H over commutative rings k are defined in the literature sometimes as comodule maps H k similarly to our Definition 2.9, sometimes by the analogue of the weaker invariant condition (2.23). For any Hopf algebroid A we have R( A) = L((A op cop) ) and R(A ) = L( (A op cop)). If the antipode is bijective then also R( A) = R((A cop ) ) = L( (A op )). SCHOLIUM Let A = (A L, A R,S) be a Hopf algebroid. The following properties of an element ρ A are equivalent: (1.a) ρ R( A), (1.b) π R s L ρ L( A), (1.c) s L ρ(as(b (1) )) b (2) = t L ρ(a (2) S(b)) a (1) a,b A.

131 INTEGRAL THEORY FOR HOPF ALGEBROIDS 575 The following properties of an element ρ A are equivalent: (2.a) ρ R(A ), (2.b) π R t L ρ L(A ), (2.c) t L ρ (ab (1) )S(b (2) ) = s L ρ (a (1) b) a (2) a,b A. The following properties of an element λ A are equivalent: (3.a) λ L(A ), (3.b) π L s R λ R(A ), (3.c) a (1) s R λ (S(a (2) )b) = b (2) t R λ (S(a)b (1) ) a,b A. The following properties of an element λ A are equivalent: (4.a) λ L( A), (4.b) π L t R λ R( A), (4.c) S(a (1) )t R λ(a (2) b) = b (1) s R λ(ab (2) ) a,b A. In particular, for ρ R( A) the element ρ S belongs to R(A ) and for λ L(A ) the element λ S belongs to L( A). 3. Maschke Type Theorems The most classical version of Maschke s theorem [22] considers group algebras over fields. It states that the group algebra of a finite group G over a field F is semisimple if and only if the characteristic of F does not divide the order of G. This result has been generalized to finite-dimensional Hopf algebras H over fields F by Sweedler [32] proving that H is a separable F -algebra if and only if it is semisimple and if and only if there exists a normalized left integral in H. The proof goes as follows. It is a classical result that a separable algebra over a field is semi-simple. If H is semi-simple then, in particular, the H -module on F, defined in terms of the counit, is projective. This means that the counit, as an H -module map H F, splits. Its right inverse maps the unit of F into a normalized integral. Finally, in terms of a normalized integral one can construct an H -bilinear right inverse for the multiplication map H F H H. The only difficulty in the generalization of Maschke s theorem to Hopf algebras over commutative rings comes from the fact that in the case of an algebra A over a commutative base ring k, separability does not imply the semi-simplicity of A in the sense [28] that every (left or right) A-module was projective. It implies [15, 17], however, that every A-module is (A, k)-projective, i.e. that every epimorphism of A-modules which is k-split, is also A-split. In order to avoid confusion, we will say that the k-algebra A is semi-simple [28] if it is an Artinian semi-simple ring, i.e. if any A-module is projective. By the terminology of [15] we call A a (left or right) semi-simple extension of k if any (left or right) A-module is (A, k)- projective.

132 576 GABRIELLA BÖHM Since the counit of a Hopf algebra H over a commutative ring k is a split epimorphism of k-modules, the Maschke theorem generalizes to this case in the following form [12, 20]. The extension k H is separable if and only if it is (left and right) semi-simple and if and only if there exist normalized (left and right) integrals in H. In this section we investigate the properties of the total algebra of a Hopf algebroid, as an extension of the base algebra, that are equivalent to the existence of normalized integrals in the Hopf algebroid. Dually, we investigate also the properties of the coring over the base algebra, underlying a Hopf algebroid, that are equivalent to the existence of normalized integrals on the Hopf algebroid (in any of the four possible senses). A Maschke-type theorem on certain Hopf algebroids can be obtained also by the application of [37, Theorem 4.2]. Notice, however, that the Hopf algebroids occurring this way are only the Frobenius Hopf algebroids (discussed in Section 4 below), that is the Hopf algebroids possessing nondegenerate integrals (which are called Frobenius integrals in [37]). The following Theorem 3.1 generalizes results from [12, Proposition 4.7] and [20, Theorem 3.3]. THEOREM 3.1 (Maschke Theorem for Hopf algebroids). The following assertions on a Hopf algebroid A = (A L, A R,S)are equivalent: (1.a) The extension s R : R A is separable. That is, the multiplication map A R R A A splits as an A A bimodule map. (1.b) The extension t R : R op A is separable. That is, the multiplication map R A A R A splits as an A A bimodule map. (1.c) The extension s L : L A is separable. That is, the multiplication map A L L A A splits as an A A bimodule map. (1.d) The extension t L : L op A is separable. That is, the multiplication map L A A L A splits as an A A bimodule map. (2.a) The extension s R : R A is right semi-simple. That is, any right A-module is (A, R)-projective. (2.b) The extension t R : R op A is right semi-simple. That is, any right A-module is (A, R op )-projective. (2.c) The extension s L : L A is left semi-simple. That is, any left A-module is (A, L)-projective. (2.d) The extension t L : L op A is left semi-simple. That is, any left A-module is (A, L op )-projective. (3.a) There exists a normalized left integral in A. That is, an element l L(A) such that π L (l) = 1 L. (3.b) There exists a normalized right integral in A. That is, an element R(A) such that π R ( ) = 1 R. (4.a) The epimorphism π R : A R splits as a right A-module map. (4.b) The epimorphism π L : A L splits as a left A-module map.

133 INTEGRAL THEORY FOR HOPF ALGEBROIDS 577 Proof. (1.a) (2.a), (1.b) (2.b), (1.c) (2.c) and (1.d) (2.d) It is proven in [17, Proposition 2.6] that a separable extension is both left and right semi-simple. (2.a) (4.a) ((2.b) (4.a)) The epimorphism π R is split as a right (left) R-module map by s R (by t R ), hence it is split as a right A-module map. (4.a) (3.b) Let ν: R A be the right inverse of π R in M A.Then := ν(1 R ) is a normalized right integral in A. (3.a) (3.b) By part (2) of Proposition 2.3 the antipode takes a normalized left/right integral to a normalized right/left integral. (3.a) (1.a) and (3.b) (1.b) If l is a normalized left integral in A then, by Scholium 2.8, the required right inverse of the multiplication map A R R A A is givenbythea Abimodule map a al (1) S(l (2) ) l (1) S(l (2) )a. Similarly, if is a normalized right integral in A then the right inverse of the multiplication map R A A R A is given by a as( (1) ) (2) S( (1) ) (2) a. The proof is completed by applying the above arguments to the Hopf algebroid A op cop. Let us make a comment on the semi-simplicity of the algebra A (cf. [17, Proposition 1.3]). If R is a semi-simple algebra and the equivalent conditions of Theorem 3.1 hold true, then A being a semi-simple extension of a semi-simple algebra is a semi-simple algebra. On the other hand, notice that condition (4.a) in Theorem 3.1 is equivalent to the projectivity of the right A-module R. Hence if A is a semi-simple k-algebra then the equivalent conditions of the theorem hold true. It is not true, however, that the semi-simplicity of the total algebra implies the semi-simplicity of the base algebra (which was shown by Lomp to be the case in Hopf algebras [20]). A counterexample can be constructed as follows: If B is a Frobenius algebra over a commutative ring k then A: = End k (B) has a Hopf algebroid structure over the base B [6]. If B is a Frobenius algebra over a field which can be non-semi-simple! then A is a Hopf algebroid with semi-simple total algebra. The following Theorem 3.2 can be considered as a dual of Theorem 3.1 in the sense that it speaks about corings over the base algebras instead of algebra extensions. It is important to emphasize, however, that the two theorems are independent results. Even in the case of Hopf algebroids such that all module structures (2.16) are finitely generated and projective, the duals are not known to be Hopf algebroids. Recall that the dual notion of that of a relative projective module is the relative injective comodule. Namely, a comodule M for an R-coring A is called (A, R)- injective [10, 18.18] if any monomorphism of A-comodules from M, which splits as an R-module map, splits also as an A-comodule map. THEOREM 3.2 (Dual Maschke Theorem for Hopf algebroids). The following assertions on a Hopf algebroid A = (A L, A R,S)are equivalent:

134 578 GABRIELLA BÖHM (1.a) The R-coring ( R A R,γ R,π R ) is coseparable. That is, the comultiplication γ R : A A R R A splits as an A R A R bicomodule map. (1.b) The L-coring ( L A L,γ L,π L ) is coseparable. That is, the comultiplication γ L : A A L L A splits as an A L A L bicomodule map. (2.a) Any right A R -comodule is (A R,R)-injective. (2.b) Any left A R -comodule is (A R,R)-injective. (2.c) Any left A L -comodule is (A L,L)-injective. (2.d) Any right A L -comodule is (A L,L)-injective. (3.a) There exists a normalized left s-integral on A. That is, an element λ L(A ) such that λ (1 A ) = 1 R. (3.b) There exists a normalized left t-integral on A. That is, an element λ L( A) such that λ(1 A ) = 1 R. (3.c) There exists a normalized right s-integral on A. That is, an element ρ R( A) such that ρ(1 A ) = 1 L. (3.d) There exists a normalized right t-integral on A. That is, an element ρ R(A ) such that ρ (1 A ) = 1 L. (4.a) The monomorphism s R : R A splits as a right A R -comodule map. (4.b) The monomorphism t R : R A splits as a left A R -comodule map. (4.c) The monomorphism s L : L A splits as a left A L -comodule map. (4.d) The monomorphism t L : L A splits as a right A L -comodule map. Proof. (1.a) (2.a), (2.b) is proven in [10, 26.1]. (2.a) (4.a) ((2.b) (4.b)) The monomorphism s R (t R ) is split as a right (left) R-module map by π R hence it is split as a right (left) A R -comodule map. (4.a) (3.a) and (4.b) (3.b) The left inverse λ of s R in the category of right A R -comodules is a normalized s-integral on A R by its very definition. Similarly, the left inverse λ of t R in the category of left A R -comodules is a normalized t-integral on A R. (3.a) (3.b) If λ is a normalized s-integral on A R then λ S is a normalized t-integral on A R by Scholium (3.b) (1.a) In terms of the normalized t-integral λ on A R the required right inverse of the coproduct γ R is constructed as the map A R R A A, a b t R λ(as(b (1) )) b (2). It is checked to be an A R A R bicomodule map using that by Scholium 2.10, (4.b) and (1.c) we have t R λ(as(b (1) )) b (2) = a (1) s R π R [t R λ(a (2) S(b (1) )) b (2) ] for all a,b in A. (3.a) (3.d) follows from Scholium 2.10, (2.b). The remaining equivalences are proven by applying the above arguments to the Hopf algebroid A op cop. The proofs of Theorem 3.1 and 3.2 can be unified if one formulates them as equivalent statements on the forgetful functors from the category of A-modules,

135 INTEGRAL THEORY FOR HOPF ALGEBROIDS 579 and from the category of A L or A R -comodules, respectively, to the category of L- or R-modules as it is done in the case of Hopf algebras over commutative rings in [12]. We believe (together with the referee), however, that the above formulation in terms of algebra extensions and corings, respectively, is more appealing. 4. Frobenius Hopf Algebroids and Nondegenerate Integrals A left or right integral l in a Hopf algebra (H,,ɛ,S)over a commutative ring k is called nondegenerate [19] if the maps Hom k (H, k) H, φ (φ H) (l) and Hom k (H, k) H, φ (H φ) (l) are bijective. The notion of nondegenerate integrals is made relevant by the Larson Sweedler Theorem [19] stating that a free and finite-dimensional bialgebra over a principal ideal domain is a Hopf algebra if and only if there exists a nondegenerate left integral in H. The Larson Sweedler Theorem has been extended by Pareigis [27] to Hopf algebras over commutative rings with trivial Picard group. He proved also that a bialgebra over an arbitrary commutative ring k, which is a Frobenius k-algebra, is a Hopf algebra if and only if there exists a Frobenius functional ψ: H k satisfying (H ψ) = 1 H ψ( ). As a matter of fact, based on the results of [27] the following variant of [13, 3.2 Theorem 31] can be proven: THEOREM 4.1. The following properties of a Hopf algebra (H,,ɛ,S)over a commutative ring k are equivalent: (1) H is a Frobenius k-algebra. (2) There exists a nondegenerate left integral in H. (3) There exists a nondegenerate right integral in H. (4) There exists a nondegenerate left integral on H. That is, a Frobenius functional ψ: H k satisfying (H ψ) = 1 H ψ( ). (5) There exists a nondegenerate right integral on H. That is, a Frobenius functional ψ: H k satisfying (ψ H) = 1 H ψ( ). The main subject of the present section is the generalization of Theorem 4.1 to Hopf algebroids. The most important tool in the proof of Theorem 4.1 is the Fundamental Theorem for Hopf modules [19]. A very general form of it has been proven by Brzeziński [7, Theorem 5.6], see also [10, 28.19] in the framework of corings. It can be applied in our setting as follows.

136 580 GABRIELLA BÖHM Hopf modules over bialgebroids are examples of Doi Koppinen modules over algebras, studied in [8]. A left-left Hopf module over a left bialgebroid A L = (A,L,s L,t L,γ L,π L ) is a left comodule for the comonoid (A, γ L,π L ) in the category of left A-modules. That is, a pair (M, τ) where M is a left A-module, hence aleftl-module L M via s L. The pair ( L M,τ) is a left A L -comodule such that τ: M A L L M is a left A-module map to the module a (b m): = a (1) b a (2) m for a A, b m A L L M. The right right Hopf modules over a right bialgebroid A R are the left left Hopf modules over (A R ) op cop. It follows from [8, Proposition 4.1] that the left left Hopf modules over A L are the left comodules over the A-coring W := (A L L A, γ L L A, π L L A), (4.1) where the A A bimodule structure is given by a (b c) d: = a (1) b a (2) cd for a,d A, b c A L L A. The coring (4.1) was studied in [2]. It was shown to possess a group-like element 1 A 1 A A L L A and corresponding coinvariant subalgebra t L (L) in A. The coring (4.1) is Galois (w.r.t. the group-like element 1 A 1 A ) if and only if A L is a L -Hopf algebra in the sense of [30]. Since in a Hopf algebroid A = (A L, A R,S) the left bialgebroid A L is a L -Hopf algebra by Proposition 2.4, the A-coring (4.1) is Galois in this case. Denote the category of left left Hopf modules over A L (i.e. of left comodules over the coring (4.1)) by W M. The application of [7, Theorem 5.6] results that if A = (A L, A R,S) is a Hopf algebroid, such that the module L A is faithfully flat, then the functor G: W M M L, (M, τ) Coinv(M) L := {m M τ(m) = 1 A m A L L M} (4.2) (where the right L-module structure on Coinv(M) is given via t L ) and the induction functor F : M L W M, N L ( L A N L,γ L N L ) (4.3) (where the left A-module structure on L A N L is given by left multiplication in the first factor) are inverse equivalences. In the case of Hopf algebras H over commutative rings k, these arguments lead to the Fundamental Theorem only for faithfully flat Hopf algebras. The proof of the Fundamental Theorem in [19], however, does not rely on any assumption on the k-module structure of H. Since the Hopf algebroid structure is more restrictive than the L -Hopf algebra structure, one hopes to prove the Fundamental Theorem for Hopf algebroids also under milder assumptions using the whole strength of the Hopf algebroid structure.

137 INTEGRAL THEORY FOR HOPF ALGEBROIDS 581 THEOREM 4.2 (Fundamental Theorem for Hopf algebroids). Let A = (A L, A R,S) be a Hopf algebroid and W the A-coring (4.1). The functors G: W M M L in (4.2) and F : M L W M in (4.3) are inverse equivalences. Proof. We construct the natural isomorphisms α: F G W M and β: G F M L.Themap α M : L A Coinv(M) L M, a m a m is a left W-comodule map and natural in M. The isomorphism property is proven by constructing the inverse α 1 M : M L A Coinv(M) L, m m (1) 1 S(m (2) 1 ) m 0, where we used the standard notation τ(m) = m 1 m 0. It requires some work to check that α 1 M (m) belongs to L A Coinv(M) L. Let us introduce the right L- submodule X of A L L A L L M as { X := a i b i m i A L L A L L M i a i t L (l) b i m i = } a i b i s L (l) m i l L i i with L-module structure [ i a i b i m i ] l := i a i t R π R t L (l) b i m i, and the map ω: A L L A L L M M, a i b i m i S[s L π L (a i )b i ] m i. i i Making M a right L-module via t L, the restriction of ω becomes a right L-module map X M L. The image of the map ω (A L τ): A L L M M lies in Coinv(M), sinceforanya m A L L M we have τ ω (A L τ)(a m) = S(m (2) 1 )m 0 1 S[s L π L (a) m (1) 1 ] m 0 0 = s R π R (m 1 (2) ) S[s L π L (a) m 1 (1) ] m 0 = 1 A ω (A L τ)(a m). Since α 1 M =[L A ω (A L τ)] (γ R L M) τ, it follows that α 1 M (m) belongs to L A Coinv(M) L for all m M, as stated. The coinvariants of the left W-comodule L A N L are the elements of Coinv( L A N L ) { = a i n i L A N L a i n i = i i i s R π R (a i ) n i },

138 582 GABRIELLA BÖHM hence the map β N :Coinv( L A N L ) N, a i n i n i π L S(a i ) n i π L (a i ) i i i is a right L-module map and is natural in N. It is an isomorphism with inverse β 1 N : N Coinv(L A N L ), n 1 A n. An analogous result for right right Hopf modules over A R can be obtained by applying Theorem 4.2 to the Hopf algebroid A op cop. PROPOSITION 4.3. Let A = (A L, A R,S) be a Hopf algebroid and (M, τ) a left left Hopf module over A L.ThenCoinv(M) is a k-direct summand of M. Proof. The canonical inclusion Coinv(M) M is split by the k-module map E M : M Coinv(M), m S(m 1 ) m 0. (4.4) As the next step towards our goal, let us assume that A = (A L, A R,S)is a Hopf algebroid such that the module A R and hence by Lemma 2.6 also A L is finitely generated and projective. Under this assumption we are going to equip A with the structures of a left left Hopf module over A L and a right right Hopf module over A R. Let {b i } A and {β i } A be dual bases for the module A L.Aleft A L -comodule structure on A can be introduced via the L-module structure LA : l φ := φ S s L (l) and the left coaction for l L, φ A τ L : A A L L A, φ b i χ 1 (β i )φ. (4.5) i Similarly, a right A R -comodule structure on A can be introduced by the right R-module structure A R : φ r := φ s R (r) and the right coaction for r R, φ A τ R : A A R R A, φ i χ 1 (β i )φ S(b i ), (4.6) where χ: A A is the algebra anti-isomorphism (2.22). PROPOSITION 4.4. Let A = (A L, A R,S) be a Hopf algebroid such that the module A R is finitely generated and projective.

139 INTEGRAL THEORY FOR HOPF ALGEBROIDS 583 (1) Introduce the left A-module AA : a φ := φ S(a) for a A, φ A. Then ( A A,τ L ) whereτ L is the map (4.5) is a left left Hopf module over A L. (2) Introduce the right A-module A A : φ a: = φ a for a A, φ A. Then (A A,τ R) whereτ R is the map (4.6) is a right right Hopf module over A R. The coinvariants of both Hopf modules ( A A,τ L ) and (A A,τ R) are the elements of L(A ). Proof. (1)Wehavetoshowthatτ L is a left A-module map. That is, for all a A and φ A, b i χ 1 (β i )(φ S(a)) = a (1) b i (χ 1 (β i )φ ) S(a (2) ) (4.7) i i as elements of A L L A.Sinceforanyφ A and a A, χ 1 (β i ) S[s L φ (a (1) b i )a (2) ]=χ 1 (φ ) s L π L (a), i the following identity holds true in A L L A for all a A: a (1) b i χ 1 (β i ) S(a (2)) i = i,j t L β j (a (1)b i )b j χ 1 (β i ) S(a (2)) = i,j = j b j χ 1 (β i ) S[s L β j (a (1)b i )a (2) ] b j χ 1 (β j ) s L π L (a). (4.8) Since for all φ,ψ A and a A, (φ ψ ) a= (φ a (2) )(ψ a (1) ), (4.9) the identity (4.8) is equivalent to (4.7). (2) We have to show that τ R is a right A-module map. That is, for all a A and φ A, χ 1 (β i )(φ a) S(b i ) = (χ 1 (β i )φ ) a (1) S(b i )a (2) (4.10) i i

140 584 GABRIELLA BÖHM as elements of A R R A. Recall from the proof of Lemma 2.6 that the dual bases, {b i } Aand {β i} A for the module A L, and the dual bases, {k j } Aand {κj } A for A R, are related to each other by b i β i = k j(1) χ[s π R (k j(2) )κj ] as elements of A L L A. i j This implies that τ R (φ ) = j s π R (k j(2) )κj φ S(k j(1) ). The following identity holds true in A R R A for all a A: [s π R (k j(2) )κj ] a(1) S(k j(1) )a (2) j = j s π R (a (1) (2)k j(2) )κ j S(a(1) (1)k j(1) )a (2) = j = j = j = j s π R [s R π R (a (1) (2) )k j(2) ]κj S(a (2) (1)k j(1) )a (2) [s π R (k j(2) )κj ] s R π R (a (1) (2) (2) ) S(a (1) k j(1) )a (2) s π R (k j(2) )κj S(k j(1))s R π R (a) [s π R (k j(2) )κj ] t R π R (a) S(k j(1) ). (4.11) Here we used the identity (s (r)φ ) a= s (r) (φ a)for r R, φ A and a A, the property of the dual bases j k j κj a= j akj κj for all a A as elements of A R R A, the right analogue of the bialgebroid axiom (2.6) and the Hopf algebroid axioms (2.19) and (2.20). In view of (4.9) the identity (4.11) is equivalent to (4.10). In the cases of the Hopf modules ( A A,τ L ) and (A A,τ R) a projection onto the coinvariants is given by the map (4.4) and its right right version, respectively, both yielding E A : A Coinv(A ), φ χ 1 (β i )φ S 2 (b i ). (4.12) i Alefts-integral λ on A is a coinvariant, since it is an invariant of the left regular A -module and so for all a A, E A (λ )(a) = χ 1 (β i )(1 A)λ (S 2 (b i )a) i = λ [S 2 (t L β i (1 A)b i )a] =λ (a). On the other hand, for all a A, S(b i )(a β i ) = S[t L β i (a (1)) b i ]a (2) = s R π R (a), (4.13) i

141 INTEGRAL THEORY FOR HOPF ALGEBROIDS 585 hence for all φ A, E A (φ ) a = a (2) t R π R {[φ S 2 (b i )a (1) ] β i } i = i = i = i,j = i,j = j t R π R S 2 (b (2) i )a (2) t R π R {[φ S 2 (b (1) i )a (1) ] β i } S(b i(2) )(S 2 (b i(1) )a) (2) t R π R {[φ (S 2 (b i(1) )a) (1) ] β i } S(b i(2) ){[φ S 2 (t L β j (b i(1)) b j )a] β i } S(b i β j ){[φ S 2 (b j )a] β i } s R π R {[φ S 2 (b j )a] β j }=s R E A (φ )(a). That is, any coinvariant is an s-integral on A R. Here we used (4.12), the right analogue of (2.1), the identity t R π R S 2 = S s R π R, (2.20), the right analogue of (2.3), the identity γ R [(φ a) ψ ]=(φ a (1) ) ψ a (2), holding true for all a A, φ A and ψ A, the right L-linearity of the map (φ ) ψ : A L A L and (4.13). The application of Theorem 4.2 to the Hopf modules of Proposition 4.4 results in isomorphisms α L : L A L(A ) L A, a λ λ S(a) and (4.14) α R : R L(A ) A R A, λ a λ a (4.15) of left left Hopf modules over A L and of right right Hopf modules over A R,respectively. (The right L-module structure on L(A ) is given by λ l := λ s L (l) and the left R-module structure is given by r λ := λ t R (r) seethe explanation after (4.2).) COROLLARY 4.5. For a Hopf algebroid A = (A L, A R,S), such that any of the modules A R, R A, L A and A L is finitely generated and projective, there exist nonzero elements in all of L(A ), L( A), R( A) and R(A ). Proof. Suppose that the module A R (equivalently, by Proposition 2.6 the module A L ) is finitely generated and projective. It follows from Proposition 4.4 and Theorem 4.2 that the map (4.14) is an isomorphism, hence there exist nonzero elements in L(A ). For any element λ of L(A ), λ S is a (possibly zero) element of L( A) by Scholium Now we claim that it is excluded by the bijectivity of the map (4.14) that λ S = 0forallλ L(A ). For if so, then by the surjectivity of the

142 586 GABRIELLA BÖHM map (4.14) we have φ (1 A ) = 0forallφ A. But this is impossible, since π R (1 A ) = 1 R, by definition. It follows from Scholium 2.10, (3.b) and (4.b) that also R( A) and R(A ) must contain nonzero elements. The case when the module L A (equivalently, by Proposition 2.6 the module R A) is finitely generated and projective can be treated by applying the same arguments to the Hopf algebroid A op cop. Since none of the duals of a Hopf algebroid is known to be a Hopf algebroid, it does not follow from Theorem 4.2, however, that for a Hopf algebroid, in which the total algebra is finitely generated and projective as a module over the base algebra, also L(A) and R(A) contain nonzero elements. At the moment we do not know under what necessary conditions the existence of nonzero integrals in a Hopf algebroid follows. It is well known [27, Proposition 4] that the antipode of a finitely generated and projective Hopf algebra over a commutative ring is bijective. We do not know whether a result of the same strength holds true on Hopf algebroids. Our present understanding on this question is formulated in PROPOSITION 4.6. The following statements on a Hopf algebroid A = (A L, A R,S)are equivalent: (1) The antipode S is bijective and any of the modules L A, A L, A R and R A is finitely generated and projective. (2) There exists an invariant k x k λ k of the left A-module R A L(A ) R defined via left multiplication in the first factor with respect to A L, satisfying k λ k (x k) = 1 R. (The right R-module structure of L(A ) is defined by the restriction of the one of (A ) R,i.e.asλ r := λ ( t R (r)).) Proof. For any invariant k x k λ k of the left A-module R A L(A ) R and any element a A the identities S(a)x (1) k x (2) k λ k = k k ax (1) k S(x (2) k ) λ k = k k x (1) k ax (2) k λ k and x (1) k S(x (2) k )a λ k hold true as identities in R A R R A L(A ) R and in R A R R A L(A ) R, respectively. (2) (1) In terms of the invariant k x k λ k the inverse of the antipode is constructed explicitly as S 1 : A A, a k (λ k a) x k.

143 INTEGRAL THEORY FOR HOPF ALGEBROIDS 587 The dual bases {b i } Aand { β i } A for the module R A are introduced by the requirement that β i b i = λ k (S( )x(1) k ) x (2) k i k as elements of A R R A. Together with Lemma 2.6 this proves the implication (2) (1). (1) (2) If S is bijective then in the case of the Hopf algebroid A cop the isomorphism (4.14) takes the form α cop L : AL L L( A) A, a λ λ S 1 (a), where the left L-module structure on L( A) is defined by l λ := λ t L (l). In terms of k x k λ k := (α cop L ) 1 (π R ) the required invariant of R A L(A ) R is given by k x k λ k S 1. In any Hopf algebroid A = (A L, A R,S), in which the module A L is finitely generated and projective, the extensions s R : R A and t L : L op A satisfy the left depth two (or D2, for short) condition and the extensions t R : R op A and s L : L A satisfy the right D2 condition of [18]. If furthermore S is bijective then all the four extensions satisfy both the left and the right D2 conditions. This means [18, Lemma 3.7] in the case of s R : R A, for example, the existence of finite sets (the so called D2 quasi-bases) {d k } A R R A, {δ k } R End R ( R A R ), {f l } A R R A and {φ l } R End R ( R A R ) satisfying d k m A (δ k R A)(u) = u k and m A (A R φ l )(u) f l = u l for all elements u in A R R A,wheretheA A bimodule structure on A R R A is defined by left multiplication in the first factor and right multiplication in the second factor. The D2 quasi-bases for the extension s R : R A can be constructed in terms of the invariants i x i λ i : = α 1 L (π R) and j x j λ j := (αcop L ) 1 (π R ) via the requirements that d k δ k = x (1) i(1) S(x (2) i(1) ) [λ i S(x i(2) )] k i and φ l f l l = [x j(1) π L s R λ j S 1 ] j x j(2)(1) S(x j(2)(2) )

144 588 GABRIELLA BÖHM as elements of A R R A L L [ R End R ( R A R )] and of [ R End R ( R A R )] L L A R R A, respectively. (The L L bimodule structure on R End R ( R A R ) is given by l 1 l 2 = s L (l 1 )( )s L (l 2 ) for l 1,l 2 L, R End R ( R A R ).) The D2 property of the extensions t R : R op A, s L : L A and t L : L op A follows by applying these formulae to the Hopf algebroids A cop, A op cop and A op, respectively. The following theorem, characterizing Frobenius Hopf algebroids A = (A L, A R,S) that is, Hopf algebroids such that the extensions, given by the source and target maps of the bialgebroids A L and A R, are Frobenius extensions is the main result of this section. Recall that for a homomorphism s: R A of k-algebras the canonical R A bimodule R A A is a 1-cell in the additive bicategory of [k-algebras, bimodules, bimodule maps], possessing a right dual, the bimodule A A R.IfAis finitely generated and projective as a left R-module, then R A A possesses also a left dual, the bimodule A[ R Hom(A, R)] R defined as a φ r = φ( a)r for r R, a A, φ R Hom(A, R). A monomorphism of k-algebras s: R A is called a Frobenius extension if the module R A is finitely generated and projective and the left and right duals AA R and A[ R Hom(A, R)] R of the bimodule R A A are isomorphic. Equivalently, if A R is finitely generated and projective and the left and right duals RA A and R[Hom R (A, R)] A of the bimodule A A R are isomorphic. This property holds if and only if there exists a Frobenius system (ψ, i u i v i ),whereψ: A R is an R R bimodule map and i u i v i is an element of A R A such that s ψ(au i )v i = a = u i s ψ(v i a) for all a A. i i THEOREM 4.7. The following statements on a Hopf algebroid A = (A L, A R,S) are equivalent: (1.a) The map s R : R A is a Frobenius extension of k-algebras. (1.b) The map t R : R op A is a Frobenius extension of k-algebras. (1.c) The map s L : L A is a Frobenius extension of k-algebras. (1.d) The map t L : L op A is a Frobenius extension of k-algebras. (2.a) The module A R is finitely generated and projective and the module L(A ) L, defined by λ l := λ s L (l), is free of rank 1. (2.b) S is bijective, the module R A is finitely generated and projective and the module L L( A), defined by l λ := λ t L (l), is free of rank 1.

145 INTEGRAL THEORY FOR HOPF ALGEBROIDS 589 (2.c) The module L A is finitely generated and projective and the module R R( A), defined by r ρ := s R (r) ρ, is free of rank 1. (2.d) S is bijective, the module A L is finitely generated and projective and the module R(A ) R, defined by ρ r: = t R (r) ρ, is free of rank 1. (3.a) The module A R is finitely generated and projective and there exists an element λ L(A ) such that the map F : A A, a λ a (4.16) is bijective. (3.b) S is bijective, the module R A is finitely generated and projective and there exists an element λ L( A) such that the map A A, a λ ais bijective. (3.c) The module L A is finitely generated and projective and there exists an element ρ R( A) such that the map A A, a a ρ is bijective. (3.d) S is bijective, the module A L is finitely generated and projective and there exists an element ρ R(A ) such that the map A A, a a ρ is bijective. (4.a) There exists a left integral l L(A) such that the map F : A A, φ φ l (4.17) is bijective. (4.b) S is bijective and there exists a left integral l L(A) such that the map F : A A, φ φ l (4.18) is bijective. (4.c) There exists a right integral R(A) such that the map A A, φ φ is bijective. (4.d) S is bijective and there exists a right integral R(A) such that the map A A, φ φ is bijective. In particular, the integrals λ, λ, ρ and ρ on A satisfying the condition in (3.a), (3.b), (3.c) and (3.d), respectively, are Frobenius functionals themselves for the extensions s R : R A, t R : R op A, s L : L A and t L : L op A, respectively. What is more, under the equivalent conditions of the theorem the left integrals l L(A) satisfying the conditions in (4.a) and (4.b) can be chosen to be equal, that is, to be a nondegenerate left integral in A. Similarly, the right integrals R(A) satisfying the conditions in (4.c) and (4.d) can be chosen to be equal, that is to be a nondegenerate right integral in A. Proof. (4.a) (1.a) In terms of the left integral l in (4.a) define λ := F 1 (1 A ) A. We claim that λ is a left s-integral on A. The element l λ R L(A) L(A ) R is an invariant of the left A-module R A L(A ) R,

146 590 GABRIELLA BÖHM hence by Proposition 4.6 the antipode is bijective and the modules A R and R A are finitely generated and projective. Since for all φ A, φ λ = F 1 (φ 1 A ) = F 1 (s φ (1 A ) 1 A ) = s π (φ )λ, λ is an s-integral on A R, so in particular an R R bimodule map R A R R. Since for all a A, l (2) t R λ (S(a)l (1) ) = a, we have F 1 (a) = λ S(a)hence l (1) s R λ S(l (2) ) = 1 A. A Frobenius system for the extension s R : R A is provided by (λ,l (1) S(l (2) )). (1.a) (2.a) The module A R is finitely generated and projective by assumption. In terms of a Frobenius system (ψ, i u i v i ) for the extension s R : R A one constructs an isomorphism of right L-modules as [ ] κ: L(A ) L, λ π L s R λ (u i )v i (4.19) i with inverse κ 1 : L L(A ), l E A (ψ s L (l)), (4.20) where E A is the map (4.12). The right L-linearity of κ follows from the property of the Frobenius system (ψ, i u i v i ) that i au i v i = i u i v i a for all a A, the bialgebroid axiom (2.5), and left R-linearity of the map λ : R A R and the right L-linearity of π L : L A L. The maps κ and κ 1 are mutual inverses as κ 1 κ(λ ) = [χ 1 (β j )ψ] s L π L (s R λ (u i )v i )S 2 (b j ) i,j = i,j = i,j [χ 1 (β j )ψ] S2 (b (2) j )t R π R [t R π R S(s R λ (u i )v i )S 2 (b (1) j )] [χ 1 (β j )ψ] S2 (b (2) j )s L π L [S(b (1) j )s R λ (u i )v i ]=λ, (4.21) where in the first step we used (4.9), in the second step the fact that by Proposition 2.3 we have s L π L = t R π R S, then the right analogue of (2.5) and finally in the last step the identity in R L(A ) A R : [χ 1 (β j )ψ] S2 (b (2) j ) S(b (1) j )s R λ (u i )v i i,j = α 1 R ( ) ψ s R λ (u i )v i = λ 1 A, i

147 INTEGRAL THEORY FOR HOPF ALGEBROIDS 591 which follows from the explicit form of the inverse of the map (4.15). In a similar way, also κ κ 1 (l) = i,j = i,j = i,j = i,j = i,j π L [s R (χ 1 (β j )ψ)(s L(l)S 2 (b j )u i )v i ] π L [s R (χ 1 (β j ) ψ)(s L(l)u i )v i S 2 (b j )] π L [s R (χ 1 (β j ) ψ)(s L(l)u i )v i t L π L S 2 (b j )] π L [s R (χ 1 (β j ) ψ)(s L(l) t L π L S 2 (b j )u i )v i ] π L {s R [(χ 1 (β j ) t L π L S 2 (b j ))ψ](s L (l)u i )v i } = l, where in the last step we used that j χ 1 (β j ) t L π L S 2 (b j ) = χ 1 ( j βj t π L (b j )) = π R. (2.a) (3.a) If κ: L(A ) L L is an isomorphism of L-modules then π R s L κ: R L(A ) R is an isomorphism of R-modules. Introduce the cyclic and separating generator λ := κ 1 (1 L ) for the module L(A ) L.ThemapF in (4.16) is equal to α R (κ 1 π L t R A R ) whereα R is the isomorphism (4.15) hence bijective. (3.a) (4.a), (4.b) A Frobenius system for the extension s R : R A is given in terms of the dual bases {b i } Aand {βi } A for the module A R as (λ, i b i F 1 (βi )). The element l := i b i t L π L F 1 (βi ) isaleftintegralina. Usingthe identities [ ] λ l = s R λ b i t L π L F 1 (βi ) i = t L π L [ i s R λ (b i ) F 1 (β i ) ] = 1 A, l (1) S(l (2) ) = i = i = i b i s R λ [F 1 (β i )l(1) ] S(l (2) ) b i S[l (2) t R λ (l (1) )]F 1 (β i ) b i F 1 (β i ) one checks that the inverse of the map F in (4.17) is given by F S. This implies, in particular, that S is bijective.

148 592 GABRIELLA BÖHM The inverse of the map F in (4.18) defined in terms of the same left integral l isthemap A A, a λ S S 1 (a). (1.a) (1.d) The datum (ψ, i u i v i ) is a Frobenius system for the extension s R : R A if and only if (π L s R ψ, i u i v i ) is a Frobenius system for t L : L op A, whereπ L s R : R L op was claimed to be an isomorphism of k-algebras in part (1) of Proposition 2.3. (1.a) (1.c) We have already seen that (1.a) (3.a) S is bijective. If the datum (ψ, i u i v i ) is a Frobenius system for the extension s R : R A then (π L s R ψ S 1,S(v i ) S(u i )) is a Frobenius system for s L : L A. (4.c) (1.c) (2.c) (3.c) (4.c), (1.c) (1.b) and (1.c) (1.a) follow by applying (4.a) (1.a) (2.a) (3.a) (4.a), (1.a) (1.d) and (1.a) (1.c) to the Hopf algebroid A op cop. (1.b) (2.b) (3.b) (4.b) (1.b) We have seen that (1.b) (1.c) S is bijective. Hence we can apply (1.a) (2.a) (3.a) (4.a) (1.a) to the Hopf algebroid A cop. (1.d) (2.d) (3.d) (4.d) (1.d) follows by applying (1.b) (2.b) (3.b) (4.b) (1.b) to the Hopf algebroid A op It is proven in [5, Theorem 5.17] that under the equivalent conditions of Theorem 4.7 the duals, A, A, A and A of the Hopf algebroid A, possess (anti-) isomorphic Hopf algebroid structures. The Hopf algebroids, satisfying the equivalent conditions of Theorem 4.7, provide examples of distributive Frobenius double algebras [37]. (Notice that the integrals, which we call nondegenerate, are called Frobenius integrals in [37].) Our result naturally raises the question, under what conditions on the base algebra the equivalent conditions of Theorem 4.7 hold true. That is, what is the generalization of Pareigis condition the triviality of the Picard group of the commutative base ring of a Hopf algebra to the noncommutative base algebra of a Hopf algebroid. We are going to return to this problem in a different publication. cop. 5. The Quasi-Frobenius Property It is known [27, Theorem added in proof], that any finitely generated projective Hopf algebra over a commutative ring k is (both a left and a right) quasi-frobenius extension of k in the sense of [23]. In this section we examine in what Hopf algebroids the total algebra is (a left or a right) quasi-frobenius extension of the base algebra. The quasi-frobenius property of an extension s: R A of k-algebras has been introduced by Müller [23] as a weakening of the Frobenius property (see the paragraph preceding Theorem 4.7). The extension s: R A is left quasi-frobenius (or left QF, for short) if the module R A is finitely generated and projective (hence the

149 INTEGRAL THEORY FOR HOPF ALGEBROIDS 593 bimodule R A A possesses both a right dual A A R and a left dual A [ R Hom(A, R)] R ) and the bimodule A A R is a direct summand in a finite direct sum of copies of A[ R Hom(A, R)] R. The extension s: R A is right QF if s, considered as a map R op A op,is a left QF extension. That is, if the module A R is finitely generated and projective and the left dual bimodule R A A is a direct summand in a finite direct sum of copies of the right dual bimodule R [Hom R (A, R)] A. To our knowledge it is not known whether the notions of left and right QF extensions are equivalent (except in particular cases, such as central extensions, where the answer turns out to be affirmative [29]; and Frobenius extensions, which are also both left and right QF [23]). A powerful characterization of a Frobenius extension s: R A is the existence of a Frobenius system see the paragraph preceding Theorem 4.7. In the following lemma a generalization to quasi-frobenius extensions is introduced: LEMMA 5.1. (1) An algebra extension s: R A is left QF if and only if the module R A is finitely generated and projective and there exist finite sets {ψ k } R Hom R (A, R) and { i uk i vk i } A Asatisfying R u k i s ψ k (vi k ) = 1 A i,k and i,k au k i vk i = u k i vk i a for all a A. The datum {ψ k, i uk i vi k } is called a left QF-system for the extension s: R A. (2) An algebra extension s: R A is right QF if and only if the module A R is finitely generated and projective and there exist finite sets {ψ k } R Hom R (A, R) and { i uk i vk i } A Asatisfying R s ψ k (u k i )vk i = 1 A i,k and i,k au k i vk i = u k i vk i a for all a A. The datum {ψ k, i uk i vi k } is called a right QF-system for the extension s: R A. Proof. Let us spell out the proof in the case (1). Suppose that there exists a left QF system {ψ k, i uk i vi k} for the extension s: R A. The bimodule AA R

150 594 GABRIELLA BÖHM is a direct summand in a finite direct sum of copies of A [ R Hom(A, R)] R by the existence of A R bimodule maps k : R Hom(A, R) A, φ i u k i s φ(v k i ) and k : A RHom(A, R), a ψ k ( a) satisfying k k k = A. Conversely, in terms of the A R bimodule maps { k : R Hom(A, R) A} and { k : A RHom(A, R)}, satisfying k k k = A, and the dual bases, {b j } Aand {β j } R Hom(A, R) for the module R A, a left QF system can be constructed as ψ k := k (1 A) R Hom R (A, R) and i u k i vk i := j k (β j ) b j A R A. Lemma 5.1 implies, in particular, that for a left/right QF extension R A, A is finitely generated and projective also as a right/left R-module. THEOREM 5.2. The following properties of a Hopf algebroid A = (A L, A R,S) are equivalent: (1.a) s R : R A is a left QF extension. (1.b) t L : L op A is a left QF extension. (1.c) The modules A R and L(A ) L defined by λ l := λ s L (l) are finitely generated and projective. (1.d) The module A R is finitely generated and projective and the module L(A ) L is flat. (1.e) The module A R is finitely generated and projective and the invariants of the left A-module L A L(A ) L defined via left multiplication in the first factor with respect to A L are the elements of L L(A) L(A ) L. (1.f) There exist finite sets {l k } L(A) and {λ k } L(A ) satisfying k λ k S(l k ) = 1 R. (1.g) The left A-module A A definedbya φ := φ S(a) is finitely generated and projective with generator set {λ k } L(A ). The following properties of A are also equivalent: (2.a) s L : L A is a right QF extension. (2.b) t R : R op A is a right QF extension. (2.c) The modules L A and R R( A) defined by r ρ := s R (r) ρ are finitely generated and projective. (2.d) The module L A is finitely generated and projective and the module R R( A) is flat.

151 INTEGRAL THEORY FOR HOPF ALGEBROIDS 595 (2.e) The module L A is finitely generated and projective and the invariants of the right A-module R R( A) A R defined via right multiplication in the second factor with respect to A R are the elements of R R( A) R(A) R. (2.f) There exist finite sets { k } R(A) and { ρ k } R( A) satisfying k ρ k S( k ) = 1 L. (2.g) The right A-module A A defined by φ a := S(a) φ is finitely generated and projective with generator set { ρ k } R( A). If furthermore the antipode is bijective, then the conditions (1.a) (1.g) and (2.a) (2.g) are equivalent to each other and also to (1.h) The left A-module on A defined by φ a := φ a is finitely generated and projective with generator set {l k } L(A). (2.h) The right A -module on A definedbya φ := a φ is finitely generated and projective with generator set { k } R(A). Proof. (1.a) (1.b) It follows from part (1) of Proposition 2.3 that the module A L is finitely generated and projective if and only if R A is, and the datum {ψ k, i uk i v k i } is a left QF system for the extension s R: R A if and only if {π L s R ψ k, i uk i vk i } is a left QF system for t L: L op A. (1.a) (1.c) The module A R is finitely generated and projective by Lemma 5.1. In terms of the left QF system, {ψ k, i uk i v k i } for the extension s R: R A, the dual bases for the module L(A ) L are given with the help of the map (4.12) as {E A (ψ k )} L(A ) and {κ k := π L [ i s R (u k i )vk i ]} Hom L(L(A ) L,L). The right L-linearity of the maps κ k : L(A ) L is checked similarly to the right L-linearity of the map (4.19). Notice that for any R R bimodule map ψ: R A R R we have E A (ψ) s L (l) = j = j = j = j = j [χ 1 (β j )ψ] s L(l)S 2 (b j ) [χ 1 (t π L t R π R t L (l) β j )ψ] S2 (b j ) [χ 1 (β j )t π R t L (l) ψ] S 2 (b j ) [χ 1 (β j )s π R t L (l) ψ] S 2 (b j ) [χ 1 (s π L t R π R t L (l) β j )ψ] S2 (b j ) = E A (ψ s L (l)) for all l L, where in the first step we used (4.12) and (4.9), in the second step the property of the dual bases {b j } A and {β j } A that j βj s L (l)b j = j t (l)β j b j for all l L as elements of L A A L, in the third step the identity

152 596 GABRIELLA BÖHM χ 1 t = t π R s L, in the fourth step the fact that by the left R-linearity of ψ we have t (r)ψ = s (r)ψ for all r R, inthefifthstepχ 1 s = s π R s L, and finally j βj b j s L (l) = j s (l)β j b j, holding true for all l L as an identity in L A A L. The dual basis property of the sets {E A (ψ k )} and {κ k } is verified by the property that i,k E A (ψ k s L κ k (λ )) = λ for all λ L(A ), which is checked similarly to (4.21). (1.c) (1.d) is a standard result. (1.d) (1.e) If the module A R equivalently, by Lemma 2.6 the module A L is finitely generated and projective then the invariants of any left A-module M with respect to A L are the elements of the kernel of the map ( ) ζ M : M L A M L, m β i b i m π L m, i where the right L module M L is defined via t L,andthesets{b i } A and {β i } A are dual bases for the module A L. The map ζ A, corresponding to the left regular A-module, is a left L-module map L A L A L A L and ζl A L(A ) L = ζ A L(A ) L. Since tensoring with L(A ) L is an exact functor by assumption, it preserves the kernels, that is the invariants in this case. (1.e) (1.f) With the help of the map (4.14) introduce l k λ k := α 1 L (π R) Inv( L A L(A ) L ) L L(A) L(A ) L. k It satisfies k λ k S(l k) = α L α 1 L (π R)(1 A ) = 1 R. (1.f) (1.a) In terms of the sets {l k } L(A) and {λ k } L(A ) aleftqf system for the extension s R : R A can be constructed as {λ k,l(1) k S(l (2) k )}. The module A R is finitely generated and projective since there exist dual bases {b i } Aand {βi } A defined by i b i βi = k l(1) k λ k [S(l(2) k ) ], as elements of A R R A. The module A L is finitely generated and projective by Lemma 2.6, hence so is R A. (1.f) (1.g) In terms of the sets {l k } L(A) and {λ k } L(A ) the dual bases for the module A A are given by {λ k } L(A ) and { l k } AHom( A A,A). (1.g) (1.f) In terms of the dual bases {λ k } L(A ) and { k } AHom( A A,A)one defines the required left integrals l k := k (π R ) in A. The equivalence of the conditions (2.a) (2.g) follows by applying the above results to the Hopf algebroid A op cop. Now assume that S is bijective. Then (1.f) (2.f) follows from Scholium (1.f) (1.h) Scholium 2.8, (1.b) and Scholium 2.10, (3.c) can be used to show that in terms of the sets {l k } L(A) and {λ k } L(A ) the dual bases for the

153 INTEGRAL THEORY FOR HOPF ALGEBROIDS 597 left A-module on A are given by {l k } L(A) and {λ k S S 1 ( )} AHom(A, A). (1.h) (1.f) Let {l k } L(A) and {χ k } AHom(A, A) be dual bases for the left A-module A. Since for all a A we have k l(1) k s R χ k (a)(l (2) ) = a, the module A R, and hence by Proposition 2.6 also R A, is finitely generated and projective. For any value of the index k the element χ k (1 A ) is an invariant of the left regular A-module, hence a t-integral on A R. By Scholium 2.10 the elements λ k := χ k(1 A ) S 1 are s-integrals on A R, satisfying [ ] λ k S(l k) = π R χ k (1 A ) l k = 1 R. k k (2.f) (2.h) follows by applying (1.f) (1.h) to the Hopf algebroid A op cop. If the antipode of a Hopf algebroid A = (A L, A R,S)is bijective then the application of Theorem 5.2 to the Hopf algebroid A op results in equivalent conditions under which the extensions s R : R A and t L : L op A are right QF, and s L : L A and t R : R op A are left QF. In order to show that in contrast to Hopf algebras over commutative rings not any finitely generated projective Hopf algebroid is quasi-frobenius, let us give here an example (with bijective antipode) such that the total algebra is finitely generated and projective as a module over the base algebra (in all the four senses listed in (2.16)) and the total algebra is neither a left nor a right QF extension of the base algebra. The example is taken from [21, Example 3.1] where it is shown that for any algebra B over a commutative ring k the k-algebra A := B k B op has a left bialgebroid structure, A L, over the base B with structural maps s L : B A, b b 1 B, t L : B op A, b 1 B b, γ L : A A B B A, b 1 b 2 (b 1 1 B ) (1 B b 2 ), π L : A B, b 1 b 2 b 1 b 2. (5.1) The bialgebroid A L satisfies the Hopf algebroid axioms of [21] with the involutive antipode S, equal to the flip map S: B k Bop B op k B, b 1 b 2 b 2 b 1. (5.2) The reader may check that A has a Hopf algebroid structure also in the sense of this paper with left bialgebroid structure (5.1), antipode (5.2) and right bialgebroid structure A R = (A, B op,s s L,S t L,(S S) γ op L S,π L S). If B is finitely generated and projective as a k-module then all modules A Bop, B op A, A B and B A are finitely generated and projective, and vice versa. What is more, we have

154 598 GABRIELLA BÖHM LEMMA 5.3. Let B be an algebra over the commutative ring k with trivial center. The following statements are equivalent: (1) The extension k B is left QF. (2) The extension k B is right QF. (3) The extension B B k Bop, b b 1 B is left QF. (4) The extension B B k Bop, b b 1 B is right QF. The equivalence (1) (2) is proven in [29] and the rest can be proven using the techniques of quasi-frobenius systems. In view of Lemma 5.3 it is easy to construct a finitely generated projective Hopf algebroid which is not QF. Let us choose, for example, B to be the algebra of n n upper triangle matrices with entries in the commutative ring k. ThenB has trivial center and it is neither a left nor a right QF extension of k, hence A = B k B op is neither a left nor a right QF extension of B. Acknowledgements I am grateful to the referee for the careful study of the paper and for pointing out an error in the earlier version. His or her constructive comments lead to a significant improvement of the paper. This work was supported by the Hungarian Scientific Research Fund OTKA T , T , FKFP 0043/2001 and the Bolyai János Fellowship. References 1. Böhm, G.: An alternative notion of Hopf algebroid, In: S. Caenepeel and F. Van Oystaeyen (eds), Hopf Algebras in Non-Commutative Geometry and Physics, Marcel Dekker, New York, 2004, pp Böhm, G.: Internal bialgebroids, entwining structures and corings, In: Contemp. Math. 376, Amer. Math. Soc., Providence, 2005, pp Böhm, G., Nill, F. and Szlachányi, K.: Weak Hopf algebras I: Integral theory and C -structure, J. Algebra 221 (1999), Böhm, G. and Szlachányi, K.: A coassociative C -quantum group with non-integral dimensions, Lett. Math. Phys. 35 (1996), Böhm, G. and Szlachányi, K.: Hopf algebroids with bijective antipodes: Axioms, integrals and duals, J. Algebra 274 (2004), Böhm, G. and Szlachányi, K.: Hopf algebroid symmetry of Frobenius extensions of depth 2, Comm. Algebra 32 (2004), Brzeziński, T.: The structure of corings, Algebra Represent. Theory 5 (2002), Brzeziński, T., Caenepeel, S. and Militaru, G.: Doi-Koppinen modules for quantum-groupoids, J. Pure Appl. Algebra 175 (2002), Brzeziński, T. and Militaru, G.: Bialgebroids, R -bialgebras and duality, J. Algebra 251 (2002), Brzeziński, T. and Wisbauer, R.: Corings and Comodules, London Math. Soc. Lecture Note Ser. 309, Cambridge Univ. Press, Bulacu, D. and Caenepeel, S.: Integrals for (dual) quasi-hopf algebras. Applications, J. Algebra 266 (2003),

155 INTEGRAL THEORY FOR HOPF ALGEBROIDS Caenepeel, S. and Militaru, G.: Maschke functors, semisimple functors and separable functors of the second kind, J. Pure Appl. Algebra 178 (2003), Caenepeel, S., Militaru, G. and Zhu, S.: Frobenius and separable functors for generalized module categories and non-linear equations, In: Lecture Notes in Math. 1787, Springer, New York, Drinfeld, V. G.: Quasi-Hopf algebras, Leningrad J. Math. 1 (1990), Hattori, A.: Semisimple algebras over a commutative ring, J. Math. Soc. Japan 15 (1963), Hausser, F. and Nill, F.: Integral theory for quasi-hopf algebras, arxiv:math.qa/ Hirata, K. and Sugano, K.: On semisimple extensions and separable extensions over noncommutative rings, J. Math. Soc. Japan 18 (1966), Kadison, L. and Szlachányi, K.: Bialgebroid actions on depth two extensions and duality, Adv. Math. 179 (2003), Larson, R. G. and Sweedler, M. E.: An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), Lomp, C.: Integrals in Hopf algebras over rings, arxiv:math.ra/ , Comm. Algebra 32 (2004), Lu, J.-H.: Hopf algebroids and quantum groupoids, Internat. J. Math. 7 (1996), Maschke, H.: Über den arithmetischen Character der Coefficienten der Substitutionen endlicher linearen Substitutiongruppen, Math. Ann. 50 (1898), Müller, B.: Quasi-Frobenius Erweiterungen, Math. Zeit. 85 (1964), Năstăsescu, C., Van den Bergh, M. and Van Oystaeyen, F.: Separable functors applied to graded rings, J. Algebra 123 (1989), Panaite, F.: A Maschke type theorem for quasi-hopf algebras, In: S. Caenepeel and A. Verschoren (eds), Rings, Hopf Algebras and Brauer Groups, Marcel Dekker, New York, Panaite, F. and Van Oystaeyen, F.: Existence of integrals for finite-dimensional quasi-hopf algebras, Bull. Belg. Math. Soc. Simon Steven 7(2) (2000), Pareigis, B.: When Hopf algebras are Frobenius algebras, J. Algebra 18 (1971), Pierce, R. S.: Associative Algebras, Springer, New York, Rosenberg, Chase, as referred to in [23]. 30. Schauenburg, P.: Duals and doubles of quantum groupoids ( R -Hopf Algebras), In: Contemp. Math. 267, Amer. Math. Soc., Providence, 2000, pp Schauenburg, P.: Weak Hopf algebras and quantum groupoids, Banach Center Publications 61 (2003), Sweedler, M. E.: Hopf Algebras, Benjamin, New York, Sweedler, M. E.: Integrals for Hopf algebras, Ann. Math. 89 (1969), Szlachányi, K.: Finite quantum groupoids and inclusions of finite type, Fields Institute Commun. 30 (2001), Szlachányi, K.: Galois actions by finite quantum groupoids, In: L. Vainerman (ed.), Locally Compact Quantum Groups and Groupoids, IRMA Lect. Math. Theoret. Phys. 2, de Guyter, Berlin, Szlachányi, K.: The monoidal Eilenberg Moore construction and bialgebroids, J. Pure Appl. Algebra 182 (2003), Szlachányi, K.: The double algebraic viewpoint of finite quantum groupoids, J. Algebra 280 (2004), Takeuchi, M.: Groups of algebras over A Ā, J. Math. Soc. Japan 29 (1977), Takeuchi, M.: Morita theory Formal ring laws and monoidal equivalences of categories of bimodules, J. Math. Soc. Japan 39 (1987), Vecsernyés, P.: Larson Sweedler theorem and the role of grouplike elements in weak Hopf algebras, J. Algebra 270 (2003), Xu, P.: Quantum groupoids, Comm. Math. Phys. 216 (2001),

156 Algebr Represent Theor DOI /s Erratum to: Integral Theory for Hopf Algebroids [Algebra Represent. Theory (2005) 8: ] Gabriella Böhm Springer Science + Business Media B.V Abstract Because of similar errors in Lemma 2.6 and Theorem 4.2, we need to add stronger assumptions in several statements. At several points in the paper we repeated essentially the same incorrect step, thanks to which the to-be-dual-bases in the proof of Lemma 2.6 (1.a) (1.b) and (2.a) (2.b), and also the map [ L A ω (A L τ)] (γ R L M) τ in the proof of Theorem 4.2, are ill-defined. As a consequence of these errors, the published proofs of Lemma 2.6, Theorem 4.2 and Proposition 4.4(2) are not correct. Since in this way also some unjustified claims were used to prove them, Corollary 4.5, Proposition 4.6, some considerations on page 587, Theorems 4.7 and 5.2 become valid only by adding some finitely generated projectivity assumptions. Although the obtained results are somewhat weaker than their (eventually incorrect) versions in the original paper, the most interesting cases are still covered. Below we go through the necessary corrections. In Lemma 2.6, only the equivalences (1.a) (2.b) and (1.b) (2.a) are justified, for a Hopf algebroid with a bijective antipode. Theorem 4.2 is replaced by Theorem 1 below. In its formulation Sweedler s index notation τ(m) = m 1 m 0 (with implicit summation) is used, for the left coaction τ : M A L L M of the constituent left L-bialgebroid A L in a Hopf algebroid A, on a left A L -comodule M and m M. The online version of the original article can be found at Presented by Susan Montgomery. G. Böhm (B) Research Institute for Particle and Nuclear Physics, 1525 Budapest 114, P.O.B. 49, Budapest, Hungary G.Bohm@rmki.kfki.hu

157 G. Böhm Theorem 1 Let A = (A L, A R, S) be a Hopf algebroid and W be the A-coring (4.1). Assume that the kernel of the maps M A L L M, m ( m 1 m 0 ) (1A m) (1) is preserved by the functor L A :M L M L,foranyM W M (e.g. L Aisaflat module). Then the functors (4.2) and (4.3) are inverse equivalences. Proof We only need to check that α 1 M (m) = m 1 (1) S(m 1 (2) ) m 0 belongs to L A Coinv(M) L, the rest of the proof in the paper is valid. By the assumption that the kernel of Eq. 1 is preserved by the functor L A :M L M L, we need to show only that m 1 (1) ( S ( m 1 (2) ) m 0 ) 1 ( S ( m 1 (2) ) m 0 ) 0 = m 1 (1) 1 A S ( m 1 (2) ) m 0, (2) as elements of L A A L L M L,forallm M. Compose the well defined map A R R A L L A A R R A, a b c a S(b)c with the equal maps (γ R L A) γ L = (A R γ L ) γ R : A A R R A L L A (cf. (2.18)) in order to conclude that, for any a A, a (1) (1) S ( a (1) (2) ) a (2) = a (1) S ( a (2) (1)) a (2) (2) = a (1) s R π R ( a (2) ) = a 1 A. (3) In Eq. 3, in the second equality (2.20) was used and the last equality follows by the counitality of γ R.UsingtheleftA-linearity of the coaction τ : M W A M = A L L M, anti-comultiplicativity of the antipode (cf. Proposition 2.3(2)), coassociativity of τ and γ R and finally Eq. 3, the left hand side of Eq. 2 is computed to be equal to m (1) 2 S ( (2) m ) 2 (1) m 1 S ( (2) m ) 2 (2) m 0 ( = m (1) (2) 2 S m (2)) ( (2) 2 m 1 S m (1)) 2 m 0 ( ) ( = m 1 (1)(1) (1) S m 1 (1)(2) m 1 (2) S m 1 (1)(1) (2)) m 0 = m 1 (1) 1 A S ( m 1 (2) ) m 0. Thus it follows that α 1 M (m) belongs to L A Coinv(M) L for all m M, as stated. In the arxiv version of [1] a more restrictive notion of a comodule of a Hopf algebroid is studied, cf. [1, arxiv version, Definition 2.20]. The total algebra A of any Hopf algebroid A can be regarded as a monoid in the monoidal category of A-comodules in this more restrictive sense. In this setting, the category of A-modules in the category of A-comodules, and the category of modules for the base algebra L of A, were proven to be equivalent without any further (equalizer preserving) assumption, see [1, Theorem 3.27 and Remark 3.28]. That is, in the arxiv version of [1] a weaker statement is proven under weaker assumptions, compared to Theorem 1 above. Since the original version of Lemma 2.6 turned out to be incorrect, so is the proof of Proposition 4.4(2) built on it. In order to replace Proposition 4.4(2) by a justified

158 Corrigendum to Integral Theory for Hopf Algebroids claim, take a Hopf algebroid A, such that the module R A is finitely generated and projective, and let {k j } A and { κ j } A be dual bases for it. Alternatively to (4.6), arighta R -comodule structure on A can be introduced by the right R-action and the right coaction A R : φ r : = φ s R (r) for r R, φ A τ R : A A R R A φ i χ 1 (π L t R κ j S)φ k j, (4) where χ : A A is the algebra anti-isomorphism (2.22). Note that if both modules A L and R A are finitely generated and projective, then the dual bases {b i } A, {β i } A for A L,and{k j } A,{ κ j } A for R A, are related via the identity β i S(b i) = π L t R κ j S k j i j in A R R A. Hence in this case the coaction (4) is equal to (4.6). Proposition 4.4 is replaced by the following. Proposition 2 Let A = (A L, A R, S) be a Hopf algebroid. (1) Introduce the left A-module AA : a φ : = φ S(a) for a A, φ A. If the module A L is finitely generated and projective, then ( A A,τ L ) where τ L is the map (4.5) is a left-left Hopf module over A L. (2) Introduce the right A-module A A : φ a: = φ a fora A, φ A. If the module R A is finitely generated and projective, then (A A,τ R ) where τ R is the map (4) is a right-right Hopf module over A R. The coinvariants of both Hopf modules ( A A,τ L ) and (A A,τ R ) are the elements of L(A ). Proof Part (1) of Proposition 2 is proven in the paper, and part (2) follows by similar steps. Since Theorem 1 and Proposition 2 contain stronger assumptions than their original counterparts, we need stronger assumptions than those in the paper to conclude that the maps (4.14) and (4.15) are isomorphisms. Applying Theorem 1 to the Hopf modules in Proposition 2, the following are obtained. The map (4.14) is an isomorphism of left-left Hopf modules over A L provided that the module A L is finitely generated and projective, and the kernel of the map L A A L L A A L L A L, a φ a τ L (φ ) a 1 A φ is equal to L A L(A ) L. The map (4.14) is an isomorphism, in particular, if both modules A L and L A are finitely generated and projective. The map (4.15) is an

159 G. Böhm isomorphism of right-right Hopf modules over A R provided that the module R A is finitely generated and projective, and the kernel of R A A R R A R R A A R, φ a τ R (φ ) a φ 1 A a is equal to R L(A ) A R. The map (4.15) is an isomorphism, in particular, if both modules R A and A R are finitely generated and projective. The proofs of Corollary 4.5, Proposition 4.6, considerations about depth 2 properties on page 587, Theorems 4.7 and 5.2 all resort to the bijectivity of (4.14) and/or (4.15). Hence, as it is explained above, they are valid only under more restrictive assumptions. In each case, it has to be added to the hypothes is that A = (A L, A R, S) is a Hopf algebroid such that all of the modules A R, R A, L A and A L are finitely generated and projective. Accordingly, all claims about finitely generated projectivity of these modules need to be removed in the statements (together with their verifications in the proofs). Based on the corrected form of Theorem 4.7, we obtain the following generalization of Theorem 4.1. Corollary 3 For any Hopf algebroid A = (A L, A R, S), the following assertions are equivalent. (1.a) (1.b) (2.a) (2.b) (2.c) (2.d) (3.a) (3.b) Both maps s R : R A and t R : R op A are Frobenius extensions of k-algebras. Both maps s L : L A and t L : L op A are Frobenius extensions of k-algebras. The module A R is finitely generated and projective and there exists an element λ L(A ) such that the map F : A A, a λ a is bijective. S is bijective, the module R A is finitely generated and projective and there exists an element λ L( A) such that the map A A, a λ aisbijective. The module L A is finitely generated and projective and there exists an element ρ R( A) such that the map A A, a a ρ is bijective. S is bijective, the module A L is finitely generated and projective and there exists an element ρ R(A ) such that the map A A, a a ρ is bijective. There exists a non-degenerate left integral, that is an element l L(A) such that both maps F : A A, φ φ land F : A A, φ φ lare bijective. There exists a non-degenerate right integral, that is, an element R(A) such that both maps A A, φ φ and A A, φ φ are bijective. Proof (1.a) (1.b): This follows by the same reasoning used to prove (1.a) (1.d) and (1.b) (1.c) in Theorem 4.7. (1.a) (2.a): Sinces R : R A is a Frobenius extension by assumption, the modules A R and R A (hence also A L ) are finitely generated and projective by definition. Similarly, since t R : R op A is a Frobenius extension, the modules R A and L A are finitely generated and projective. Thus this implication follows by (the corrected form of) Theorem 4.7 (1.a) (3.a).

160 Corrigendum to Integral Theory for Hopf Algebroids (2.a) (3.a) and S is bijective: This is proven by repeating the same steps used to prove (3.a) (4.a) and (4.b) in Theorem 4.7. (3.a) (1.a) and S is bijective: Putting λ := F 1 (1 A ),themapa (λ a) l is checked to be the inverse of S. For any r R, F (rλ ( )) = t R (r) = F (λ s R (r)). So by the bijectivity of F, we conclude that λ is a left R-module map R A R. Therefore the module R A is finitely generated and projective with dual basis λ (S( )l (1) ) l (2) A R R A. Since the antipode is bijective, the module A L is finitely generated and projective by Lemma 2.6 (2.b) (1.a). Applying the same reasoning to the Hopf algebroid A cop, we conclude that by the bijectivity of F also the modules A R and L A are finitely generated and projective. Hence the claim follows by (the corrected form of) Theorem 4.7, (4.a) (1.a) and (1.b). (1.b) (2.c) (3.b): This follows by applying (1.a) (2.a) (3.a) to the Hopf algebroid A op cop. (1.a) (2.b): Since we proved that (1.a) implies the bijectivity of the antipode, we can apply (1.a) (2.a) to the Hopf algebroid A cop. (1.b) (2.d): This follows by applying (1.a) (2.b) to the Hopf algebroid A op cop. Finally, we would like to correct some regrettable typos in the paper. In both parts of Lemma 5.1, instead of the condition i,k auk i vi k = i,k uk i vi ka,foralla A, the conditions i auk i vi k = i uk i vi k a need to hold, for all possible values of k and all a A. (In the published version also the summation symbols are missing on the right hand sides.) In the computation on page 595, the first and the last expressions are interchanged. In the penultimate line on page 597, the phrase and vice versa has to be erased. References 1. Böhm, G.: Galois extensions over commutative and non-commutative base. In: Caenepeel, S., Van Oystaeyen, F. (eds.) New Techniques in Hopf Algebras and Graded Ring Theory. (2006)

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163 Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. LI, (2005) Galois theory for Hopf algebroids GABRIELLA B()HM (*) SUNTO - Un'estensione B c A di algebre su un anello commutativo k ~ una 7-/-estensione per un L-bialgebroide 7/se A ~ una 7-/-comodulo algebra e B ~ la sottoalgebra dei suoi coinvarianti. Essa ~ 7-/-Galois se l'applicazione canonicaa B --~ A 7-I ~ un isomorfismo o, equivalentemente, se il coanello canonico (A 7/ : A) ~ un coanello di Galois. Nel caso di un algebroide di Hopf 7-/= (~L, ~R, S) si dimostra che ogni ~R-estensione una 7-/L-estensione. Se rantipode ~ biiettivo allora si dimostra che anche le nozioni di estensioni ~-/R-Galois e 7-/L-Galois coincidono. I risultati per le strutture biiettive entwining sono estesi alle strutture entwining su algebre non commutative, al fine di dimostrare un teorema simile al Teorema dii Kreimer- Takeuchi per un Hopf algebroide 7-/proiettivo finitamente generato con antipode biiettivo. I1 teorema afferma che ogni estensione 7-/-Galois B c A ~ proiettiva e se A ~ k-piatto allora la suriettivit~ dell'applicazione canonica ~ sufficiente a garantire la propriet~ di Galois. La teoria di Morita, sviluppata per i coanelli da Caenepeel, Vercruysse e Wang, viene applicata per ottenere criteri equivalenti per la propriet/~ di Galois per estensioni di algebroidi di Hopf. Questo conduce a risultati analoghi, per algebroidi di Hopf, a quelli ottenuti da Doi per estensioni di algebre di Hopf e da Cohen Fishman e Montgomery nel caso degli algebroidi di Hopf Frobenius. ABSTRACT - nil extension B c A of algebras over a commutative ring k is an 7-/-extension for an L-bialgebroid 7-/if A is an 7-/-comodule algebra and B is the subalgebra of its coinvariants. It is ~-/-Galois if the canonical map A A --~ A 7-/ is an isomorphism or, equivalently, if the canonical coring (A 7-/ : A) is a Galois coring. In the case of a Hopf algebroid 7l = (~-/L, 7-/R, S) any 7-/R~extension is shown to be also an 7-/L-extension. If the antipode is bijective then also the notions of 7-/R-Galois extensions and of ~/L-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroid 7-/with bijective antipode. It states that any ~-Galois extension B c A is projective, and if A is k-flat then already the surjectivity of the canonical map implies the Galois property. (*) Indirizzo dell'autore: Research Institute for Particle and Nuclear Physics, Budapest, H-1525 Budapest 114, P.0.B. 49, Hungary; G.Bohm@rmki.kfki.hu

164 234 GABRIELLA BOHM The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang, is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery Introduction. An extension B c A of algebras over a commutative ring k is an H-extension for a k-bialgebra H ifa is a right H-comodule algebra and B is the subalgebra of its coinvariants i.e. of elements b c A such that b{0) b{1) = b 1H -- where the map A ~ A H, a~a(o) a(1) is the coaction of H on A (summation understood). An H-extension B c A is H-Galois if the canonical map A A-+ A H, a a'~aaio I all ) is an isomorphism of k-modules. In many cases it is technically much easier to check the surjectivity of the canonical map than its injectivity. A powerful tool in the study of H-extensions is the Kreimer-Takeuchi theorem [25] stating that if H is a finitely generated projective Hopf algebra then the surjectivity of the canonical map implies its bijectivity and also the fact that A is projective both as a left and as a right B-module. The proof of the Kreimer-Takeuchi theorem went through both simplification and generalization in the papers [30, 32, 9, 31]. In the present paper we adopt the method of Brzezifiski [9] and of Schauenburg and Schneider [31], who used the following observation. A comodule algebra A for a bialgebra H determines a canonical entwining structure [11] consisting of the algebra A, the coalgebra underlying the bialgebra H, and the entwining map H h a ~ a(ol ha(l). In the case when the bialgebra H possesses a skew antipode, this entwining map is a bijection. The proof of (a wide generalization of) the Kreimer-Takeuchi theorem both in [9] and in [31] is based on the study of bijective entwining structures, under slightly different assumptions. In Section 4 below we are going to show that these arguments can be repeated almost without modification by using entwining structures over non-commutative algebras [4]. In the paper [21] Doi constructed a Morita context for an H-extension B c A. If H is finitely generated and projective as a k-module then the surjectivity of one of the connecting maps is equivalent to the projectivity and the Galois property of the extension B c A, while the strictness of the Morita context is equivalent to faithful flatness and the Galois property. This observation made it possible to use all results of Morita theory for characterizing H-Galois extensions. In the case when H is a finite dimensional Hopf algebra over a field (or a Frobenius Hopf algebra over a commutative ring), the Morita context of Doi is equivalent to another Morita context, introduced by Cohen, Fishman and Montgomery [20]. One of the most beautiful applications of the theory of corings [14] is the observation [8] that the Galois property of an H-extension B c A is equivalent to the Galois property of a canonical A-coring A H. In [18] the construction of the

165 GALO[S THEORY FOR HOPF ALGEBROIDS 235 Morita context by Doi has been extended to any A-coring C possessing a grouplike element (i.e. such that A is a C-comodule). In the case when C is a finitely generated projective A-module (or an A-progenerator, see [15]) the application of Morita theory results then several equivalent criteria for the Galois property of the coring C and the projectivity (or faithful flatness) of A as a module for the subalgebra of coinvariants in A. In the case when the A-dual algebra of the coring C is a Frobenius extension of A, also the Morita context in [20] has been generalized to the general setting of corings and the precise relation of the two Morita contexts has been explained. The notion of bialgebra extensions has been generalized to bialgebroids by Kadison [23] as an extension B c A of k-algebras such that A is a comodule algebra and B is the subalgebra of coinvariants. The Galois property of a bialgebroid extension can be formulated also as the Galois property of a canonical coring. This implies that the general theory, developed in [18], can be applied also to bialgebroid extensions. In the present paper we study Hopf algebroid extensions. The notion of Hopf algebroids has been introduced in [7, 3] and studied further in [5]. It consists of two compatible (left and right) bialgebroid structures on the same algebra which are related by the antipode. It is shown in Section 3 below that any comodule for the constituent right bialgebroid ~R of the Hopf algebroid possesses a unique comodule structure for the constituent left bialgebroid 7-tL in such a way that the ~R-coaction is ~L-COlinear and the ~-/L-coaction is ~R-Colinear. In particular, an ~tr-comodule algebra possesses a distinguished ~L-comodule algebra structure and the 7-tR- and the ~Lcoinvariants coincide. What is more, -just as in the case of Hopf algebras - if ~/is a Hopf algebroid with bijective antipode then the canonical entwining structure (over the non-commutative base algebra of 7-t), associated to an ~-extension, is bijective. This fact is used to prove a Kreimer-Takeuchi type theorem in Section 4. In Section 3 we show that if the antipode of the Hopf algebroid ~ is bijective then an ~- extension is Galois for the left bialgebroid structure of ~ if and only if it is Galois for its right bialgebroid structure. In Section 5 we apply the Morita theory for corings to a Hopf algebroid extension B c A, looked at as a right bialgebroid extension. In the finitely generated projective case this results equivalent criteria, under which it is a projective right bialgebroid Galois extension. Similarly, we can look at B c A as a left bialgebroid extension and obtain equivalent conditions for its projective left bialgebroid Galois property. Making use of the results about Hopf algebroid extensions in Section 3, and the Kreimer-Takeuchi type theorem proven in Section 4, we conclude that if ~ is a finitely generated projective Hopf algebroid with bijective antipode then the two equivalent sets of conditions are equivalent also to each other. In the case of Frobenius Hopf algebroids [5] we obtain a direct generalization of ([20], Theorem 1.2).

166 236 GABRIELLA B()HM Throughout the paper k is a commutative ring. By an algebra R = (R,/~, J7) we mean an associative unital k-algebra. Instead of the unit homomorphism ~/we use sometimes the unit element 1R: = ~/(lk). We denote by RM, MR and R.MR the categories of left, right, and bimodules for R, respectively. For the k-module of morphisms in RM, MR and RA4R we write RHom(, ), HomR(, ) and RHomR(, ), respectively. Acknowledgment. I would like to thank Tomasz Brzezifiski and Korn~l Szlachgnyi for useful discussions. This work was supported by the Hungarian Scientific Research Fund OTKA - T and T and the Bolyai Jfinos Fellowship Preliminaries Bialgebroids and Hopf algebroids. L-bialgebroids [26, 37, 34, 28] or, what were shown in [12] to be equivalent to it, XL-bialgebras [36] are generalizations of bialgebras to the case of non-commutative base algebras. This means that instead of coalgebras and algebras over commutative rings one works with corings and rings over non-commutative base algebras. Recall that a coring over a k-algebra L is a comonoid in L./t4L while an L-ring is a monoid in L34L. The notion of L-rings is equivalent to a pair, consisting of a k-algebra A and an algebra homomorphism L --* A. DEFINITION 2.1. A left bialgebroid is a 6-tuple 7-I = (H, L, s, t, ),, z:), where H and L are k-algebras. H is an L ~)L~ via the algebra homomorphisms s : L ~ H and t : L op ~ H, the images of which are required to commute in H. In terms of the maps s and t one equips H with an L-L bimodule structure as (2.1) I.h. l': = s(1)t(l')h for h ~ H, l, l' E L. The triple (H,y~u) is an L-coring. Introducing Sweedler's notation ),(h) = h(1) ~ h(2) for h c H the axioms (2.2) (2.3) (2.4) (2.5) (2.6) are required for all I E L and h, h' E H. h(1)t(1) ~i ~ h(.2) - h(1) ~ h(2)s(1) y(1h) =IH ~ 1H ~,(hh') =),(h)i,(h') 7~(1H) =IL n(h s o u(h')) = u(hh') = 7~(h t o 7~(h'))

167 GALOIS THEORY FOR HOPF ALGEBROIDS 237 Notice that - although H ~ H is not an algebra - the axiom (2.4) makes sense in view of (2.2). The bimodule (2.1) is defined in terms of multiplication by s and t on the left. The R-R bimodule structure in a right bialgebroid 7-I = (H, R, s, t, ~,, 7:) is defined in terms of multiplication on the right. For the details we refer to [24]. The opposite of a left bialgebroid %/= (H, L, s, t, 7', ~) is the right bialgebroid 7-I ~ = (H ~ L, t, s, ~,, ~) where H ~ is the algebra opposite to H. Its co-opposite is the left bialgebroid %loop = (H,L~ s, ~/ol~ ~) where y~ H i~ H is the opposite coproduct h ~ h(2) L~ h(1). It has been observed in [24] that for a left bialgebroid 7-I = (H, L, s, t, ),~ ~) such that H is finitely generated and projective as a left or right L-module the corresponding L-dual possesses a right bialgebroid structure over the base algebra L. Analogously, the R-duals of a finitely generated projective right bialgebroid possess left bialgebroid structures. For the explicit forms of the dual bialgebroid structures consult [24]. Before defining the notion of Hopf algebroid let us introduce some notations. Analogous notations were used already in [7, 5]. When dealing with an L ~ L~ H we have to face the situation that H carries different module structures over the base algebra L. In this situation the usual notation H ~ H would be ambiguous. Therefore we make the following notational convention. In terms of the algebra homomorphisms s : L -~ H and t : L ~ ~ H (with commuting images in H) we introduce four L-modules (2.7) LH : HL : H L : LH : 1. h: = s(1)h h. l: = t(1)h h. l = hs(1) I. h = ht(l). This convention can be memorized as left indices stand for left modules and right indices for right modules. Upper indices for modules defined in terms of right multiplication and lower indices for the ones defined in terms of left multiplication. In writing L-module tensor products we write out explicitly the module structures of the factors that are taking part in the tensor products, and do not put marks under the symbol E.g. we write HL Q LH. In writing elements of tensor product modules we do not distinguish between the various module tensor products. That is, we write both h ~ h' c HL LH and g ~ g' c H L LH, for example. A left L-module can be considered as a right L~ and sometimes we want to take a module tensor product over L ~ In this case we use the name of the corresponding L-module and the fact that the tensor product is taken over

168 238 GABRIELLA BOHM L ~ should be clear from the order of the factors. In writing multiple tensor products we use different types of letters to denote which module structures take part in the same tensor product. DEFINITION 2.2. A Hop]" algebroid 7-I = (7-~L, 7-~R, S) consists of a left bialgebroid 7-lL = (H,L, SL, tl,~l,7~l) and a right bialgebroid 7-IR = (H,R, sr, tr, 7R, ZR), with common total algebra H, and a k-module map S : H ~ H, called the antipode, such that the following axioms hold true: i) SL O T~L O tr = tr, tl O T~L O SR = SR and 8R O T~R O tl : tl, tr O ZR O SL : SL ii) (yl RH) ~ yr = (HL TR) ~ TL as mapsh-~hl R RH and (YR LH) o 7L = ( HR 7L) o YR as maps H --* H R RHL LH iii) S is both an L - L bimodule map LHL --~ LH L and an R - R bimodule map RH R --~ RH R iv) gh o (S LH) o 7L = SR o 7~ R and gh o (H R S) o 7R -= 8L o 7~ L. If 7-/= (7-/L, 7-/R, S) is a Hopf algebroid then so is 7-/~~ = ((7/R)c~ (7-tL)~ p, S) and if S is bijective then also 7-leop = ((7-lL)~op, (7~R)~op, S -1) and 7-l ~ = ((~R) ~ (~L) ~ S-t). We are going to use the following variant of Sweedler's convention. For a Hopf algebroid 7-/= (7-/L, ~R, S) we use the notation 75(h) = h(1) h(2) with lower indices and 7R(h) = h (1) h (2) with upper indices for h c H in the case of the coproducts of ~L and ~R, respectively. The axioms (Definition 2.2 (ii)) read in this notation as h~ h(1)(2) h (2) = h(t) h(2) h(2) (2) h(t) (1) h(1) h(2) = h (t) h(2)(d h(2)(2) for h E H. It is proven in ([5], Proposition 2.3) that the base algebras L and R of the left and right bialgebroids ~L and 7-/R in a Hopf algebroid 7-/are anti-isomorphic via any of the isomorphisms UL o SR and ~L o tr. The antipode is a homomorphism of left bialgebroids ~L -~ (~R)co ~ and also (~R)e~ p -~ ~L in the sense that it is an anti-algebra endomorphism of H and the pair of maps (S, UL o SR) is a coring homomorphism from the R~ (H, y~ ur) to the L-coring (H, YL, UL) and the pair of algebra homomorphisms (S, ur o SL) is a coring homomorphism (H, yl, ~L) ~ (H, y~p,.r). By ([5], Lemma 2.6) the module HL is finitely generated and projective if and only ifh R is and the module LH is finitely generated and projective if and only if RH is. If the antipode is bijective then the finitely generated projectivity of all the four listed modules are equivalent properties. Therefore we term a Hopf algebroid 7-/, the antipode of which is bijective and one (and hence, afortiori, all)

169 GALOIS THEORY FOR HOPF ALGEBROIDS 239 of the modules H R, RH, HL and LH is finitely generated and projective, as a finitely generated projective Hopf algebroid with bijective antipode. The left integrals in a left bialgebroid 7-/L are defined ([7], Definition 5.1) as the invariants of the left regular H-module i.e. the elements of s VhEH}. By ([5], Scholium 2.8) an element 6 of a Hopf algebroid ~ is a left integral if and only if h6 (1) ~ S(6 (2)) = 6 (1) ~ Sff(2))h for all h E H. A left integral 6 in a Hopf algebroid ~/is called non-degenerate ([7], Definition 5.3) ff both maps 6R:H*:=HOmR(HR,R)- r162 tror (1)) and R6 : *H: = RHom(RH, R) ---* H *r ~--~*r ~ g =~ 6 (1) SR o *r (2)) are isomorphisms. By ([7], Proposition 5.10) for a non-degenerate left integral in a Hopf algebroid 7-/also the maps ~L : H,: = HomL(H~L) -~ H r ~ r = SL o r 6(2) and L 6 :,Y: : LHom(LH, L) ~ H,r ~.r ~ tl o,r 6(1) are isomorphisms. It is shown in ([5], Theorem 4.7) that the existence of a nondegenerate left integral in a Hopf algebroid ~-/is equivalent to the Frobenius property of any of the four extensions SR : R -~ H, tr : R ~ ~ H, SL : L ~ H and tl : L ~ --~ H and it implies the bijectivity of the antipode. What is more, if the Hopf algebroid 7-/ possesses a non-degenerate left integral then also the four duals H*, *H, H, and,h possess (anti-) isomorphic Hopf algebroid structures with nondegenerate integrals ([7], Theorem 5.17 and Proposition 5.19). Motivated by these results we term a Hopf algebroid possessing a non-degenerate left integral as a Frobenius Hopf algebroid. Recall from [35] that a Frobenius Hopf algebroid is equivalent to a distributive Frobenius double algebra Module and comodule algebras. The category H.A/~ of left modules for the total algebra H of a left bialgebroid (H, L, s, t, ),, u) is a monoidal category. As a matter of fact, any H-module is an L- L bimodule via s and t. The monoidal product in H.A/~ is the L-module tensor product with H-module structure h.(m~n):=h(1).m~h(2).n and the monoidal unit is L with H-module structure forheh, m~n~m~n

170 240 GABRIELLA BOHM h. h = u(hs(l)) for h E H, 1 c L. A left H-module algebra is defined as a monoid in the monoidal category HM. A left H -module algebra A is in particular an L-ring via the homomorphism L ~A l~l. 1A ==- 1A "l. The invariants of A are the elements of AH:={aEA[h.a=sozc(h).a YhcH}. Just the same way, the category MH of right modules for the total algebra H of a right R-bialgebroid is a monoidal category with monoidal product the R- module tensor product and monoidal unit R. A right H-module algebra is a monoid in A4H. A right module algebra is in particular an R-ring. The invariants are defined analogously to the left case in terms of the counit. By a comodule for a left bialgebroid 7-/= (H, L, s, t, y, r) we mean a comodule for the L-coring (H,),, ~). Recall that the category of left 7~-comodules is also a monoidal category in the following way. Any left H-comodule (M, r) can be equipped with a right L-module structure via (2.8) m. h = ~(m(_lls(l)) m(ol for m E M, l c L where m(-l/~ m(0/stands for r(m) (summation understood). Indeed, (2.12) is the unique right action via which M becomes an L-L bimodule and r becomes an L-L bimodule map from LML to the Takeuchi product H XL M. Recall from [36] that H XL M is the L-L submodule of LH~ LM the elements ~i hi ~ mi of which satisfy (2.9) ~ hi ~ mi. l = E hit(l) ~ mi for l E L. i i This observation amounts to saying that our definition of left ~-/-comodules is equivalent to ([28], Definition 5.5). Hence, without loss of generality, from now on we can think of a comodule in this latter sense. On the basis of ([28], Definition 5.5) the category HM of left 7-I -comodules was shown in ([28], Proposition 5.6) to be monoidal. The monoidal product is the L-module tensor product with comodule structure M ~ N ~ H ~ M ~ N and the monoidal unit is L with comodule structure m ~ n~m(_l)n(1) ~ re(o) i~ > n(o) L --~ L ~ H ~_ H l~s(1). Following ([28], Definition 5.7), a left comodule algebra for a left bialgebroid ~ is

171 GALOIS THEORY FOR HOPF ALGEBROIDS 241 a monoid in the monoidal category HA/[. Notice that - in view of the equivalence of the two definitions of H -comodules - this definition of comodule algebras is equivalent to ([10], Definition 3.4). By similar arguments also the category of right ~-/-comodules - that is of right comodules for the L-coring (H, y, ~) - is monoidal. The monoidal product is the L- module tensor product with coaction M? N -+ M ~ N? H m ~ n~m<0> ~ n<0> ~ n<l>m<l) and the monoidal unit is L with comodule structure L -+ H lilt(l), where the left L-module structure of a right (H, y, u)-comodule M is defined as I. m: = re<o> 9 u(m<l>s(1)). Similarly to the case of left comodule algebras we expect the coaction of a right comodule algebra A to be multiplicative - i.e. such f Q f!! ; that (aa)<o> L (aa)<1> = a<o>a<o> ~ a(l>a(l> for a, a c A. Therefore we consider the monoidal category (A4H) ~ the monoidal structure of which is the opposite of A4 H (i.e. it comes from the monoidal structure of Lo~MLo,). A right 7-/-comodule algebra is defined as a monoid in the category (A/tH) ~ Notice that a left %/-comodule algebra is in particular an L-ring while a right %/-comodule algebra is an L~ The coinvariants of a left (right) }/-comodule algebra (A, v) are the elements of the subalgebra A~~ { a E A ] ~(a) = IH ~ a } ( A~~ { a ~ A ] v(a) = a ~ lh ) ). Recall that for a left L-bialgebroid ~-/the left and right L-duals.H and H. are rings. The category M.H of right.h-modules is a full subcategory in the category M H of right 7/-comodules and the two categories are equivalent if and only if the module LH is finitely generated and projective. The category MH. is a full subcategory in HAd and they are equivalent if and only if the module HL is finitely generated and projective. The left and right comodules, comodule algebras and their coinvariants for a right bialgebroid are defined analogously Entwining structures over non-commutative algebras. Entwining structures over non-commutative algebras were introduced in [4] as mixed distributive laws in the bicategory of [Algebras, Bimodules, Bimodule maps]. This definition is clearly equivalent to a monad in the bicategory of corings i.e. the bicategory of comonads [33] in the bicategory of [Algebras, Bimodules, Bimodule maps]. Explicitly, we have

172 242 GABRIELLA BOHM DEFINITION 2.3. An entwining structure over an algebra R is a triple (A, C, ~,) wherea = (A,/~, ~]) is an R-ring, C = (C, A, e) is an R-coring and ~ is an R- R bimodule map C ~ A --, A ~ C satisfying ~,o(c~)=~c (A ~e)o~=e~a ~ C) o (A or ~) o (~ ~ A) = ~ o (C ~ it) (A R e d) o ~ = (V ~ C) o (C ~ ~,) o (d ~ A). An entwining structure is bijective if ~ is an isomorphism. It is shown in ([4], Example 4.5) that an entwining structure (A, C, ~) over the algebra R determines an A-coring structure on A ~ C with A-A bimodule structure al. (a ~ c). a2 = ala~,(c or a2) for al, a2 E A, a ~ c E A ~ C, coproduct A ~ d and counit A ~ e. DEFINITION 2.4. A right entwined module over an R-entwining structure (A, C, ~) is a right comodule over the correspondinga-coringa ~ C. Explicitly, it is a triple (M, p, r), where (M, p) is a right A-module, making M in particular a right R-module. The pair (M, r) is a right C-comodule such that ~ is a right A- module map i.e. rop=(p~c)o(m ~,)o(t~a). A morphism of entwined modules is a morphism of comodules for the A-coring A ~ C, that is, ana-linear and C-colinear map. The category of entwined modules will be denoted by MA c. By ([14], (2)) the forgetful functor A4A C -~ A4A possesses a right adjoint, the functor (2.10) ~C:A4A~A4 C (M,p)H(M~C, (p~c)o(m~,), M~), where the right R-module structure of M comes from its A-module structure. What is more, by the self-duality of the notion of R-entwining structures, also ([14], 32.8 (3)) extends to entwining structures over non-commutative algebras. That is, also the forgetful functor MA C -~ A4 C possesses a left adjoint, the functor (2.11) ~A:.A4C~A.4 C (M,r)~(Me~A, (M~/~), (M~)o(r~A)).

173 GALOIS THEORY FOR HOPF ALGEBROIDS 243 This implies, in particular, that both A }~ C and C ~ A are entwined modules. The morphism ~ becomes a morphism of entwined modules. If the R-coring C possesses a grouplike element, e, then also the A-coring A ~ C possesses a grouplike element, 1A ~ e. Hence A is an entwined module via the right regular A-action and the C-coaction A ~ A ~ C a~(e ~ a). The coinvariants of an entwined module are defined as its coinvariants as a comodule for the A-coring A~C, w.r.t, the grouplike element 1d ~r e. This is the same as its C-coinvariants w.r.t.e. In this case also the functors (2.12) ( )coc : 3dcA ~.MA~OC and ~ A: fl/ia~oc ~ j~c - ACOC are adjoints ([14], 28.8). (The entwined module structure of NAG A for a right X~ N is defined via the second tensor factor). The unit and the eounit of the adjunction are tln: N ~ (N AG A)C~ n~n A~oC 1A and I~M : Mc~ A ~ M m A2c a~m 9 a A coc for any right At~ N and entwined module M Morita theory for corings. In the paper [18] a Morita context (A "c, *C,A, *C "c, v,/z) has been associated to an A-coring C possessing a grouplike element e. Here the ring *C = AHom(C,A) is the left A-dual of the A-coring C with multiplication (fg)(c) = g(c(1).f(ca))). The invariants of a right *C-module M are defined with the help of the grouplike element e as the elements of M*C:={ m~mfm.f =m.[e(_)f(e)] VfE*C}. In terms of the grouplike element e the k-module A can be equipped with a right *C-module structure as a.f: =f(e. a) for a E A, f 6 *C. The ring A "c is the subring of *C-invariants of A i.e. A *c={bca]f(e.b)=bf(e) VfE*C}. A is an A "c - *C bimodule via b.a.f:=bf(e.a)=f(e.ba) forbca*c,a~a, f E*C.

174 244 GABRIELLA BOHM *C *C is the k-module of *C-invariants of the right regular *C-module i.e. *C *C = { q E *C f f(c(1), q(c(2))) =f(q(c). e) Vf c *C, c ~ C }. It is a *C - A *C bimodule via (f. q- b)(c): = q(c(1).f(c(2)))b forf c *C, q ~ *C *C, b E A *c, c c C. The connecting maps v and/l are given as v : A *c~ *C *C ~ A *C a ~ qhq(e, a) and IL : *C *C A~, A ~ *C q A~" a~( c~q(c)a ). In [18] the following theorem has been proven. THEOREM 2.5. Let C be an A-coring possessing a grouplike element e, and let (A"c, *C,A, *C*c, ~, ~) be the Morita context associated to it. (1) ([18], Theorem 3.5). If C is finitely generated and projective as a left A- module (hence the categories A/[ c and M.c are isomorphic and the C-coinvariants coincide with the *C-invariants) then the following assertions are equivalent. (a) The map ll is surjective (and, a fortior~ bijective). (b) The functor ( _ )c~ : MC ---* A4AcOC is fully faithful. (c) A is a right *C-generator. (d) A is projective as a left At~ and the map *C --~ AcooEnd (A) fh( ahf(e, a) ) is an algebra anti-isomorphism. (e) A is projective as a left At~ and the A-coring C with grouplike element e is a Galois coring. (2) ([18], Theorem 2.7). If the algebra extension A -~ *C a~( c~e(c)a ) is a Frobenius extension with Frobenius system (~, ui ~ vi) then the Morita context (A *c,*c,a, *C *c,v,z) is equivalent to the Morita context (A*C, *C,A,A, v', ll') via the isomorphism A ~ *C *C a~( c~ ~vi[c. aui(e)] ).

175 GALOIS THEORY FOR HOPF ALGEBROIDS Hopf algebroid extensions. An extension B ca is an ~/R -extension for a right bialgebroid ~R = (H, R, s, t, y, 7:) ifa is a right 7-/R-comodule algebra and B is the subalgebra of 7-/R-coinvariants of A. In this situation - denoting the R-coring (H, y, 7:) by C - the triple (A, ~-IR, C) is a Doi-Koppinen datum over R in the sense of ([10], Definition 3.6). This implies that the R-R bimodule map (3.13) ~ : H ~ A -~ A ~ H h ~ alia (~ ~ ha (11 gives rise to an entwining structure (A, C, ~,) over R. Hence the R-R bimodule A ~ H possesses an A-coring structure with A-A bimodule structure al. (a ~ h). a2 = alaa~ ~ ~ ha~ 11 for al, a2 c A, a ~ h c A ~ H, coproduct A ~ y and counit A ~ 7:. This coring possesses a grouplike element 1A ~ 1H. The ~R-extension B c A was termed ~R-Galois in [23] if the A-coring A ~ H, associated to it above, is a Galois coring. This property is equivalent to the bijectivity of the canonical map (3.14) canr : A ~ A ---* A ~ H a ~ a' ~ aa '(~ ~ a '(1). Analogously, in the case of a right comodule algebra A for the left bialgebroid ~-IL = (H, L, s, t, ~, 7:) the ~L-Galois property of the extension A c~ c A means the bijectivity of the canonical map (3.15) canl : A A ~ A ~ H a ~ a'~a(o)a' ~ a(1). AeO~L This is equivalent to the Galois property of the A-coring A ~ H with A-A bimodule structure al. (a ~ h). a2 = al(o)aa2 ~ al(llh for al, a2 c A, a ~ h E A ~ H, coproduct A ~ y, counit A ~ 7: and grouplike element 1A ~ 1H. (Recall from Subsection 2.2 that by our convention A is an L~ in this case.) A Hopf algebroid ~l = (~~L,~-~R,S) possesses two compatible bialgebroid structures, ~-~L and 7-/R, on the same total algebra H. It turns out that the notions of comodules (comodule algebras) for the constituent left bialgebroid 7-/L and for the constituent right bialgebroid ~'~R are related as follows. PROPOSITION 3.1. right ~R-comodule. structure with (1) Let ~ = (~L, ~~R, S) be a Hopf algebroid and (M, rr) a Making M an L-L bimodule using its R-R bimodule (3.16) l. m. l' = 7:R o tl(l'). m. 7:R o tl(1),

176 246 GABRIELLA B(~HM there exists a unique right?-ll-coaction rl on M such that TR is right }-~Lcolinear and rl is right 7-lR-Colinear. That is, the identities (3.17) (rr ~ H) o VL = (M ~ YL) o r R (3.18) (T L ~ H) o ZR = (M ~ 7R) o T L hold true. Furthermore, the ~R-coinvariants of M coincide with its %ll-coinvariants. (2) Let 7-l = (7-/L, 7-tR, S) be a Hopf algebroid and (A, rr) a right ~tr-comodule algebra. Then A is a right 7-lL-comodule algebra with the 7-lL-coaction TL in (1). (3) Let 7-/= (7-/L, 7-lR, S) be a Hopf algebroid and (M, r~r ) and (N, r~r) two right 7~R-comodules. Considering the right ~L-comodule structures of M and N as in (1), any right 7~R-Colinear map is also ~L-Colinear. PROOF. Denote vr(m) ---- m <~ ~ m <1) for m E M. Recall from Subsection 2.2 that M is an R-R bimodule with (3.19) r. m. r' = m (~. ~R(SR(r)m(1)sR(r~)) -- m (~ 9 7~R(SR(r)m(1))r ', and an L-L bimodule with (3.16). The required right 7-/L-coaction on M is given by the map (3.20) ZL : M -~ M ~ H m~m (~ 9 rtr(m(1)(1)) ~ L m(1)(2). In order to see that the map (3.20) is well defined, we have to show that it makes sense to apply the map (M~7:R) ~H to the element m (0) R o m(1)(1) L o m(1)(2) of M ~ H ~ H. This can be seen, on one hand, by using the left R-linearity of 7L (i.e. the identity 7L(htR(r)) = h(1)tr(r) ~ h(2)) and the left R-linearity of UR. On the other hand, the image of rr is in the Takeuchi product M (see Subsection 2.2), hence m (~ ~m(1)(1)lore(l)(2 ) is an element of (M ~H. Since the restriction of the map M ~ UR : M ~ H -~ M to M H is right L-linear, the expression on the right hand side of (3.20) makes sense. In order to see that M is a right ~/-comodule we have to check the counitality and the coassociativity of the map (3.20). It goes as follows. (M ~ ~L) o ~L(m) =(m (~ 9 7~R(m{1)(1))) 9 ~L(m(1)(2)) =m (~176 -~R[tL o 7tL(m(1) (2))m(~ (1)s R o nr(m(1)(1))] ----m (~ - ~R[tL o ul(mo)(2))m(1)(1)(1)s R o UR(m(1)(1)(2))] = m, where the second equality follows from (3.16) and (3.19), the third one from the coassociativity of ~R and the Hopf algebroid axiom (Definition 2.2 (ii)) and the last one from the counitality of the right and left coproducts YR and 75 together

177 with the counitality of rr- GALOIS THEORY FOR HOPF ALGEBROIDS 247 (M ~ YL) o ~L(m) = m (~ - ~R(m(1)(1)) ~ m(1)(2)(1) ~ m(1>(2)(2) (1) {1} (2), =m (0) 9 7~R(m{1>(1)) ~ L /(1)(2)(1 ) 8R 0 7CR(m (2)(1)) m(1)(2)(2) = re(o} (0>. 7~R(m(0} (1}(1)) L e re(o} (1}(2)8R o 7~R(m(1}(1)) 0 L m(1>(2) = (~L ~ H) o ~L(m), where the second equality follows from the counitality of YR, the penultimate one from the Hopf algebroid axiom (Definition 2.2 (ii)) and the coassociativity of ~R and the last one from the right R-linearity of ~R and of YL (i.e. of ~L). The 7-/R-colinearity of Z" L is checked as o -. ~<0> <I> m<1> H) ZR(m) -- m (~176 7~R(m<0> (1)(1)) L "'~ (2) -- m (0). R(/{1>(1)) L Q m(1)(2) (1) m(1>(2) (2) = (M ~ YR) o ~L(m), where the second equality follows from the coassociativity of "t" R and the Hopf algebroid axiom (Definition 2.2 (ii)). Also "C R is ~L-Colinear as o vl(m ) = m (~176 ~ m(~ <l)8 R o 7~R(m(1)(1)) ~ L m<1>(2 ) = m(1)(1)(1)sr o 7~R(/(1}(1)(2) ) /(1}(2 ) = (M ~ YL) o VR(m), where the first equality follows from the right R-linearity of ~R, the second one from the coassociativity of ~R and the Hopf algebroid axiom (Definition 2.2 (ii)) and the last one from the counitality of YR. The 1-/L-coaction (3.20) is unique since any right 1-/R-Colinear right ~L-COaCtion m H m<o> ~ m(l> on M satisfies m<o) L ~ m(1) = m<o) <~ 9 7~R(m(0) (1)) ~ L m(1) = m <~ 9 7~R(m<l>(1)) L ~ re(l)(2) ~-- l:i(m). It is clear from (3.20) that any ~R-coinvariant in M is ~/-coinvariant. Writing ~L(m) ---- m(o> ~ m<l) and using the right 1-/u-colinearity of ~'L the relation (3.20) can be inverted to yield m<o> ~/(1) _-- m<o) 9 7~5(/(1) (1)) ~ re(l> (2), which shows that any ~/-coinvariant in M is also 1-/s-coinvariant. (2) Let (A, VR) be a right ~R-comodule algebra. Then by (1) A is an 7-/5-co-

178 248 GABRIELLA BOHM module. It remains to check that the ~L-coaction ~L on A is multiplicative. Indeed, for a, a' c A vl(a)vl(a') =[a {~ 9 7~R(a(1)(1))][a '(0). 7~R(a'(1)(1))] a{1)(2)a'(1)(2 ) = a(~ 9 a'(o). 7~R(a'(1}(1)) ] L 0 a{1}(2)a'(1}(2 ) = a(~ '(~ rcr (tr o R(a (1)(1))a'(1)(1))] ~ a {1)(2)a'(1)(2) = (a(0}a'(o)). 7~R(a(1)(1)a'(1)(1)) ~ a(1}(2)a'(1}(2) = ~L(aa'), where the second equality follows from the fact that A is an R-ring, the third one from the fact that the image of rr is in the Takeuchi product A XR H, the fourth one from the right bialgebroid version of axiom (2.6) and the R-ring structure of A and the last one from the multiplicativity of VR and of ~L- (3) Suppose that f:m ~ N is a right 7tR-Colinear map. Denote the 7tLcoactions on M and on N as in (1) by vi(m) = m(0> ~ re(l) for m E M and r~l (n) = n(0) ~ n(1} for n E N, respectively. Then f(m)<o) ~f(m)(l> = f(m)(o} (0}. ~R(f(m)(o)(1}) =f(m)< ~. 7~R(f(m)(1)(1)) ~f(m)(l> ~f(m)<l>(2) = f(m(~ 7:R(m(1}(1)) L o re(l)(2) =f(m (~. 7oR(m(1}(1))) o L m(1}(2 ) = f(m(o> (0>. UR (m(o> <1))) ~ re(l> = f(m(o)) ~ m(1}, where the first and the last equalities follow from the counitality of ~R, the second and the penultimate ones from the right %//-cohnearity of ~R, the third one from the 7/s-colinearity off and the fourth one from the right R-linearity off. 9 In view of Proposition 3.1 the usage of both the :Hi- and the 7tR-comodule structures of a comodule for a Hopf algebroid (7-/L, ~R, S) is redundant. On the other hand, it turns out that a symmetric treatment makes the theory more transparent. Therefore we make the following DEFINITION 3.2. A right comodulefor the Hopfalgebroid ~ = (~-/L, ~a, S) is a triple (M, ~L, ~R), where the pair (M, rr) is a right 7~R-comodule and (M, ~i) is a right ~L-comodule such that the R-R and the L-L bimodule structures of M are related via (3.16) and the compatibility relations ( ) hold true.

179 GALOIS THEORY FOR HOPF ALGEBROIDS 249 The right 7-/-comodule (A, rl, rs) is a right 7-l-comodule algebra if (A, ~R) is a right ~-/R-comodule algebra. Equivalently, if (A, rl) is a right ~-/L-comodule algebra. An extension B c A is an 7t-extension if A is a right ~-/-comodule algebra such that B is the coinciding subalgebra of the HL- and of the ~-ts-coinvariants. An/-/-extension B c A is ~-l-galois if it is both 7-/L-Galois and?-/r-galois. We follow the convention of using upper indices to denote the components of the coaction of a right bialgebroid and lower indices in the case of a left bialgebroid. LEMMA 3.3. Let 7-I be a Hopf algebroid with bijective antipode and let B c A be an H-extension. The canonical map cans : A ~ A -~ A or H a ~ a'~aa '(~ ~ a '(t) is injective/surjective/bijective if and only if the canonical map cann : A ~ A ~ A ~ H a ~ a'~a(o)a' ~ a(1) is injective/surjective/bijective. PROOF. For any right ~-comodule (M, %, rr) there exists an isomorphism with inverse q5 M : M ~ H ~ M e L H 9 i 1 : M ~ H -~ M ~ H m ~ h~m(o) ~ mll)s(h) m ~ h~m (~ ~ S l(h)m{1). Using the 7-/s-colinearity of %, the Hopf algebron axiom (Definition 2.2 (iv)), the fact that the image of % is in the Takeuchi product A XL H, the L~ structure of A and the counitality of % one checks that for an ~-/-comodule algebra (A, TL, TR) the identity r o cans = canl holds true. 9 EXAMPLE 3.4. A k-bialgebra (H, IL, tl, A, ~) is an example both of a left- and of a right bialgebroid via the correspondence B: = (H, k, 0, t/, A, ~). A Hopf algebra (H,/~, 0, A, s, S) is an example of a Hopf algebroid via ~: = (g, B, S). A right H-comodule (algebra) (A, T) is an example of a right B-comodule (algebra) and gives rise to an ~-comodule (algebra) via (A, ~, ~). The H-coinvariants are obviously the same as the g-coinvariants and the extension A c~ C A is H-Galois if and only if it is g-galois. EXAMPLE 3.5. A weak bialgebra ([27, 6]) has been shown to determine a left bialgebroid in ([22, 29, 34]). Weak comodule algebras ([2, 16, 17]) over a weak

180 250 GABRIELLA BOHM bialgebra have been shown in ([10], Proposition 3.9) to be equivalent to comodule algebras for the corresponding left bialgebroid. Now just the same way as the weak bialgebra determines a left bialgebroid, it determines also a right bialgebroid, and a weak Hopf algebra determines a Hopf algebroid ([7], Example 4.8). A weak comodule algebra for the weak bialgebra is equivalent also to a comodule algebra for the corresponding right bialgebroid, and a weak comodule algebra for a weak Hopf algebra is equivalent to a comoduie algebra for the corresponding Hopf algebroid. The coinvariants of a weak comodule algebra for a weak bialgebra are the same as its coinvariants for the corresponding left or right bialgebroid. A weak bialgebra extension is Galois in the sense of [17] if and only if the corresponding right bialgebroid extension is Galois. In the case of a weak Hopf algebra with bijective antipode this is equivalent also to the Galois property of the corresponding left bialgebroid extension. EXAMPLE 3.6. The total algebra H of a right bialgebroid 7-/R = (H, R, s, t, y, ~) is a right 7-/R-comodule algebra via y. We have H c~ = t(r). The total algebra H of a left bialgebroid ~L = (H, L, s, t, y, ~) is a right 7-tLeomodule algebra via y and H c~ = s(l). For a Hopf algebroid 7-I = (~L,~IR,S) the total algebra is then a right 7-lcomodule algebra and the canonical map is bijective with inverse canr : RH HR --~ H R RH h ~ h'~-~,hh '(1) ~ h '(2) can~ ~ : H R Q RH ~ RH HR h ~ h'~-~hs(hll )) ~ h'(2). That is, the extension tr : R ~ ~ H is 7-/R-Galois. If the antipode S is bijective then also the canonical map canl : H L Q LH ~ I-1L LH h ~ h'hh(1)h' ~ h(e) is bijective with inverse cani 1 : HL LH --~ H L LH h ~ h'~h '(2) ~ S-l(h'(1))h, that is also the extension SL : L -~ H is 7~L-Galois. EXAMPLE 3.7. Let 7-/be a Hopf algebroid and A a right?-/r-module algebra. The smash product algebra [24] A )~ H is defined as the k-module A R Q RH with multiplication (a >~ h)(a' ~ h'): = a'(a. h '(l)) ~ hh '(2).

181 GALOIS THEORY FOR HOPF ALGEBROIDS 251 With this definition A ~ H is an R-ring via the homomorphism or an L~ R-~ AxH rh ia)~sl~(r) via the anti-homomorphism L-~ AxH l~-~ la)~tl(1). One can introduce right ~tl- and right ~n-comodule structures on A ~ H via ~S: = A ~ YL and TR: = A ~ YR, respectively. The triple (A H, ~L, rn) is a right 7-/- comodule algebra. We have (A ~ H) c~nr = {a x 1H}a~A and the canonical map cann : (A)~ H) ~ (A)~ H) _~ A n RHR RH --~ (A x H) ~ H ~- A R RH is bijective with inverse a ~ h a o h' ~-* a ~ h'h (1) ~ h (2) can~ 1 : A R RHR RH ~ A R RHR RH a ~ h ~ h'~-*a ~ h'(2) OR hs(hil)). This means that the extension A c A )~ H is ~R-Galois. If the antipode of 7-/is bijective then it is also 7-/n-Galois. EXAMPLE 3.8. In ([23], Theorem 5.1) Kadison has shown that a depth 2 extension B c A of k-algebras - if it is balanced or faithfully flat - is a Galois extension for the right bialgebroid, constructed in [24] on the total algebra (A ~ A) B (the centralizer of B in the canonical bimodule A ~ A). Recall that if the extension B c A is in addition a Frobenius extension, (A ~ A) B possesses a Frobenius Hopf algebroid structure [7]. Extending the result of [23] considerably, B~lint and Szlach~nyi have shown in ([1], Theorem 3.7) that an extension B c A of k-algebras is 7-/-Galois for some Frobenius Hopf algebroid 7/if and only if it is a balanced depth 2 Frobenius extension A Kreimer-Takeuchi type theorem for Hopf algebroids. In this section we investigate ~-extensions for finitely generated projective Hopf algebroids 7-I with bijective antipode. We show that for an ~-Galois extension B c A the algebra A is projective both as a left and as a right B-module and - under the additional assumption that (A ~ A) c~ ~- A ~ B, see below - the surjectivity of the canonical map (3.18) implies its bijectivity. This is a generalization of the classical theorem for finitely generated projective Hopf algebras by Kreimer and Takeuchi ([25], Theorem 1.7). Recently Brzezifiski [9] and Schauenburg and Schneider [31] have used new

182 252 GABRIELLA BOHM ideas to prove the Kreimer-Takeuchi theorem and generalizations of it. Their arguments are formulated in terms of entwining structures [11] over a commutative ring. In what follows we claim that their line of reasoning can be applied almost without modification to entwining structures over non-commutative algebras so to prove a Kreimer-Takeuchi type theorem for Hopf algebroids. As we have seen at the beginning of Section 3, a right comodule algebra A for a right R -bialgebroid 7-/R determines an entwining structure (3.17) over R. LEMMA 4.1. Let ~ (~-[L, ~R, S) be a Hopf algebroid with bijective antipode and A a right ~l-comodule algebra. The map ~ in (3.17), corresponding to the right ~R-comodule algebra structure of A, is an isomorphism. PROOF. The inverse of ~ is constructed using the 7-/L-comodule algebra structure of A as 1 :A~H-~H~A a~h~hs l(a(ll)~a(o ). 9 Motivated by Lemma 4.1 we study bijective entwining structures over R. We are going to generalize ([31], Theorem 3.1) which is, indeed, a variant of ([9], Theorem 4.4). Recall that for any entwined module M and any k-module V the k- module V ~ M is an entwined module via the second tensor factor. The elements of V ~ M C~ form a subset of (V ~ M) C~ We have V ~ M c~ = (V ~ M) c~ if, for example, the k-module V is flat. PROPOSITION 4.2. Let (A, C, ~) be a bijective entwining structure over the algebra R, such that the R-coring C possesses a grouplike element e. Denote the corresponding right C-coaction on A by alo I ~ a(l/: = ~(e ~ a) and the subring of its coinvariants by B. Assume that C is fiat as a left (right) R-module and projective as a right (left) C-comodule. (1) Suppose that (A ~ A) c~ = A ~ B and the canonical map I t I (4.21) can : A ~ A ~ A or C a ~ a ~aa(o I ~ all I is surjective. Under these assumptions the canonical map (4.25) is bijective. (2) If the canonical map (4.25) is bijective then A is projective as a right (left) B-module. PROOF. The proof is actually the same as the proof of ([31], Theorem 3.1), so we present only a sketchy proof here. (1) Let us use the assumption that C is projective as a right C-comodule. By the bijectivity of ~, and the adjunction (2.15) for any entwined module M we have HomAc(A ~ C, M) _~ HomC(C, M). By the flatness of the left R-module C the

183 GALOIS THEORY FOR HOPF ALGEBROIDS 253 projectivity of the right C-comodule C implies the projectivity of the entwined module A ~ C. Hence the surjective map A ~ A ---* A ~ C,~)! a "~ a' ~ aalo I R a(l> is a split epimorphism of entwined modules. This means that A ~ C is a direct summand of A ~ A. Notice that (A ~ CY ~ ~_ A via the isomorphism a :A ~ (A ~ C) c~ alia ~ e, hence the canonical map (4.25) is related to the unit of the adjunction (2.16) as can = t~a~ic o (a ~ A). This means that can is bijective provided I~A~A is bijective. Tensoring a k-free resolution...p~ -~ P,-1 --~ ~P1 ~-~ P0 ~176 A---~0 of A with A over k we can write A ~ A as the cokernel of the morphism 01 ~ A : P1 ~ A -~ P0 ~ A between entwined modules that are direct sums of copies of A. The functoriality of I~ implies that /~A~:A o [(00 ~ B) ~ A] = (0o ~ A) o (P0 ~/~A)- Since #A is an isomorphism we can use the universality of the cokernel to define the inverse of tta~ A as the unique map r : A ~ A ~ (A ~ AY ~ ~ A = (A ~ B) ~ A which satisfies r o (00 ~ A) = [(0o ~ B) ~ A] o (P0 ~//A1) - This proves the bijectivity of ZA~A, hence of the canonical map (4.25). If C is projective as a left C-comodule and flat as a right R-module then ~ has to be replaced with its inverse in the above line of reasoning. (2) The projectivity of the right B-module A is proven from the projectivity of the right C-comodule C as follows. By the bijectivity of the canonical map (4.25) and the adjunction (2.16) for any entwined module M we have HomC(C, M) ~_ -~ HomB(A, Me~ hence the functor HomB(A, ( )coc):.a4c ~.h~k is exact. Letf : ~ A be an epimorphism in MB for some index set I. Thenf ~ A is an epimorphism in MA C which is mapped by the functor Horns(A, ( _ yoc) to the epimorphism HOmB(A,f) in AJk. In order to prove the projectivity of the left B-module A from the projectivity of the left C-comodule C replace ~ by its inverse in the above arguments. 9 As it has been explained to us by Tomasz Brzezifiski, there exists an alternative, more general argument proving the bijectivity of the canonical map (4.25) from its split surjectivity in M C. Namely, the application of ([13], Theorem 2.1) to the A-coring A ~ C, corresponding to the R-entwining structure (A, C, ~,) with

184 254 GABRIELLA BOHM grouplike element e c C, and its comodule M = A, implies the claim. Indeed, in this case the condition b) of ([13], Theorem 2.1) reduces to (A ~ A) c~ ~- A ~ A c~ We have formulated Proposition 4.2 (1) in terms of the assumption (A ~ A) c~ = A ~ X ~ It has the advantage that in certain situations (e.g. ifk is a field) it obviously holds true. We do not know, however, whether it is also a necessary condition for the claim of Proposition 4.2 (1). On the other hand, notice that if the canonical map (4.25) is an isomorphism in MA c then (4.22) (A ~ A) c~ ~- (A ~ C) c~ = A, where B stands for A c~ as before. The application of ([38], Proposition 5.1) to the A-coring A ~ C, corresponding to the R-entwining structure (A, C, ~) with grouplike element e ~ C, and its comodule M = A, implies that the bijectivity of the canonical map (4.25) follows from its split surjectivity also under the (sufficient and necessary) condition (4.26). Let us turn to the application of Proposition 4.2 to Hopf algebroid extensions. Let 7-/be a finitely generated projective Hopf algebroid with bijective antipode. It follows from the Fundamental Theorem for Hopf algebroids ([5], Theorem 4.2) and the existence of a Hopf module structure on H* = HomR(H, R) with coinvariants s ([5], Proposition 4.4) that for such a Hopf algebroid the map (4.23) al : LH ff~(g*) L --~ H* h ~,t* H,F ~- S(h) =- ~*[S(h) _ ] is an isomorphism of Hopf modules, hence in particular of left H-modules. (The left H-module structure on LH s L is given by left multiplication in the first tensor factor, and on H* by h-r = r ~ S(h) =- r _ ].) This implies that the element a~l(~r) is an invariant of the left H-module s L. The elements {xi} c H and {~} c s satisfying ~i xi ~ ~ = a~l(rcr) can be used to construct dual bases for the left *H-module on H defined as *r ~ h:= h ~ SR o *r As a matter of fact, for ~*E s we have 4" o S E s ([5], Scholium 2.10), and *H is a right H module via *r ~ h = *r _ ). We leave it to the reader to check that the sets {x/} c H and {~ o S ~ S-1( _ )} c.hhom(h, *H) are dual bases, showing that H is a finitely generated and projective *H-module. Since the antipode was assumed to be bijective, we can apply the same argument to the co-opposite Hopf algebroid?-tcop to conclude on the finitely generated projectivity of the left H*-module on H defined as r h:= = h (2) tr o r Now the projeetivity of H as a left *H module implies that it is projective as a right 7-/R-comodule and its projectivity as a left H*-module implies that it is projective as a left %/R-comodule. These observations allow for the application of Proposition 4.2 to the entwining structure (3.17) - which is bijective by Lemma 4.1.

185 GALOIS THEORY FOR HOPF ALGEBROIDS 255 COROLLARY 4.3. Let ~-I be a finitely generated projective Hopf algebroid with bijective antipode and let B c A be an 7-l-extension. (1) Suppose that (A ~ A) c~ = A ~ B (e.g. A is k-flat). If the canonical map canr : A ~ A ~ A ~ H a ~ a~aa '(~ ~ a '(1) is surjective then the extension B c A is ~R-Galois (equivalently, ~L-Galois). (2) If the extension B c A is 7-l-Galois then A is projective both as a left and as a right B-module Morita theory for Hopf algebroid extensions. As we have seen at the beginning of Section 3, an ~R-extension B c A for a right R-bialgebroid ~R determines an A-coring structure on A ~ H. One can apply the Morita theory for corings, developed in [18], to this coring. In particular, if A is finitely generated and projective as a left R-module, then Theorem 2.5 (1) can be used to obtain criteria which are equivalent to the ~R- Galois property of the extension B c A and the projectivity of the left B- module A. Analogously, one can apply Theorem 2.5 (1) to obtain criteria which are equivalent to the Galois property of an ~L-extension B c A for a finitely generated projective left bialgebroid ~L and the projectivity of the right B- module A. Applying the results of the previous sections we prove that if 7-/= (~tl, ~R, S) is a finitely generated projective Hopf algebroid with bijective antipode and B c A is an ~t-extension then the equivalent conditions, derived from Theorem 2.5 (1) forb c A as an ~tr-extension, on one hand, and as an ~L-extension, on the other hand, are equivalent also to each other. Let ~R = (H, R, s, t, y, ~) be a right bialgebroid such that H is finitely generated and projective as a left R-module and let A be a right ~R-comodule algebra. As a first step, let us identify the Morita context (A *C, *C,A, *C *C, ~,/~), associated to the A-coring C = A ~ H (as it is explained in Subsection 2.4). Recall that A - being a right 7-/R-comodule - is also a left *H-module via *r a = a (~ - *r Since the module RH is finitely generated and projective, *H possesses a left bialgebroid structure. Let us introduce the smash product algebra *H ~< A as the k-module *H ~ A (where the right R-module structure of *H is given by (*r = *r with multiplication (5.24) (*r ~ a)(*~ ~( a'): = *~Y(1)*r ~( (*~(2)" a)a'. With this definition *H ~ A is an A-ring via the homomorphism ia :A-~ *H~A a~u~(a.

186 256 GABRIELLA B(~HM Define a right-right relative (A, HR) -module to be an entwined module for the R -entwining structure (3.17), i.e. a right comodule for the A-coring A ~ H. This means a right A-module (and hence in particular a right R-module) and a right HR-comodule M such that the compatibility condition (m. a) (~ ~ (m- a) (1) = m (~ - a (~ ~ m{1)a {1) holds true for any m E M and a E A. Clearly, A is itself a relative (A, HR)- module. It is straightforward to check that the category M H of relative (A, HR)- modules is isomorphic to the category M.H ~ A of right *H ~< A-modules and the *H ~< A-invariants are the same as the ~R-coinvariants. This fact can be easily understood in view of LEMMA 5.1. Let ~-~R be a right R- bialgebroid such that H is finitely generated and projective as a left R-module and let A be a right %lr-comodule algebra. The left A-dual algebra of the A-coring A ~ H is isomorphic to the smash product algebra * H ~< A. PROOF. The required algebra isomorphism is constructed as *H ~( A ~ AHom(A ~ H,A) *r ~( a~( a' ~ h~a'(*r a)). The Morita context, associated to the 7-/R-extension B ca, (B, *H ~< A,A, (*H ~ A) c~ ~R, I~R) with connecting maps is then (5.25) VR :A *~A (*H ~< A) c~ ~ B a'.~a ( E*r ~< ai)~ E (*r a')ai i i (5.26) #R : (*H ~ A) c~ ~ A ~ *H ~( A ( E*r ~< ai) ~ a'h E *r ~( aia'. i i Since H is finitely generated and projective as a left R-module, so is the left A- module A ~ H. Hence part (1) in Theorem 2.5 implies PROPOSITION 5.2. Let ~-~R be a right R- bialgebroid such that H is finitely generated and projective as a left R-module and let B c A be an ~R-extension. The following assertions are equivalent. (a) The map/a R in (5.30) is surjective (and, a fortiori, bijective). (b) The functor ( _ )conr : A, IH ~ A, IB is fully faithful. (c) A is a right *H ~( A-generator. (d) A is projective as a left B-module and the map (5.27) *H~A ~ BEnd(A) *r all( a'~(*r a')a ) is an algebra anti-isomorphism. (e) A is projective as a left B-module and the extension B c A is ~R-Galois.

187 GALOIS THEORY FOR HOPF ALGEBROIDS 257 The arguments, leading to Theorem 5.2, can be repeated by replacing the right bialgebroid ~R with a left L-bialgebroid ~-~L such that H is finitely generated and projective as a left L-module. Indeed, in this case the left L-dual,,H, possesses a right bialgebroid structure and A is a right,h-module via a., r = a(0) 9 r The right A-dual of the A-coring A ~ H is isomorphic to the smash product algebra,h ~< A, which is defined as the k-module,h ~ A, (where the right L-module structure on,h is given by (,r 1)(h) =,r with multiplication (,r ~< a)(,~ ~< a'): =,~,r ~( a(a'.,r Since A is an L~ a left A-module is in particular a right L-module and we have the isomorphism (A ~ H) ~ M -~ M ~ H for any left A-module M. A left comodule for the A-coring A ~ H is then equivalent to a left A-module (hence in particular a right L-module) and a right?-/l-comodule M such that the compatibility condition (a- m)(o) ~ (a. m)(1) = a(0) - m(o) ~ a(1)m(1) holds true for m ~ M and a ~ A. Such modules are called left-right relative (A,?-/L)-modules and their category is denoted by AM H. It follows that the category A./~ H is isomorphic also to,h~am, the category of left,h ~< A-modules. The Morita context associated to the 7-/L-extension B ca is (B,,H~<A, A, (,H ~< A) c~, ~L, #L) with connecting maps (5.28) ~)L : (,H ~< A) c~,~a A ~ B (5.29) PL : A ~ (,H ~< A) c~ ~,H ~< A Part (1) of Theorem 2.5 implies (E,r ai),~a a'~ea~(a'.,r i) i i a' ~( E *r ~< ai)~ E *r ~< a'ai. i i PROPOSITION 5.3. Let 7-IL be a left L- bialgebroid such that H is finitely generated and projective as a left L-module and let B c A be an ~L-extension. The following assertions are equivalent. (a) The map tt L in (5.33) is surjective (and, afortiori, bijective). (b) The functor ( _ )cowl : Aj~4H ~ B~4 is fully faithful. (c) A is a left,h ~< A-generator. (d) A is projective as a right B-module and the map (5.30),H ~< A -~ EndB(A),r ~< a ~ ( a'~a(a'.,r ) is an algebra isomorphism. (e) A is projective as a right B-module and the extension B c A is 7-~ L-GalOis.

188 258 GABRIELLA BOHM Combining Proposition 5.2, 5.3, Corollary 4.3 and Lemma 3.3 we can state our main result. THEOREM 5.4. Let ~ = (%ll, %lr, S) be a finitely generated projective Hopf algebroid with bijective antipode and let B c A be an %l-extension. The following assertions are equivalent. (a) The extension B c A is 7-IR -Galois. (b) A is projective as a left B-module and the map (5.31) is an algebra antiisomorphism. (c) A is a right *H ~ A-generator. (d) The functor ( _ )co~r :/~4 H ~ A/IB is.fully faithful. (e) The map tt R in (5.30) is surjective (and, afortiori, bijective). (a') The extension B c A is %ll -Galois. (b') A is projective as a right B-module and the map (5.34) is an algebra isomorphism. (c') A is a left.h ~ A-generator. (d') The functor ( _ )cou~ : A.MH ~ B.A/[ is fully faithful. (e') The map I~L in (5.33) is surjective (and, a fortior~ bijective). PROOF. It follows from part (2) of Corollary 4.3 that (a) is equivalent to any of the assertions (5.31) A is projective as a left B module and the extension B c A is ~R-Galois. (5.32) A is projective as a right B module and the extension B c A is 7-/R-Galois. By Lemma 3.3 the assertions (a) and (a') are equivalent and (5.36) is equivalent to (5.33) A is projective as a right B module and the extension B c A is ~-~L--Galois. The rest of the proof follows from Theorem 5.2 and 5.3. Let us mention that in ([15], Theorem 4.7) a stronger version of Theorem 2.5 (1) has been proven. Its application to bialgebroid extensions implies PROPOSITION 5.5. Let ~-~R be a right R- bialgebroid such that H is finitely generated and projective as a left R-module and let B c A be an %lr-extension. The following assertions are equivalent. (a) The Morita context (B, *H ~ A,A, (*H ~ A) tour, OR,/~R) is strict. (b) The functor (_)c~ ~ A.4B is an equivalence with inverse _ ~A:MB---*Mt~. (c) The map (5.31) is an algebra anti-isomorphism and A is a left B-progenerator. (d) A is faithfully flat as a left B-module and the extension B c A is %lr- Galois. 9

189 GALOIS THEORY FOR HOPF ALGEBROIDS 259 Also the analogue of Proposition 5.5 for left bialgebroid extensions can be proven. We can not prove, however, that for an H-extension, in the case of any finitely generated projective Hopf algebroid 7-I with bijective antipode, the equivalent conditions in Proposition 5.5 and their counterparts on the left bialgebroid of H are equivalent also to each other (as it was seen to be the case with Proposition 5.2 and 5.3). The Morita context (X C, *C,A~ *C *C, o, ]L), associated in [18] to an A-coring C with a grouplike element, is the generalization of the Morita context, associated to a bialgebra extension in [21]. In the case of a finite dimensional Hopf algebra over a field (or a Frobenius Hopf algebra over a commutative ring) another Morita context has been associated to a Hopf algebra extension in [20, 19]. The relation of the two Morita contexts is of the type described in part (2) of Theorem 2.5. In order to see what is the analogue of the Morita context of Cohen, Fishman and Montgomery in the case of Hopf algebroids, in the rest of the section we assume that 7-I is a Frobenius Hopf algebroid. LEMMA 5.6. Let H be a Frobenius Hopf algebroid and A a left HL-module algebra. Consider the smash product algebra H ~( A, which is the k-module H ~ A with multiplication The extension (h ~< a)(g ~ a'): = g(1)h ~ (g(2)" a)a'. i:a--~h~(a ah1h~a is a Frobenius extension. PROOF. Recall (from Subsection 2.1) that a Frobenius Hopf algebroid possesses non-degenerate left integrals. Let us f'lx such an integral ~ and denote by p. the unique element in H., for which ~ ~ p. =_ SL o p,(~(1))~(2) = 1H. A Frobenius functional 9 : H ~< A -~ A is given by h ~( a ~ p.(h), a. A Hopf algebroid calculation shows that it is an A-A bimodule map and possesses a dual basis (S(g(2)) ~< 1A) A e (~(1) ~< 1A). 9 Recall that for a Frobenius Hopf algebroid ~/also the left bialgebroid *H possesses a Frobenius Hopf algebroid structure. Applying Lemma 5.6 together with part (2) of Theorem 2.5 we conclude that the Morita context (A c~ * H ~< A,A, (* H ~< A) c~ or, IXR), associated to the right HR -comodule algebra structure of a right H-eomodule algebra A for the Frobenius Hopf algebroid H, is equivalent to the Morita context (A c~ *H ~< A,A,A, o', IX') with connecting maps (5.34) ~' : A.~A A ~ A c~ a.~ a'~*~. (aa')

190 260 GABRIELLA B(3HM (5.35) /[ :A ~ A ~*H~A aa, % d~-*(zr~a)(*,~la)(z~r~<a'), where "4 is a non-degenerate left integral in *H. COROLLARY 5.7. If we add to the conditions of Theorem 5.4 the requirement that 7t be a Frobenius Hopf algebroid then we can add to the equivalent assertions (a)-(e') also (f) For any non-degenerate left integral " 4 in the Frobenius Hopf algebroid * H the map A ~ A ~ *H ~< A is surjective (and, afortiori, bijective). a ~ a'~(7~r ~ a)(*~ ~ 1A)(7~ R P(a') By ([7], Lemma 5.14) for any non-degenerate left integral g in a Hopf algebroid ~/, for p.:=g~l(1h)c H, and any element h of H we have [(l)h ~ ~(2) = t?(1) ~ [(2)tL o p,(~h(1))s(h(2)). This implies that the image of the map (5.39) is an ideal in *H ~( A. Hence - just as it has been proven for finite dimensional Hopf algebras in ([20], Corollary 1.3) - we see that it is true also for?-/-extensions B c A for a Frobenius Hopf algebroid ~ that if the k-algebra *H ~( A is simple then the extension B c A is ~-Galois. REFERENCES [1] I. BALINT - K. SZLAC~I, Finitary Galois extensions over noncommutative bases, arxiv:math.qa/ [2] G. B~nM, Doi-Hopf modules over weak Hopf algebras, Commun. Algebra 28(10) (2000), pp [3] G. BShm, An alternative notion of Hopf algebroid, in: Hopf algebras in noncommutative geometry and physics. S. Caenepeel, F. Van Oystaeyen (eds.), Lecture Notes in Pure and Appl. Math. 239, 31-54, Dekker, New York, [4] G. BOHM, Internal bialgebroids, entwining structures and corings, AMS Contemp. Math. 376 (2005), pp [5] G. BOHM, Integral theory for HopfAlgebroids, arxiv:math.qa/ v3, Algebra Represent. Theory to appear. [6] G. BOHM - F. NILL - K. SZLACHA.NYI, Weak HopfAlgebras I: Integral Theo~?] and C*- structure, J. Algebra 221 (1999), pp [7] G. B(3HM - K. SZLAC~I, Hopf algebroids with bijective antipodes: axioms, integrals and duals, J. Algebra 274 (2004), pp [8] T. BRZEZINSKI, The structure of corings, Algebra Represent. Theory 5 (2002), pp

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192 262 GABRIELLA B()HM [35] K. SZLACHANYI, The double algebraic viewpoint of finite quantum groupoids, J. Algebra 280 (2004), pp [36] M. TAKEUCHI, Groups ofalgebras overa J. Math. Soc Japan 29 (1977), pp [37] P. Xu, Quantum Groupoids, Commun. Math. Phys. 216 (2001), pp [38] R. WISBAUER, On Galois comodules, arxiv: math.ra/ v2. Pervenuto in Redazione 21 febbraio 2005.

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195 Advances in Mathematics 225 (2010) The weak theory of monads Gabriella Böhm Research Institute for Particle and Nuclear Physics, Budapest, H-1525 Budapest 114, PO Box 49, Hungary Received 24 February 2009; accepted 2 February 2010 Available online 9 March 2010 Communicated by Ross Street Abstract We construct a weak version EM w (K) of Lack and Street s 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EM w (K) and composite pre-monads in K is discussed. If K admits Eilenberg Moore constructions for monads, we define two symmetrical notions of weak liftings for monads in K. If moreover idempotent 2-cells in K split, we describe both kinds of weak lifting via an appropriate pseudo-functor EM w (K) K. Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads Elsevier Inc. All rights reserved. Keywords: 2-Category; Monad; Weak lifting 0. Introduction Many constructions, developed independently in Hopf algebra theory, turn out to fit more general situations studied in category theory. For example, crossed products with a Hopf algebra in [21,2,12] are examples of a wreath product in [16]. As another example, the comonad induced by the underlying coalgebra in a Hopf algebra H, has a lifting to the category of modules over any H -comodule algebra. So-called Hopf modules are comodules (also called coalgebras) for the lifted comonad. Galois property of an algebra extension by a Hopf algebra turns out to correspond to comonadicity of an appropriate functor [14]. address: G.Bohm@rmki.kfki.hu /$ see front matter 2010 Elsevier Inc. All rights reserved. doi: /j.aim

196 2 G. Böhm / Advances in Mathematics 225 (2010) 1 32 Motivated by noncommutative differential geometry, these constructions were extended from Hopf algebras to coalgebras (over a commutative base ring) and to corings (over an arbitrary base ring), see e.g. the pioneering paper [8]. The resulting theory turns out to fit the same categorical framework, only the occurring (co)monads have slightly more complicated forms [14]. For a study of non-tannakian monoidal categories (i.e. those that admit no strict monoidal fiber functor to the module category of some commutative ring), another direction of generalization was proposed in [6]. The essence of this approach is a weakening of the unitality of some maps and it leads to the replacing of a Hopf algebra by a weak Hopf algebra. In the last decade many Hopf algebraic constructions were extended to the weak setting. Weak crossed products were studied e.g. in [9,13] and [18]. Weak Galois theory was developed, among other papers, in [9,8]. However, just because in these generalizations one deals with non-unital maps, they do not fit the categorical framework of (co)monads, their wreath products and liftings. The aim of the current paper is to provide a categorical framework for weak constructions. For this purpose, in Section 1 we construct, for any 2-category K, a 2-category EM w (K) which contains the 2-category EM(K) in [16] as a vertically full 2-subcategory. In EM w (K) 0-cells are thesameasinem(k), i.e. monads in K. Since we aim to describe constructions in terms of nonunital maps, in the definition of a 1-cell in EM w (K) we impose the same compatibility condition with the multiplications of monads which is required in EM(K), but we relax the compatibility condition in EM(K) with the units of monads. Certainly, without compensating it with some other requirements, we would not obtain a 2-category. We show that imposing one further axiom on the 2-cells in addition to the axiom in EM(K), EM w (K) becomes a 2-category, with the same horizontal and vertical composition laws used in EM(K). It was observed in [9] that smash products (and more generally crossed products [18]) by weak bialgebras are not unital algebras. Motivated by the definition of a pre-unit in [9], we study pre-monads (defined in Section 2) in any 2-category K. In Section 2 we interpret weak crossed products in [13] as monads in EM w s (K). This leads to a bijection between monads t t in EM w (K) and pre-monad structures (in K) on the composite 1-cell st with a t-linear multiplication. Starting from Section 3, we restrict our studies to 2-categories K which admit Eilenberg Moore constructions for monads (in the sense of [19]) and in which idempotent 2-cells split. These assumptions are motivated by applications to bimodules. The bicategory BIM of [Algebras; Bimodules; Bimodule maps], over a commutative, associative and unital ring k, satisfies both assumptions. However, in order to avoid technical complications caused by non-strictness of the horizontal composition in a bicategory, we prefer to restrict to 2-categories. In the examples, instead of the bicategory BIM, we can work with its image in the 2-category CAT = [Categories; Functors; Natural transformations], under the hom 2-functor BIM(k, ) : BIM CAT, which image is a 2-category with the desired properties. For a 2-category K which admits Eilenberg Moore constructions for monads, the inclusion 2-functor I : K EM(K) possesses a right 2-adjoint J, cf. [16]. In Section 3 we use the splitting property of idempotent 2-cells in K to construct a factorization of J through the inclusion 2-functor EM(K) EM w (K) and an appropriate pseudo-functor J w : EM w (K) K. Fora s monad t t in EM w (K) and any 0-cell k in K, we prove that both monads K(k, J w (s)) and K(k, ŝt) in CAT possess isomorphic Eilenberg Moore categories, where ŝt is a canonical retract monad of the pre-monad st. In a 2-category K which admits Eilenberg Moore constructions for monads, any monad k t k determines an adjunction (k f J(t),J(t) v k) in K, cf. [19]. A lifting of a 1-cell k V k

197 G. Böhm / Advances in Mathematics 225 (2010) in K for monads k t k and k t k is, by definition, a 1-cell J(t) V J(t ), such that v V = Vv, cf. [17]. In Section 4 we define a weak lifting by replacing this equality with the existence of a 2-cell v V ι Vv, possessing a retraction Vv π v V. This leads to two symmetrical notions of weak lifting of a 2-cell V ω W for the monads t and t. A weak ι-lifting V ω W is defined by the condition ι v ω = ωv ι and a weak π-lifting V ω W is defined by v ω π = π ωv.we show that any weak ι-lifting and any weak π-lifting of a 2-cell in K, if it exists, is isomorphic to the image of an appropriate 2-cell in EM w (K) under the pseudo-functor J w. Both a weak ι-lifting and a weak π-lifting are proven to strictly preserve vertical composition and to preserve horizontal composition up to a coherent isomorphism. We also give sufficient and necessary conditions for the existence of weak ι- and π-liftings of a 2-cell in K. A powerful tool to treat algebra extensions by weak bialgebras is provided by weak entwining structures in [9]. A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell tc ct relating both structures in a way which generalizes a mixed distributive law in [1]. It was observed in [8] that any weak entwining structure (in BIM) induces a comonad (called a coring in the particular case of the bicategory BIM). In Section 5 we show that in the same way as mixed distributive laws in a 2-category K provide examples of comonads in EM(K) weak entwining structures provide examples of comonads in EM w (K). Moreover, if the 2-category K satisfies the assumptions in Section 3, then the comonad in K, induced by a weak entwining structure, is an example of a weak ι-lifting of a comonad for a monad. Studying partial coactions of Hopf algebras, in [10] another generalization of a mixed distributive law, a so-called partial entwining structure was introduced. Partial entwining structures (in BIM) were proven to induce comonads as well. We show that also partial entwining structures in a 2-category K provide examples of a comonad in EM w (K). Moreover, if the 2-category K satisfies the assumptions in Section 3, then the comonad in K, induced by a partial entwining structure, is an example of a weak π-lifting of a comonad for a monad. As a final application, weak bialgebras are characterized via weak liftings. If a module H, over a commutative, associative and unital ring k, possesses both an algebra and a coalgebra structure, then it induces two monads t R = ( ) k H and t L = H k ( ), and two comonads c R = ( ) k H and c L = H k ( ) on the category of k-modules. We relate weak bialgebra structures of H to weak ι-liftings of c R and c L for t R and t L, respectively, and weak π-liftings t R and t L for c R and c L, respectively. Notation. We assume that the reader is familiar with the theory of 2-categories. For a review of the occurring notions (such as a 2-category, a 2-functor and a 2-adjunction, monads, adjunctions and Eilenberg Moore construction in a 2-category) we refer to the article [15]. In a 2-category K, horizontal composition is denoted by juxtaposition and vertical composition is denoted by, 1-cells are represented by an arrow and 2-cells are represented by. For any 2-category K, Mnd(K) denotes the 2-category of monads in K as in [19] and Cmd(K) := Mnd(K ) denotes the 2-category of comonads in K, where ( ) refers to the vertical opposite of a 2-category. We denote by EM(K) the extended 2-category of monads in [16]. We use the reduced form of 2-cells in EM(K), see [16]. 1. The 2-category EM w (K) For any 2-category K, the 2-category EM w (K) introduced in this section extends the 2- category EM(K) in [16].

198 4 G. Böhm / Advances in Mathematics 225 (2010) 1 32 Theorem 1.1. For any 2-category K, the following data constitute a 2-category, to be denoted by EM w (K). 0-cells are monads (k t μ k,tt t,k η t) in K. 1-cells (t,μ,η) (t,μ,η ) are pairs (V, ψ), consisting of a 1-cell k V k and a 2-cell t V ψ Vt in K, such that The identity 1-cell is t (k,t) t. 2-cells (V, ψ) (W, φ) are 2-cells V ϱ Wt in K, such that The identity 2-cell is (W, φ) φ η W Vμ ψt t ψ = ψ μ V. (1.1) Wμ ϱt ψ = Wμ φt t ϱ, (1.2) ϱ = Wμ φt η Wt ϱ. (1.3) (W, φ). Horizontal composition of 2-cells (V, ψ) ϱ (W, φ), (V,ψ ) ϱ (W,φ ) (for 1-cells (V, ψ), (W, φ) : t t and (V,ψ ), (W,φ ) : t t ) is given by ϱ ϱ := W Wμ W ϱt W ψ ϱ V. (1.4) Vertical composition of 2-cells (V, ψ) ϱ (W, φ) τ (U, θ) (for 1-cells (V, ψ), (W, φ) and (U, θ) : t t ) is given by τ ϱ := Uμ τt ϱ. (1.5) Proof. We verify only those axioms whose proof is different from the proof of the respective axiom for EM(K). The vertical composite of 2-cells (V, ψ) ϱ (W, φ) τ (U, θ) in EM w (K) is checked to satisfy (1.2) by the same computation used to verify that the vertical composite of 2-cells in EM(K) is a 2-cell in EM(K). In order to see that τ ϱ satisfies (1.3), use the interchange law in K, associativity of the multiplication μ of the monad t and the fact that τ satisfies (1.3): Uμ θt η Ut (τ ϱ) = Uμ θt η Ut Uμ τt ϱ = Uμ Uμt θtt η Utt τt ϱ = Uμ τt ϱ = τ ϱ. Associativity of the vertical composition follows by the same reasoning as in the case of EM(K). Using the constraint (1.1), it follows for any 1-cell t (W,φ) t in EM w (K) that Wμ φt t φ t η W = φ μ W t η W = φ and Wμ φt η Wt φ = Wμ φt t φ η t W = φ μ W η t W = φ.

199 G. Böhm / Advances in Mathematics 225 (2010) Thus the 2-cell (W, φ) φ η W (W, φ) in K satisfies (1.2) and (1.3), proving that it is a 2-cell in EM w (K). It is immediate by condition (1.3) that for any 2-cell (V, ψ) ϱ (W, φ) in EM w (K), ( φ η W ) ϱ = Wμ φt η Wt ϱ = ϱ. Using (1.2), the interchange law in K and then (1.3), one checks that also ϱ ( ψ η V ) = Wμ ϱt ψ η V = Wμ φt t ϱ η V = Wμ φt η Wt ϱ = ϱ. Hence there are identity 2-cells of the stated form. On identity 2-cells (V, ψ) ψ η V (V, ψ) and (V,ψ ) ψ η V (V,ψ ), the horizontal composite comes out as ( ψ η V ) ( ψ η V ) = V ψ ψ V η V V, where we applied (1.1) for ψ and unitality of the monad t. That is to say, the horizontal (V,ψ) composite of the 1-cells t t (V,ψ ) t is the 1-cell (V V,V ψ ψ V).Bythesame computations used in the case of EM(K), the horizontal composite of 2-cells (V, ψ) ϱ (W, φ), (V,ψ ) ϱ (W,φ ) is checked to satisfy condition (1.2). It satisfies also (1.3), as W Wμ W φt φ Wt η W Wt ( ϱ ϱ ) = W Wμ W φt φ Wt η W Wt W Wμ W ϱt W ψ ϱ V = W Wμ W Wμt W φtt W t ϱt φ Vt η W Vt W ψ ϱ V = W Wμ W Wμt W ϱtt W ψt φ Vt η W Vt W ψ ϱ V = W Wμ W ϱt W Vμ W ψt W t ψ φ t V η W t V ϱ V = W Wμ W ϱt W ψ W μ V φ t V η W t V ϱ V = W Wμ W ϱt W ψ ϱ V = ϱ ϱ. The first and last equalities follow by (1.4). In the second and fourth equalities we used the interchange law in K and associativity of the multiplication μ of the monad t. In the third equality we used that ϱ satisfies (1.2). The fifth equality is derived by using that ψ satisfies (1.1). The penultimate equality follows by using that ϱ satisfies (1.3). This proves that the horizontal composite of 2-cells is a 2-cell. Associativity of the horizontal composition in EM w (K) is checked in the same way as it is done in EM(K). Obviously, for any 2-cell (V, ψ) ϱ (W, φ), ϱ η = Wμ Wηt ϱ = ϱ. By (1.2) and (1.3), also η ϱ = Wμ ϱt ψ η V = ϱ. Hence the identity 2-cell (k, t) η (k, t) is a unit for the horizontal composition, proving the stated form (k, t) of the identity 1-cell t t. The interchange law in EM w (K) is checked in the same way as it is done in EM(K). Clearly, any 1-cell in EM(K) is a 1-cell also in EM w (K). For a 1-cell t (W,φ) t in EM(K),any 2-cell ϱ in K of target Wt satisfies (1.3). Hence 2-cells in EM w (K) between 1-cells of EM(K)

200 6 G. Böhm / Advances in Mathematics 225 (2010) 1 32 are the same as the 2-cells in EM(K). Comparing the formulae of the horizontal and vertical compositions in EM(K) and EM w (K), we conclude that EM(K) is a vertically full 2-subcategory of EM w (K). One may ask what 2-subcategory of EM w (K) plays the role of the 2-subcategory, obtained as an image of Mnd(K) in EM(K). As the lemmata below show, there seems to be no unique answer (V,ψ) to this question. This is because, for some 1-cells t t (W,φ) and t t in EM w (K) and a 2-cell V ω W in K, the 2-cells ωt ψ η V and φ η W ω in K need not be equal. Still, there are two distinguished sets of 2-cells in EM w (K) on equal footing, both closed under the horizontal and vertical compositions and both containing the identity 2-cells. Lemma 1.2. For any 2-category K, lett V ω W be a 2-cell in K. (V,ψ) t and t (W,φ) t be 1-cells in EM w (K) and (1) The following assertions are equivalent: (i) ωt ψ η V : (V, ψ) (W, φ) is a 2-cell in EM w (K); (ii) ωt ψ = Wμ φt t ωt t ψ t η V ; (iii) Wμ φt η Wt ωt ψ η V = ωt ψ η V and Wμ φt η Wt ωt ψ = Wμ φt t ωt t ψ t η V. (2) The following assertions are equivalent: (i) φ η W ω : (V, ψ) (W, φ) is a 2-cell in EM w (K); (ii) φ t ω = Wμ φt η Wt ωt ψ; (iii) Wμ φt η Wt ωt ψ η V = φ η W ω and Wμ φt η Wt ωt ψ = Wμ φt t ωt t ψ t η V. (3) The following assertions are equivalent: (i) φ η W ω and ωt ψ η V are (necessarily equal) 2-cells (V, ψ) (W, φ) in EM w (K); (ii) φ t ω = ωt ψ. Proof. (1) (i) (ii) Using that ψ satisfies (1.1) together with the unitality of the monad t, condition (1.2) for ϱ := ωt ψ η V comes out as the equality in part (ii). Hence in order to prove the equivalence of assertions (i) and (ii), we need to show that (ii) implies that ϱ satisfies (1.3). Indeed, applying the equality in part (ii) in the second step, we obtain Wμ φt η Wt ωt ψ η V = Wμ φt t ωt t ψ t η V η V = ωt ψ η V. (1.6) (ii) (iii) We have seen in the proof of equivalence (i) (ii) above, that assertion (ii) implies (1.6), i.e. the first condition in part (iii). The second condition is checked as follows: Wμ φt η Wt ωt ψ = Wμ φt η Wt ωt Vμ ψt t ψ η t V = Wμ Wμt φtt η Wtt ωtt ψt η Vt ψ = Wμ ωtt ψt η Vt ψ = ωt ψ = Wμ φt t ωt t ψ t η V. (1.7) The first equality follows by (1.1) and unitality of the monad t. In the second equality we used associativity of μ and the interchange law. The third equality is obtained by using (1.6). The

201 G. Böhm / Advances in Mathematics 225 (2010) penultimate equality follows by (1.1) and unitality of the monad t again. The last equality follows by assertion (ii). Conversely, assume that the identities in part (iii) hold. Then ωt ψ = ωt Vμ ψt t ψ η t V = Wμ Wμt φtt η Wtt ωtt ψt η Vt ψ = Wμ φt η Wt ωt ψ = Wμ φt t ωt t ψ t η V. The first equality follows by (1.1) and unitality of the monad t. In the second equality we applied the first condition in part (iii). The third equality is obtained by using the associativity of μ, the interchange law and (1.1) again. The last equality follows by the second condition in part (iii). (2) (i) (ii) Using that φ satisfies (1.1) together with the unitality of the monad t, condition (1.2) for ϱ := φ η W ω comes out as the equality in part (ii). Condition (1.3) holds for ϱ automatically, i.e. it follows by applying (1.1) for φ. (ii) (iii) If (iii) holds, then φ t ω = Wμ φt t φ t η W t ω = Wμ φt t Wμ t φt t η Wt t ωt t ψ t η V = Wμ φt t ωt t ψ t η V = Wμ φt η Wt ωt ψ. The first equality follows by applying (1.1) for φ, together with the unitality of the monad t. The second equality is obtained by the first identity in part (iii). In the penultimate equality we applied the interchange law, associativity of μ, (1.1) on φ and unitality of t. The last equality follows by the second condition in part (iii). Conversely, if assertion (ii) holds, then the first condition in part (iii) is proven by composing both sides of the equality in part (ii) by η V on the right. The second condition, i.e. equality (1.7), is proven by the following computation: Wμ φt t ωt t ψ t η V = Wμ Wμt φtt t φt t η Wt t ωt t ψ t η V = Wμ φt t φ t η W t ω = φ t ω = Wμ φt η Wt ωt ψ. The first equality follows by applying (1.1) for φ, together with the unitality of the monad t. The second equality is obtained by using the associativity of μ, the interchange law and the first condition in part (iii). The third equality follows by applying (1.1) for φ, together with the unitality of the monad t again. The last equality follows by part (ii). (3) Assume first that assertion (3)(i) holds. Then by parts (1) and (2), also (1)(ii) and (2)(ii) hold, implying (1.7). Hence ωt ψ = Wμ φt t ωt t ψ t η V = Wμ φt η Wt ωt ψ = φ t ω. Conversely, if assertion (3)(ii) holds, then Wμ φt t ωt t ψ t η V = Wμ ωtt ψt t ψ t η V = ωt ψ Wμ φt η Wt ωt ψ = Wμ φt η Wt φ t ω = φ t ω, and

202 8 G. Böhm / Advances in Mathematics 225 (2010) 1 32 where in both computations the first equality follows by (3)(ii) and the second equality follows by (1.1) and the unitality of the monad t. We conclude by parts (1) and (2) that both ωt ψ η V and φ η W ω are 2-cells in EM w (K). It follows by comparing the first identities in (1)(iii) and (2)(iii) that φ η W ω = ωt ψ η V, as stated. Next we investigate the behaviour of the correspondences in Lemma 1.2 with respect to the horizontal and vertical compositions in K and EM w (K). Lemma 1.3. For any 2-category K,let(V, ψ), (W, φ) and (U, θ) be 1-cells t t and (V,ψ ) and (W,φ ) be 1-cells t t in EM w (K). (1) If some 2-cells V ω W and V ω W in K satisfy the equivalent conditions in Lemma 1.2(1), then (ω t ψ η V ) (ωt ψ η V)= ω ωt V ψ ψ V η V V. Hence in particular also ω ω satisfies the equivalent conditions in Lemma 1.2(1). (2) If some 2-cells V ω W and V ω W in K satisfy the equivalent conditions in Lemma 1.2(2), then (φ η W ω ) (φ η W ω) = W φ φ W η W W ω ω. Hence in particular also ω ω satisfies the equivalent conditions in Lemma 1.2(2). (3) If some 2-cells V ω W κ U in K satisfy the equivalent conditions in Lemma 1.2(1), then (κt φ η W) (ωt ψ η V)= κt ωt ψ η V. Hence in particular also κ ω satisfies the equivalent conditions in Lemma 1.2(1). (4) If some 2-cells V ω W κ U in K satisfy the equivalent conditions in Lemma 1.2(2), then (θ η U κ) (φ η W ω) = θ η U κ ω. Hence in particular also κ ω satisfies the equivalent conditions in Lemma 1.2(2). Proof. (1) This compatibility with the horizontal composition follows by applying (1.1) for ψ, and unitality of the monad t. (2) This follows by using that ω obeys Lemma 1.2(2)(ii). (3) This compatibility with the vertical composition follows using that, by Lemma 1.2(1), ωt ψ η V satisfies (1.3). (4) This assertion follows by using that κ satisfies Lemma 1.2(2)(ii). The message of Lemmas 1.2 and 1.3 can be summarized as follows. Corollary 1.4. Consider an arbitrary 2-category K. (1) There is a 2-category, to be denoted by Mnd ι (K), defined by the following data: 0-cells are monads t in K; (V,ψ) 1-cells t t are the same as 1-cells in EM w (K), cf.(1.1); 2-cells (V, ψ) ω (W, φ) are 2-cells V ω W in K, satisfying the equivalent conditions in Lemma 1.2(1); horizontal and vertical compositions are the same as in K. Furthermore, there is a 2-functor G ι : Mnd ι (K) EM w (K), acting on the 0- and 1-cells as the identity map and taking a 2-cell (V, ψ) ω (W, φ) to ωt ψ η V. (2) There is a 2-category, to be denoted by Mnd π (K), defined by the following data: 0-cells are monads t in K;

203 G. Böhm / Advances in Mathematics 225 (2010) (V,ψ) 1-cells t t are the same as 1-cells in EM w (K), cf.(1.1); 2-cells (V, ψ) ω (W, φ) are 2-cells V ω W in K, satisfying the equivalent conditions in Lemma 1.2(2); horizontal and vertical compositions are the same as in K. Furthermore, there is a 2-functor G π : Mnd π (K) EM w (K), acting on the 0- and 1-cells as the identity map and taking a 2-cell (V, ψ) ω (W, φ) to φ η W ω. Clearly, both Mnd ι (K) and Mnd π (K) contain Mnd(K) as a vertically full subcategory. 2. Monads in EM w (K) and pre-monads in K It was observed in [16, p. 257] that monads in EM(K) induce monads in K. The aim of this section is to give a similar interpretation of monads in EM w (K). A monad in EM w (K) is given by a triple ((s, ψ), ν, ϑ), consisting of a 1-cell (t,μ,η) (s,ψ) (t,μ,η)and 2-cells (s, ψ) (s, ψ) ν (s, ψ) and (k, t) ϑ (s, ψ) in EM w (K), such that ν ( ν (s, ψ) ) = ν ( (s, ψ) ν ), ν ( ϑ (s, ψ) ) = (s, ψ) = ν ( (s, ψ) ϑ ). In light of Theorem 1.1, this means a 1-cell k s k, and 2-cells ts ψ st, ss ν st and k ϑ st in K, subject to the following identities. ψ μs = sμ ψt tψ, (2.1) sμ ψt tν = sμ νt sψ ψs, (2.2) sμ ψt ηst ν = ν, (2.3) sμ ψt tϑ = sμ ϑt, (2.4) sμ νt sν = sμ νt sψ νs, (2.5) sμ νt sψ ϑs = ψ ηs, (2.6) sμ νt sϑ = ψ ηs. (2.7) Condition (2.1) expresses the requirement that (s, ψ) is a 1-cell in EM w (K), (2.2) and (2.3) together mean that ν is a 2-cell in EM w (K), (2.4) means that ϑ is a 2-cell in EM w (K) (condition (1.3) on ϑ follows by the interchange law in K, (2.4) and unitality of the monad t). Conditions (2.5), (2.6) and (2.7) express associativity and unitality of the monad ((s, ψ), ν, ϑ), after being simplified using (2.1), (2.2), (2.3) and (2.4). (s,ψ) Note that a monad (t t,ν,ϑ) in EM w (K), for a monad k t k in K, is identical to a crossed product system (t, s, ψ, ν) in the monoidal category K(k, k), in the sense of [13, Definition 3.5], subject to the twisted and cocycle conditions in [13, Definitions 3.3 and 3.6], the normalization condition in [13, Proposition 3.7] and identities (11), (12) and (13) in [13, Theorem 3.11], for ϑ. The following definition is inspired by [9, Section 3.1] (see also [13, Definition 2.3]).

204 10 G. Böhm / Advances in Mathematics 225 (2010) 1 32 Definition 2.1. A pre-monad in a 2-category K is a triple (t,μ,η), consisting of a 1-cell k t k μ and 2-cells tt t and k η t, such that the following conditions hold: μ μt = μ tμ, (2.8) μ ηt = μ tη, (2.9) μ ηη = η, (2.10) μ μt ηtt = μ. (2.11) Note that if k t μ k is a 1-cell in a 2-category K and some 2-cells tt t and k η t satisfy μ μt = μ tμ and μ ηt = μ tη = μ μt ηηt as in [9, Section 3.1], then (t, μ := μ μt ηtt,η := μ ηη) is a pre-monad in the sense of Definition 2.1. The motivation for a study of pre-monads stems from the following observation. Lemma 2.2. Consider a pre-monad (t,μ,η)in an arbitrary 2-category K. (1) The 2-cell μ ηt is idempotent. (2) Assume that there exists a 1-cell ˆt and 2-cells t π ι ˆt and ˆt t in K, such that μ ηt = ι π and ˆt = π ι. Then (ˆt, ˆμ := π μ ιι, ˆη := π η) is a monad in K. Proof. The proof is an easy computation using Definition 2.1 of a pre-monad and the properties ι and π obey, so it is left to the reader. Improving [13, Theorem 3.11], we obtain the following generalization of a correspondence between monads in EM(K) and in K, observed by Lack and Street in [16]. Theorem 2.3. For any monad (k t k,μ,η) and any 1-cell k s k in an arbitrary 2-category K, there is a bijective correspondence between the following structures: (i) A monad (t (s,ψ) t,ν,ϑ) in EM w (K); (ii) A pre-monad (st, Θ, ϑ) in K, such that Θ stsμ = sμ Θt. (2.12) Proof. The proof is built on the same constructions as [13, Theorem 3.11]. Assume first that there exist 2-cells ψ and ν as in part (i). A multiplication Θ for the premonad in part (ii) is given by the same formula of a wreath product in [16, p. 256]: Θ := sμ νt ssμ sψt. (2.13) Its associativity is checked by the same computation as in the case of the wreath product in [16]. By (2.6) on one hand, and by (2.4) and (2.7) on the other, Θ ϑst = sμ ψt ηst = Θ stϑ, (2.14)

205 G. Böhm / Advances in Mathematics 225 (2010) proving (2.9). By applying (2.4) again, we conclude that Θ ϑϑ = ϑ, i.e. also (2.10) holds true. Condition (2.11) is proven by the following computation: Θ Θst ϑstst = sμ νt ssμ sψt sμst ψtst ηstst = sμ νt ssμ ssμt sψtt stψt ψtst ηstst = sμ sμt sμtt νttt sψtt ψstt tsψt ηstst = sμ sμt sμtt ψttt tνtt tsψt ηstst = sμ sμt νtt sψt = Θ. The second equality follows by (2.1). The fourth and the fifth equalities follow by (2.2) and (2.3), respectively. Condition (2.12) follows by associativity of μ. This proves that the data in part (i) determine a pre-monad as in part (ii). Conversely, assume that there is a 2-cell Θ as in part (ii). The 2-cells ψ and ν in part (i) are constructed as By (2.12), Moreover, by (2.8), (2.12) and (2.9), ψ := Θ sμst ϑtst tsη, ν := Θ sηsη. (2.15) sμ ψt = Θ sμst ϑtst and sμ νt = Θ sηst. (2.16) Θ stψ = Θ Θst stsμst stϑtst sttsη = Θ sμst Θtst ϑsttst sttsη. (2.17) With identities (2.16), (2.17), (2.12) and (2.10) at hand, (2.1) is verified as sμ ψt tψ = Θ stψ sμts ϑtts = Θ sμst Θtst stsμtst ϑϑttst ttsη = Θ sμst sμtst ϑttst ttsη = ψ μs. Use next (2.16), (2.17), (2.12) and (2.11) to compute sμ νt sψ = Θ stψ sηts = Θ Θst ϑstst stsη = Θ stsη. (2.18) In order to prove that (2.2) holds, apply (2.18), (2.8) and (2.16): sμ νt sψ ψs = Θ ψst tssη = Θ Θst sμstst ϑtstst tsηsη = sμ ψt tν. (2.19) Condition (2.3) is verified by comparing the last and third expressions in (2.19), and using (2.11): sμ ψt ηst ν = Θ Θst ϑstst sηsη = Θ sηsη = ν. Condition (2.4) is proven by using (2.16), (2.9), (2.12) and (2.10):

206 12 G. Böhm / Advances in Mathematics 225 (2010) 1 32 sμ ψt tϑ = Θ stϑ sμ ϑt = Θ stsμ ϑϑt = sμ Θt ϑϑt = sμ ϑt. Condition (2.5) follows by (2.18), (2.8) and (2.16): sμ νt sψ νs = Θ Θst sηsηsη = Θ sηst sθ ssηsη = sμ νt sν. Condition (2.6) is checked by applying (2.18): sμ νt sψ ϑs = Θ ϑst sη = Θ sμst ϑtst ηsη = ψ ηs. (2.20) Finally, (2.7) is proven by making use of (2.16), (2.9) and comparing the second and last expressions in (2.20): sμ νt sϑ = Θ stϑ sη = Θ ϑst sη = ψ ηs. This proves that the data in part (ii) determine a monad as in part (i). It remains to show that the above constructions are mutual inverses. Take 2-cells ν and ψ as in part (i). Use (2.13) to associate a 2-cell Θ as in part (ii) to them, and then use (2.15) to define 2-cells ν and ψ as in part (i) again. We obtain ν = sμ νt sψ sηs = sμ sμt νtt sνt ssϑ = sμ νt ssμ sψt stϑ ν = sμ sμt νtt sϑt ν = sμ ψt ηst ν = ν. The first equality follows by unitality of μ and the second equality follows by (2.7) and associativity of μ. The third equality is obtained by (2.5) and the fourth equality follows by (2.4) and associativity of μ. The penultimate equality is a consequence of (2.7) and the last one follows by (2.3). Also, ψ = sμ νt sψ sμs ϑts = sμ sμt νtt sψt ϑst ψ = sμ ψt tψ ηts = ψ. The first equality follows by unitality of μ, the second equality follows by (2.1) and associativity of μ, and the third equality is obtained by (2.6). The last equality follows by (2.1) and by unitality of μ. In the opposite order, start with a 2-cell Θ as in part (ii). Apply (2.15) to construct 2-cells ψ and ν as in part (i) and then apply (2.13) to obtain a 2-cell Θ as in part (ii). It satisfies Θ = sμ νt ssμ sψt = Θ sηst ssμ sψt = sμ Θt stψt sηtst = Θ sμst Θtst ϑsttst sηtst = Θ Θst ϑstst = Θ. The second equality follows by the second identity in (2.16). The third equality is obtained by (2.12). The fourth equality is a consequence of (2.17), (2.12) and unitality of μ. The penultimate equality follows by (2.12) and unitality of μ. The last equality is obtained by (2.11).

207 G. Böhm / Advances in Mathematics 225 (2010) Examples 2.4. Examples of a composite pre-monad as in Theorem 2.3(ii) are given, first of all, by wreath products in [16]. It is shown in [16, Example 3.3] that crossed products by Hopf algebras in [21,2,12] are examples of a wreath product. As it was observed by Ross Street [20] (see also [13, Example 3.18]), so are the crossed products with coalgebras in [7] and their generalizations to comonads in [22, Section 4.8]. Examples of a composite pre-monad, which are not monads, are provided by weak smash products in [9, Section 3] (for a review see [13, Example 3.16]). This includes smash products with weak bialgebras [6]. Crossed products with weak bialgebras in [18] are also shown in [13, Section 4] to provide examples. Note, however, that (weak) crossed products with bialgebroids in [5, Section 4 & Appendix] are not (pre-)monads of the kind in Theorem 2.3(ii). Let k be a commutative and associative unital ring. A k-algebra B, measured by a left bialgebroid H over a k-algebra L, determines two monads, ( ) k B on the category of k-modules and ( ) L B on the category of L-modules. In terms of the measuring :H k B B and the comultiplication h h 1 L h 2 in H, consider the left L-module map H k B B L H, h k b h 1 b L h 2. It equips the left L-module (or L k bimodule) H with the structure of a 1-cell ( ) L B ( ) LH ( ) k B in EM(CAT) (or in EM w (CAT) if is only a weak measuring). However, if L is a non-trivial k-algebra, this 1-cell has different source and target. Although in [5] the composite endofunctor k B L H on the category of k-modules is proven to carry a monad structure, it is not a composite of a monad with an endofunctor. 3. A pseudo-functor EM w (K) K Throughout this section (and the next one), we make two basic assumptions on the 2-category K we deal with: (i) Idempotent 2-cells in K split; (ii) K admits Eilenberg Moore constructions (EM constructions, for short) for monads. In more details, assumption (i) means that for any 2-cell V ϖ V in K, such that ϖ ϖ = ϖ, there exist a 1-cell ˆV and 2-cells ˆV ι V and V π ˆV, such that π ι = ˆV and ι π = ϖ.itis easy to see that the datum ( ˆV,ι,π)is unique up to an isomorphism ˆV ι V π ˆV. Property (ii) of a 2-category was introduced by Street in [19, p. 151]. It means that the inclusion 2-functor K Mnd(K) (with underlying maps k (k k k,k,k), (k V k ) (k V k, V V V), (V ϱ W) (V ϱ W) on the 0-, 1-, and 2-cells, respectively) possesses a right 2-adjoint. By [16, Section 1], property (ii) can be formulated equivalently by saying that the inclusion 2-functor I : K EM(K) possesses a right 2-adjoint J. Important properties of 2- categories admitting EM constructions for monads are formulated in the following theorem. It is recalled from [19, Theorem 2] and [16, Section 1]. Theorem 3.1. In a 2-category K which admits EM constructions for monads, any monad (k t k,μ,η) determines an adjunction (k f J(t),J(t) v k,k η vf, f v ɛ k) in K, such that

208 14 G. Böhm / Advances in Mathematics 225 (2010) 1 32 (t,μ,η) = (vf,vɛf,η). One can choose JI = K, and the isomorphism corresponding to the 2-adjunction (I, J ) is given by the mutually inverse functors K ( l,j(t) ) EM(K) ( I(l),t ), (V ω W) ( (vv, vɛv ) vω (vw, vɛw ) ), EM(K) ( I(l),t ) K ( l,j(t) ), ( (A, α) ϱ (B, β) ) ( J(A,α) J(ϱ) J(B,β) ), (3.1) for any 0-cell l and monad t in K. The notations in Theorem 3.1 are used throughout, without further explanation. Lemma 3.2. Consider a 2-category K which admits EM constructions for monads. For any (V,ψ) 1-cell (t,μ,η) (t,μ,η ) in EM w (K), the2-cell Vvɛ ψv η Vv: Vv Vv in K is idempotent, and obeys the following identities: Vμ ψt η Vt ψ = ψ; (3.2) Vvɛ ψv t Vvɛ t ψv t η Vv= Vvɛ ψv. (3.3) Proof. All statements follow easily by applying the interchange law, (1.1) and unitality of the monad t. (V,ψ) The idempotent 2-cell in Lemma 3.2, corresponding to a 1-cell (t,μ,η) (t,μ,η ) in EM w (K), is an identity 2-cell if and only if ψ η V = Vη,i.e.(V, ψ) is a 1-cell in EM(K). Our next aim is to extend the 2-functor J in Theorem 3.1 to EM w (K). Our method is reminiscent to the way J is obtained from the right adjoint of the inclusion 2-functor K Mnd(K). Lemma 3.3. Consider a 2-category K which admits EM constructions for monads and in which (V,ψ) idempotent 2-cells split. For a 1-cell (t,μ,η) (t,μ,η ) in EM w (K), denote a chosen splitting of the idempotent 2-cell in Lemma 3.2 by Vv π Ṽ ι Vv. Then (Ṽ, ψ := π Vvɛ ψv t ι) is a 1-cell IJ(t) t in EM(K). Proof. By the interchange law, ψ η Ṽ = π ι π ι = Ṽ. Furthermore, by (3.3) and (1.1), ψ t ψ = π Vvɛ ψv t Vvɛ t ψv t t ι = π Vvɛ Vμv ψtv t ψv t t ι = π Vvɛ ψv μ Vv t t ι = ψ μ Ṽ. Lemma 3.4. Consider a 2-category K which admits EM constructions for monads and in which idempotent 2-cells split. For any 2-cell (V, ψ) ϱ (W, φ) in EM w (K), ϱ := π Wvɛ ϱv ι is a 2-cell in EM(K), between the 1-cells (Ṽ, ψ) and ( W, φ) in Lemma 3.3 (where π and ι denote chosen splittings of both idempotent 2-cells in Lemma 3.2, corresponding to the 1-cells (V, ψ) and (W, φ)). Proof. Apply (3.2) (in the first equality), (1.2) (in the third equality) and (1.3) (in the penultimate equality) to conclude that

209 G. Böhm / Advances in Mathematics 225 (2010) ϱ ψ = π Wvɛ ϱv Vvɛ ψv t ι = π Wvɛ Wμv ϱtv ψv t ι = π Wvɛ Wμv φtv t ϱv t ι = π Wvɛ φv t ι t π t Wvɛ t ϱv t ι = φ t ϱ. Theorem 3.5. Consider a 2-category K which admits EM constructions for monads and in which idempotent 2-cells split. The following maps determine a pseudo-functor J w : EM w (K) K. For a 0-cell t, J w (t) := J(t). (V,ψ) For a 1-cell t t, J w (V, ψ) := J(Ṽ, ψ), where the 1-cell (Ṽ, ψ) in EM(K) is described in Lemma 3.3. That is, denoting by ι and π a chosen splitting of the idempotent 2-cell in Lemma 3.2, J w (V, ψ) is the unique 1-cell J w (t) J w (t ) in K for which v ɛ J w (V, ψ) = π Vvɛ ψv t ι. For a 2-cell (V, ψ) ϱ (W, φ), J w (ϱ) := J( ϱ), where the 2-cell ϱ in EM(K) is described in Lemma 3.4. That is, J w (ϱ) is the unique 2-cell J w (V, ψ) J w (W, φ) in K for which v J w (ϱ) = π Wvɛ ϱv ι. The pseudo-natural isomorphism class of J w is independent of the choice of the 2-cells ι and π in its construction. Proof. Let us fix splittings (π, ι) of the idempotent 2-cells in Lemma 3.2, for all 1-cells (V, ψ) in EM w (K). By construction, v J w (ψ η (V,ψ) V)= π ι π ι = Ṽ, for any 1-cell t t in EM w (K). Hence J w preserves identity 2-cells (V, ψ) ψ η V ϱ τ (V, ψ). For 2-cells (V, ψ) (W, φ) (U, θ) in EM w (K), it follows by (1.3) (applied to ϱ) that v J w (τ) v J w (ϱ) = π Uvɛ τv Wvɛ ϱv ι = π Uvɛ Uμv τtv ϱv ι = v J w (τ ϱ). We conclude by the isomorphism (3.1) that J w preserves the vertical composition. (k,t) For an identity 1-cell t t, the idempotent 2-cell in Lemma 3.2 is the identity 2-cell v by the adjunction relation vɛ ηv = v. Hence any splitting of it yields mutually inverse isomorphisms v π k ι k and k k v. They give rise to an isomorphism J w (t) = J(π k ) J(v,vɛ) J w (k, t) = J( k,π k vɛ tι k ) with the inverse J(ι k ). Thus J w preserves identity 1-cells up to isomorphism. (In particular, we can choose for the definition of J w a trivial splitting v v v v v, in which case the 1-cell ( k, t) in Lemma 3.3 is equal to (v, vɛ). Applying the isomorphism (3.1), we conclude that with this choice, J w strictly preserves identity 1-cells, i.e. J w (k, t) = J(v,vɛ) = J w (t).) In order to investigate the preservation of the horizontal composition, consider different splittings (π, ι) and (π,ι ) of the idempotent 2-cell in Lemma 3.2, for some 1-cell (V, ψ), and denote the corresponding 1-cells in Lemma 3.3 by (Ṽ, ψ) and (Ṽ, ψ ), respectively. Applying (3.3) (for (π,ι )) and (3.2) (for (π, ι)), π Vvɛ ψv t ι t π t ι = π Vvɛ ψv t ι = π ι π Vvɛ ψv t ι.

210 16 G. Böhm / Advances in Mathematics 225 (2010) 1 32 π ι Hence (Ṽ, ψ) (Ṽ, ψ ) is an iso 2-cell in EM(K), so J(Ṽ, ψ) J(π ι) J(Ṽ, ψ ) is an iso 2-cell in K. For 1-cells (V, ψ), (W, φ) : t t and (V,ψ ), (W,φ ) : t t and 2-cells (V, ψ) ϱ (W, φ) and (V,ψ ) ϱ (W,φ ) in EM w (K), the idempotent 2-cell in Lemma 3.2 corresponding to the 1-cell (V,ψ ) (V, ψ) comes out as V Vvɛ V ψv ψ Vv η V Vv. We claim that it has a splitting given by the mono 2-cell V ι ι J w (V, ψ) and the epi 2-cell π J w (V, ψ) V π, where (π, ι) and (π,ι ) are the chosen splittings of the idempotent 2-cells in Lemma 3.2, corresponding to the 1-cells (V, ψ) and (V,ψ ) in EM w (K), respectively. Indeed, by construction of J w (its action on a 1-cell (V, ψ)), (3.2) and (3.3), V ι ι J w (V, ψ) π J w (V, ψ) V π = V ι V π V Vvɛ V ψv V t ι V t π ψ Vv η V Vv = V Vvɛ V ψv ψ Vv η V Vv. (3.4) Denote by V Vv π 2 Ṽ V ι 2 V Vv the canonical splitting of this idempotent which was chosen to define J w on the 1-cell (V,ψ ) (V, ψ) = (V V,V ψ ψ V). By considerations in the previous paragraph, there are mutually inverse iso 2-cells J(π J w (V, ψ) V π ι 2 ) : J w (V V,V ψ ψ V) J w (V,ψ )J w (V, ψ) and J(π 2 V ι ι J w (V, ψ)) : J w (V,ψ )J w (V, ψ) J w (V V,V ψ ψ V). In order to see their naturality, observe that On the other hand, v J w( ϱ ) J w (ϱ) v J w( ϱ ϱ ) = π 2 W Wvɛ W Wμv W ϱtv W ψv ϱ Vv ι 2. = π J w (W, φ) W π W Wvɛ W φv ϱ Wv V ι V π V Wvɛ V ϱv V ι ι J w (V, ψ) = π J w (W, φ) W π W Wvɛ W Wμv W φtv W t ϱv ϱ Vv V ι ι J w (V, ψ) = π J w (W, φ) W π W Wvɛ W Wμv W ϱtv W ψv ϱ Vv V ι ι J w (V, ψ). The second equality follows by applying (1.3), and the third one follows by applying (1.2), for ϱ. With this information in mind, we conclude that v J w( ϱ ϱ ) π 2 V ι ι J w (V, ψ) = π 2 W Wvɛ W Wμv W ϱtv W ψv ϱ Vv V ι ι J w (V, ψ) = π 2 W ι ι J w (W, φ) v J w( ϱ ) J w (ϱ). Thus naturality of J(π 2 V ι ι J w (V, ψ)) follows by the isomorphism (3.1). It remains to check its associativity and unitality. For a further 1-cell (V,ψ ) : t t in EM w (K), use the notation V V Vv π 3 V V V ι 3 V V Vvfor the canonically split idempotent in the construction

211 G. Böhm / Advances in Mathematics 225 (2010) of J w on (V,ψ ) (V,ψ ) (V, ψ) = (V V V,V V ψ V ψ V ψ V V). By (3.4), the associativity condition π 3 V ι 2 ι J w( V V,V ψ ψ V ) v J w( V,ψ ) J ( π 2 V ι ι J w (V, ψ) ) = π 3 V ι 2 V π 2 V V ι V ι J w (V, ψ) ι J w( V,ψ ) J w (V, ψ) = π 3 V V ι V ι J w (V, ψ) ι J w( V,ψ ) J w (V, ψ) = π 3 V V ι ι 2 J w (V, ψ) π 2 J w (V, ψ) V ι J w (V, ψ) ι J w( V,ψ ) J w (V, ψ) holds. The 2-cells ι k and π k, splitting the idempotent (identity) 2-cell in Lemma 3.2 corresponding to a unit 1-cell (k, t), are mutual inverses. Hence also the unitality conditions v J ( π Vι k ιj w (k, t) ) v J w (V, ψ)j (π k ) = π Vι k Vπ k ι = v J w (V, ψ), v J ( ι k J w (V, ψ) ) v J(π k )J w (V, ψ) = ι k J w (V, ψ) π k J w (V, ψ) = v J w (V, ψ) hold. Thus we conclude by the isomorphism (3.1) that J w preserves also the horizontal composition up to a coherent family of iso 2-cells, i.e. that it is a pseudo-functor. Finally, we investigate the ambiguity of the pseudo-functor J w, caused by a free choice of the splittings of the idempotent 2-cells in Lemma 3.2. Take two collections {(π, ι)} and {(π,ι )} of splittings (indexed by the 1-cells in EM w (K)). The pseudo-functors J w and J w, associated to both families of splittings, are pseudo-naturally isomorphic via J w (t) = J w (t) and J w (V, ψ) J(π ι) J w (V, ψ), for any 0-cell t and 1-cell (V, ψ) in EM w (K). Consider a 2-category K which admits EM constructions for monads and in which idempotent (V,ψ) 2-cells split. We can regard a 1-cell t t in EM(K) as a 1-cell in EM w (K). Choosing a trivial splitting Vv Vv Vv Vv Vv of the identity 2-cell, the corresponding 1-cell IJ(t) (Ṽ, ψ) t in Lemma 3.3 comes out as the 1-cell (V v, V vɛ ψv) in EM(K). By 1-naturality of the counit of the 2-adjunction (I, J ),wehave(v J(V,ψ),v ɛ J(V,ψ))= (V v, V vɛ ψv). From this and from the isomorphism (3.1) it follows that J w (V, ψ) = J(Vv,Vvɛ ψv)= J(V,ψ). Similarly, we can regard a 2-cell (V, ψ) ϱ (W, φ) in EM(K) as a 2-cell in EM w (K). The corresponding 2-cell (Ṽ, ψ) ϱ ( W, φ) in Lemma 3.4 is equal to the 2-cell Wvɛ ϱv : (V v, Vvɛ ψv) (W v, W vɛ φv) in EM(K). By the 2-naturality condition v J(ϱ)= Wvɛ ϱv and the isomorphism (3.1) we obtain that J w (ϱ) = J(Wvɛ ϱv) = J(ϱ). Summarizing, we proved that the pseudo-functor J w : EM w (K) K in Theorem 3.5 can be chosen such that the 2-functor J : EM(K) K in Theorem 3.1 factorizes through the obvious inclusion EM(K) EM w (K) and J w. The pseudo-functor J w in Theorem 3.5 takes a monad ((s, ψ), ν, ϑ) in EM w (K) to a monad J w (s, ψ) in K, with multiplication J w (s, ψ)j w (s, ψ) = J w ((s, ψ) (s, ψ)) J w (ν) J w (s, ψ)

212 18 G. Böhm / Advances in Mathematics 225 (2010) 1 32 and unit J w (t) = J w (k, t) J w (ϑ) J w (s, ψ). Applying to this monad in K a hom 2-functor K(l, ) : K CAT (for any 0-cell l in K), we obtain a monad in CAT. Our next aim is to describe its Eilenberg Moore category. Lemma 3.6. Consider a 2-category K which admits EM constructions for monads and in which idempotent 2-cells split. Let l be a 0-cell and (k t (s,ψ) k,μ,η) be a monad in K and let t t be a 1-cell in EM w (K). There is a bijective correspondence between the following structures: (i) Pairs (l V J w (t), J w (s, ψ)v ζ V), consisting of a 1-cell V and a 2-cell ζ in K; W (ii) Pairs ((l k,tw ϱ W),sW λ W), consisting of a 1-cell I(l) (W,ϱ) t in EM(K) and (regarding (W, ϱ) as a 1-cell in EM w (K)), a 2-cell (s, ψ) (W, ϱ) λ (W, ϱ) in EM w (K). Proof. Denote by ι and π the splitting of the idempotent 2-cell in Lemma 3.2, corresponding to the 1-cell (s, ψ) in EM w (K), that was chosen to construct J w (s, ψ). For the 1-cell (W, ϱ) in EM w (K), choose the trivial splitting of the identity 2-cell W W W, so that J w (W, ϱ) = J(W,ϱ). By (3.1), there is a bijection between the 1-cells I(l) (W,ϱ) t in EM(K) as in part (ii), and the 1-cells V := J(W,ϱ)= J w (W, ϱ) : l J(t)= J w (t) in K as in part (i). In order to see that it extends to the stated bijection, take first a 2-cell λ in EM w (K) as in part (ii). Then there is a 2-cell ζ := (J w (s, ψ)v = J w ((s, ψ) (W, ϱ)) J w (λ) V)in K as in part (i). Conversely, for a 2-cell ζ in K as in part (i), put λ := vζ πv. It satisfies ϱ tλ= vɛv tvζ tπv = vζ vɛj w (s, ψ)v tπv = vζ πv svɛv ψvv tιv tπv = λ sϱ ψw. (3.5) The second equality follows by the interchange law and the third one follows by construction of the pseudo-functor J w, cf. Theorem 3.5. The last equality follows by (3.3). Together with the unitality of ϱ, this proves that λ is a 2-cell in EM w (K), as needed. The above two constructions can be seen to be mutual inverses. Take first a pair (V, ζ ) as in part (i) and iterate both constructions. The result is (V, J w (s, ψ)v = J w ((s, ψ) (W, ϱ)) J w (vζ πv) V)= (V, ζ ). In the opposite order, a datum ((W, ϱ), λ) is taken to ((W, ϱ), sw πv vj w (s, ψ)v = vj w ((s, ψ) (W, ϱ)) vj w (λ) W) = ((W, ϱ), λ ιv πv). The resulting 2-cell λ ιv πv in K is equal to λ since by (3.5) and unitality of ϱ, λ ιv πv = λ sϱ ψw ηsw = ϱ tλ ηsw = λ. (3.6) The following extends [16, Proposition 3.1] and also [9, Theorem 3.4]. Proposition 3.7. Consider a 2-category K which admits EM constructions for monads and in which idempotent 2-cells split. Let l be a 0-cell and (k t k,μ,η) be a monad in K and let (t (s,ψ) t,ν,ϑ) be a monad in EM w (K). The following categories are isomorphic: (i) The Eilenberg Moore category EM(K)(I (l), J w (s, ψ)) of the monad K(l, J w (s, ψ)) : K(l, J w (t)) K(l, J w (t));

213 G. Böhm / Advances in Mathematics 225 (2010) (ii) The Eilenberg Moore category EM(K)(I (l), ŝt) of the monad K(l, ŝt) : K(l, k) K(l, k), where the monad ŝt is obtained from the pre-monad st in Theorem 2.3 in the way described in Lemma 2.2; (iii) The category C, with objects that are pairs ((W, ϱ), λ), consisting of a 1-cell I(l) (W,ϱ) t in EM(K) and a 2-cell (s, ψ) (W, ϱ) λ (W, ϱ) in EM w (K), satisfying λ ( (s, ψ) λ ) = λ ( ν (W, ϱ) ) ; (3.7) (W, ϱ) = λ ( ϑ (W, ϱ) ) ; (3.8) morphisms ((W, ϱ), λ) ((W,ϱ ), λ ) that are 2-cells (W, ϱ) α (W,ϱ ) in EM(K) such that λ ( (s, ψ) α ) = α λ. (3.9) Proof. Denote by ι and π the splitting of the idempotent 2-cell in Lemma 3.2, corresponding to the 1-cell (s, ψ) in EM w (K), that was chosen to construct J w (s, ψ). For the 1-cell (W, ϱ) in EM w (K), choose the trivial splitting of the identity 2-cell W W W, so that J w (W, ϱ) = J(W,ϱ). Introduce shorthand notations s := J w (s, ψ) and V := J w (W, ϱ). Isomorphism of EM(K)(I (l), J w (s, ψ)) and C. In light of Lemma 3.6, any object in EM(K)(I (l), J w (s, ψ)) is of the form (J w (W, ϱ), sv = J w ((s, ψ) (W, ϱ)) J w (λ) V),fora unique 1-cell I(l) (W,ϱ) t in EM(K) and a unique 2-cell (s, ψ) (W, ϱ) λ (W, ϱ) in EM w (K). So we only need to show that λ satisfies (3.7), i.e. the equality λ sλ = λ sϱ νw (3.10) of 2-cells in K, if and only if sv = J w ((s, ψ) (W, ϱ)) J w (λ) λ satisfies (3.8), i.e. V is an associative action, and W = λ sϱ ϑw (3.11) if and only if this s-action on V is unital. Compose the associativity condition J w (λ) J w ((s, ψ) λ) = J w (λ) J w (ν (W, ϱ)) with v horizontally on the left, and compose it with the chosen split epimorphism ssw vj w (ssw,ssϱ sψw ψsw) on the right (i.e. on the top ). It yields the equality λ sλ (ssϱ sψw ψsw ηssw) = λ sϱ νw (ssϱ sψw ψsw ηssw). Making use of (3.5), the left-hand side is easily shown to be equal to λ sλ. As far as the righthand side is concerned, use associativity of ϱ (in the first equality), (2.2) and (2.3) (in the second

214 20 G. Böhm / Advances in Mathematics 225 (2010) 1 32 and third equalities, respectively) to see that λ sϱ νw ssϱ sψw ψsw ηssw = λ sϱ sμw νtw sψw ψsw ηssw = λ sϱ sμw ψtw tνw ηssw = λ sϱ νw. This proves that the s-action on V is associative if and only if (3.10) holds. Similarly, the unitality condition J w (λ) J w (ϑ (W, ϱ)) = V is equivalent to λ ιv πv sϱ ϑw = W, hence by (3.6) it is equivalent to (3.11). Thus the bijection in Lemma 3.6 restricts to a bijection between the objects in EM(K)(I (l), J w (s, ψ)) and the objects in C. For a 2-cell (W, ϱ) α (W,ϱ ) in EM(K), the condition J w (λ ) J w ((s, ψ) α) = J w (α) J w (λ) (expressing that J w (α) = J(α) is a morphism in EM(K)(I (l), J w (s, ψ))) is equivalent to λ sα ιv πv = α λ ιv πv. The right-hand side is equal to α λ by (3.6) and the left-hand side is equal to λ sα sϱ ψw ηsw = λ sϱ ψw ηsw sα = ϱ tλ ηsw sα = λ sα, using that α is a 2-cell in EM(K), (3.5) and unitality of ϱ. Hence J w (α) = J(α) is a morphism in EM(K)(I (l), J w (s, ψ)) if and only if (3.9) holds. Thus we conclude by the isomorphism (3.1) that the 2-functor J induces an (obviously functorial) bijection between the morphisms in EM(K)(I (l), J w (s, ψ)) and the morphisms in C. Isomorphism of EM(K)(I (l), ŝt) and C. In view of (2.14), we can choose ŝt = vsf as 1-cells in K. Moreover, taking axioms (2.8), (2.9) and (2.11) of a pre-monad into account, ιf πf Θ = Θ ιf st πf st = Θ stιf stπf = Θ. (3.12) For an object (l W k,vsfw γ W) in EM(K)(I (l), ŝt), put ϱ := γ πf W sμw ϑtw and λ := γ πf W sηw. (3.13) We show that ((W, ϱ), λ) is an object in C. Recall that associativity and unitality of γ read as γ vsfγ = γ πf W ΘW ιf ιf W and W = γ πf W ϑw, respectively. Hence using associativity of γ and (3.12) (in the second equality) and applying the first identity in (2.16) (in the last equality), ϱ tγ tπfw = γ vsfγ πf πf W sμstw ϑtstw = γ πf W ΘW sμstw ϑtstw = γ πf W sμw ψtw. (3.14) Moreover, apply associativity of γ and (3.12) (in the second equality) and use (2.12) (in the third equality) to obtain λ sϱ = γ vsfγ πf πf W sηstw ssμw sϑtw = γ πf W ΘW stsμw stϑtw sηtw = γ πf W sμw ΘtW stϑtw sηtw = γ πf W. (3.15)

215 G. Böhm / Advances in Mathematics 225 (2010) In the last equality we used (2.14) and that (since μ = vɛf ) the interchange law yields sμ ιf t πf t = ιf πf sμ. With these identities at hand, associativity of ϱ is checked as ϱ tϱ = ϱ tγ tπfw tsμw tϑtw = γ πf W sμw sμtw ψttw tϑtw = γ πf W sμw sμtw ϑttw = ϱ μw. The second equality follows by (3.14) and by associativity of μ. In the third equality we applied (2.4). The last equality follows by associativity of μ and the form of ϱ in (3.13). The unitality condition ϱ ηw = W follows by unitality of μ and unitality of γ. Conditions (3.5), (3.10) and (3.11) are proven by ϱ tλ= ϱ tγ tπfw tsηw = γ πf W sμw ψtw tsηw = λ sϱ ψw; λ sλ = γ vsfγ πf πf W sηsηw = γ πf W νw = λ sϱ νw; W = γ πf W ϑw = λ sϱ ϑw. In each case, the last equality follows by (3.15). In the first computation, the second equality follows by (3.14). In the second equality of the second computation we used associativity of γ together with (3.12) and we applied the expression of ν in (2.15). In the first equality of the last computation we used unitality of γ. This proves that ((W, ϱ), λ) is an object in C. Conversely, for an object ((W, ϱ), λ) in C, put It is associative as γ πf W ΘW ιf ιf W = λ sϱ ΘW ιf ιf W γ := λ sϱ ιf W. (3.16) = λ sϱ νw ssϱ sψw stsϱ ιf ιf W = λ sλ ssϱ sψw stsϱ ιf ιf W = λ sϱ stλ stsϱ ιf ιf W = γ vsfγ. The first equality follows by (3.16) and (3.12). In the second equality we substituted Θ by its expression in (2.13) and we used associativity of ϱ twice. In the third equality we applied (3.10) and in the fourth one we used (3.5). By (2.4) and unitality of μ, ιf πf ϑ = sμ ψt ηst ϑ = ϑ. Hence the unitality condition γ πf W ϑw = W follows by (3.11). This proves that (W, γ ) is an object in EM(K)(I (l), ŝt). Let us see that the above constructions are mutual inverses. Starting with an object (W, γ ) of EM(K)(I (l), ŝt) and iterating the above constructions, we re-obtain (W, γ ) by (3.15). In the opposite order, applying both constructions to an object ((W, ϱ), λ) of C, we obtain ((W, λ sϱ ιf W πf W sμw ϑtw),λ sϱ ιf W πf W sηw). Since ιf πf sμ = sμ ιf t πf t = sμ Θt ϑstt, axiom (2.10) of a pre-monad, associativity of ϱ and (3.11) imply that λ sϱ ιf W πf W sμw ϑtw = λ sϱ sμw ϑtw = λ sϱ ϑw ϱ = ϱ.

216 22 G. Böhm / Advances in Mathematics 225 (2010) 1 32 Also, by (2.14), unitality of μ, (3.5) and unitality of ϱ, λ sϱ ιf W πf W sηw = λ sϱ sμw ψtw ηsηw = λ sϱ ψw ηsw = λ. Hence we constructed a bijection between the objects of EM(K)(I (l), ŝt) and C. It is immediate by the form of the bijection between the objects that a 2-cell W α W in K is a morphism in EM(K)(I (l), ŝt) if and only if it is a morphism in C. 4. Weak liftings If K is a 2-category which admits EM constructions for monads, then liftings of 1- and 2-cells for monads in K arise as images under the right 2-adjoint J of the inclusion 2-functor K EM(K), see [17]. In this section we discuss weak liftings and the role what the pseudofunctor J w plays in their description. Definition 4.1. Consider a 2-category K which admits EM constructions for monads. We say that a 1-cell k V k in K possesses a weak lifting for some monads (k t k,μ,η) and (k t k,μ,η ) in K if there exist a 1-cell J(t) V J(t ) and a split mono 2-cell v V ι Vv (with a retraction denoted by Vv π v V ). If in a 2-category K whichadmitsem constructionsfor monadsalso idempotent 2-cellssplit, (V,ψ) then we know by Theorem 3.5 that, for every 1-cell t t in EM w (K), the underlying 1-cell k V k in K possesses a weak lifting J w (V, ψ) for t and t. As we will see later in this section, in fact in such a 2-category K, up-to an isomorphism, every weak lifting arises in this way. This extends assertions about 1-cells in [17, Lemma 3.9 and Theorem 3.10]. Definition 4.2. Consider a 2-category K which admits EM constructions for monads. Let (k t k, μ, η) and (k t k,μ,η ) be monads, and k V k and k W k be 1-cells in K, such that there exist their weak liftings (J (t) V J(t ), ι V,π V ) and (J (t) W J(t ), ι W,π W ) for t and t.fora 2-cell V ω W in K, we say that a 2-cell V ω W is a weak ι-lifting of ω if ιw v ω = ωv ι V ; a 2-cell V ω W is a weak π-lifting of ω if v ω π V = π W ωv. Throughout, indices of ι and π are omitted, as they can be reconstructed without ambiguity from the context. By the isomorphism (3.1), the weak ι-lifting or weak π-lifting of a 2-cell is unique, provided that it exists. Moreover, if a 2-cell ω in Definition 4.2 possesses both a weak ι-lifting ω and a weak π-lifting ω, then v ω = π ι v ω = π ωv ι = v ω π ι = v ω. Hence in view of the isomorphism (3.1), ω = ω.

217 G. Böhm / Advances in Mathematics 225 (2010) Proposition 4.3 below, about weak liftings of 2-cells in a (nice enough) 2-category, extends statements about 2-cells in [17, Lemma 3.9 and Theorem 10]. Therein, notions and notations introduced in Corollary 1.4 are used. Proposition 4.3. Consider a 2-category K which admits EM constructions for monads and in (V,ψ) which idempotent 2-cells split. Let t t and t (W,φ) t be 1-cells in EM w (K) and V ω W be a 2-cell in K. Denote by ι and π the splittings of both idempotent 2-cells in Lemma 3.2, corresponding to the 1-cells (V, ψ) and (W, φ), that were chosen to construct J w (V, ψ) and J w (W, φ), respectively. (1) The following assertions are equivalent: (i) ω is a 2-cell (V, ψ) (W, φ) in Mnd ι (K); (ii) ωt ψ η V : (V, ψ) (W, φ) is a 2-cell in EM w (K); (iii) π ωv ι : (v J w (V, ψ), v ɛ J w (V, ψ)) (v J w (W, φ), v ɛ J w (W, φ)) is a 2-cell in EM(K) such that ι π ωv ι = ωv ι; (iv) ω possesses a weak ι-lifting ω : (J w (V, ψ), ι, π) (J w (W, φ), ι, π). If these equivalent statements hold, then v J w G ι (ω) = π ωv ι, that is, ω = J w G ι (ω). (2) The following assertions are equivalent: (i) ω is a 2-cell (V, ψ) (W, φ) in Mnd π (K); (ii) φ η W ω : (V, ψ) (W, φ) is a 2-cell in EM w (K); (iii) π ωv ι : (v J w (V, ψ), v ɛ J w (V, ψ)) (v J w (W, φ), v ɛ J w (W, φ)) is a 2-cell in EM(K) such that π ωv ι π = π ωv; (iv) ω possesses a weak π-lifting ω : (J w (V, ψ), ι, π) (J w (W, φ), ι, π). If these equivalent statements hold, then v J w G π (ω) = π ωv ι, that is, ω = J w G π (ω). (3) The following assertions are equivalent: (i) φ t ω = ωt ψ; (ii) φ η W ω and ωt ψ η V are (necessarily equal) 2-cells (V, ψ) (W, φ) in EM w (K); (iii) π ωv ι : (v J w (V, ψ), v ɛ J w (V, ψ)) (v J w (W, φ), v ɛ J w (W, φ)) is a 2-cell in EM(K) such that ι π ωv = ωv ι π; (iv) ω possesses both a weak ι-lifting and a weak π-lifting 2-cell (J w (V, ψ), ι, π) (J w (W, φ), ι, π) (which are necessarily equal). Proof. (1) (i) (ii) This equivalence follows by Lemma 1.2(1). (ii) (iii) The 2-cell π ωv ι in K is a 2-cell in EM(K) if and only if v ɛ J w (W, φ) t π t ωv t ι = π ωv ι v ɛ J w (V, ψ). Compose this equality by f horizontally on the right, and compose it vertically by ιf on the left and by t πf t Vη on the right. By virtue of (3.2), (3.3) and the adjunction relation ɛf fη= f, the resulting equivalent condition is identical to (1.7). Property ι π ωv ι = ωv ι is equivalent to ι π ωv ι π = ωv ι π, what is easily seen to be equivalent to (1.6). Thus we conclude by Lemma 1.2(1)(i) (iii). (iii) (iv) By the isomorphism (3.1), π ωv ι is a 2-cell in EM(K) if and only if there is a 2-cell J w (V, ψ) ω J w (W, φ) in K such that v ω = π ωv ι. Clearly, ι π ωv ι = ωv ι if and only if ω is a weak ι-lifting of ω. If the equivalent statements (i) (iv) hold, then v J w( ωt ψ η V ) = π Wvɛ ωtv ψv η Vv ι = π ωv ι π ι = π ωv ι.

218 24 G. Böhm / Advances in Mathematics 225 (2010) 1 32 (2) (i) (ii) This equivalence follows by Lemma 1.2(2). (ii) (iii) As we have seen in the proof of part (1), π ωv ι is a 2-cell in EM(K) if and only if (1.7) holds. Property π ωv ι π = π ωv is equivalent to ι π ωv ι π = ι π ωv hence to the first condition in Lemma 1.2(2)(iii). Thus we conclude by Lemma 1.2(2)(i) (iii). (iii) (iv) is proven by the same reasoning as in part (1). If the equivalent statements (i) (iv) hold, then v J w( φ η W ω ) = π Wvɛ φv η Wv ωv ι = π ι π ωv ι = π ωv ι. (3) These equivalences follow immediately by Lemma 1.2(3) and parts (1) and (2) in the current theorem. For suggesting the following theorem, the author is grateful to the referee. Consider a 2-category K which admits EM constructions for monads. To any monads (k t k, μ, η) and (k k t,μ,η ) in K, we can associate categories Lift ι (t, t ) and Lift π (t, t ), as follows. In both categories objects are quadruples (V, V,ι,π)such that the 1-cell J(t) V J(t ) in K is a weak lifting of the 1-cell k V k, corresponding to the split monic 2-cell v V ι Vv, with a retraction Vv π v V. Morphisms (V, V,ι,π) (W, W,ι,π) are pairs of 2-cells (ω, ω) in K such that V ω W is a weak ι-lifting, respectively, a weak π-lifting, of V ω W. Composition of morphisms is defined via component-wise composition of 2-cells in K. Theorem 4.4. For any 2-category K which admits EM constructions for monads and in which idempotent 2-cells split, and for any monads (k t k,μ,η) and (k t k,μ,η ) in K, the following assertions hold. (1) Lift ι (t, t ) is equivalent to the category Mnd ι (K)(t, t ). (2) Lift π (t, t ) is equivalent to the category Mnd π (K)(t, t ). For t = t, these equivalences are also strong monoidal, with respect to the monoidal structure of Lift ι/π (t, t) induced by the horizontal composition in K. (V,ψ) Proof. (1) For any 1-cell t t in EM w (K), denote by Vv π c v J w ι (V, ψ) c Vv the chosen splitting of the idempotent 2-cell in Lemma 3.2, used to construct J w (V, ψ). By Corollary 1.4 and Theorem 3.5, there is a pseudo-functor J w G ι : Mnd ι (K) K. By Proposition 4.3(1)(i) (iv), it induces a functor G : Mnd ι (K)(t, t ) Lift ι (t, t ), with object map (V, ψ) (V, J w (V, ψ), ι c,π c ) and morphism map ω (ω, J w G ι (ω)). In the opposite direction, consider a functor F : Lift ι (t, t ) Mnd ι (K)(t, t ) with the object map (V, V,ι,π) ( V,ψ := ιf v ɛ Vf t πf t Vη: t V Vt ) (4.1) and morphism map (ω, ω) ω. By the form of ψ in (4.1) and the adjunction relation vɛ ηv = v, it follows that Vvɛ ψv = ι v ɛ V t π. (4.2)

219 G. Böhm / Advances in Mathematics 225 (2010) Using (4.2) together with the form of ψ in (4.1) and naturality, it is easily checked that ψ satisfies (1.1), i.e. (V, ψ) is an object in Mnd ι (K)(t, t ). By (4.2), the associated idempotent in Lemma 3.2 obeys ι c π c = Vvɛ ψv η Vv= ι π. Applying (4.2) together with (3.2) and (3.3), respectively, we conclude that v ɛ J w (V, ψ) t π c t ι = π c ι v ɛ V and π ι c v ɛ J w (V, ψ) = v ɛ V t π t ι c. That is, there are 2-cells (v J w (V, ψ), v ɛ J w (V, ψ)) π c ι π ι c (v V,v ɛ V) and (v V,v ɛ V) (v J w (V, ψ), v ɛ J w (V, ψ)) in EM(K). By (3.1) they induce mutually inverse isomorphisms J w (V, ψ) J(π ι c) V and V J(π c ι) J w (V, ψ) in K. Both of these 2-cells are weak ι-liftings of the identity 2-cell V V V. Hence, for any morphism (V, V,ι,π) (ω,ω) (W, W, ι,π) in Lift ι (t, t ), the composite J(π c ι) ω J(π ι c ) : J w (V, ψ) J w (W, φ) is a weak ι c -lifting of ω (where both ψ and φ are defined via (4.1)). Thus it follows by Proposition 4.3(1)(iv) (i) that ω is a morphism (V, ψ) (W, φ) in Mnd ι (K)(t, t ). This proves that F is a well-defined functor. For any object (V, ψ) in Mnd ι (K)(t, t ), we obtain FG(V,ψ)= (V, ψ) by (3.2), (3.3) and unitality of μ. Evidently, also FG(ω) = ω. For any object (V, V,ι,π) of Lift ι (t, t ), we obtain GF (V, V,ι,π)= (V, J w (V, ψ), ι c,π c ). The mutually inverse isomorphisms (V, V,ι,π) (V,J (π c ι)) (V, J w (V, ψ), ι c,π c ) and (V, J w (V, ψ), ι c,π c ) (V,J (π ι c)) (V, V,ι,π)in Lift ι (t, t ) define, in turn, mutually inverse natural isomorphisms between the identity functor and GF. Indeed, for any morphism (V, V,ι,π) (ω,ω) (W, W,ι,π) in Lift ι (t, t ), we conclude by Corollary 1.4 and Theorem 3.5 that v J w G ι (ω) π c ι = π c Wvɛ ωtv ψv η Vv ι = π c ωv ι = π c ι v ω. Hence naturality follows by the isomorphism (3.1). It remains to prove strong monoidality of G in the t = t case. Recall from the proof of Theorem 3.5 that the coherent natural isomorphisms J w (V,ψ )J w (V, ψ) j V,V J w ((V,ψ ) (V, ψ)) and J w (t) j 0 J w (k, t), rendering J w (hence J w G ι ) a pseudo-functor, arise as weak ι-liftings of identity 2-cells, for any 1-cells (V,ψ ), (V, ψ) : t t in EM w (K). Hence they induce a strong monoidal structure G(V,ψ )G(V, ψ) (V V,j V,V ) G((V,ψ ) (V, ψ)) and (k, J w (t)) (k,j 0) G(k, t) of G. Part (2) is proven symmetrically. 5. Applications In this section we collect from the literature several situations where weak liftings occur. The following corollary is a consequence of Proposition 4.3 and Theorem 4.4. Corollary 5.1. Consider a 2-category K which admits EM constructions for monads and in which idempotent 2-cells split. Let (k t k,μ,η) be a monad and (k c k,δ,ε) be a comonad in K. (1) The following assertions are equivalent:

220 26 G. Böhm / Advances in Mathematics 225 (2010) 1 32 (i) There exists a comonad ((c, ψ), δ, ε) in Mnd ι (K). That is, there exists a 1-cell t (c,ψ) t in EM w (K), satisfying δt ψ = ccμ cψt ψct tδt tψ tηc; (5.1) εt ψ = μ tεt tψ tηc. (5.2) (ii) There is a comonad (t (c,ψ) t,δt ψ ηc,εt ψ ηc) in EM w (K). (iii) There are a comonad (J w (t) c J w (t), δ, ε)and a split monic 2-cell vc ι cv in K such that δ is a weak ι-lifting of δ and ε is a weak ι-lifting of ε. If these equivalent statements hold, then we say shortly that the comonad (c, δ, ε)is a weak ι-lifting of the comonad (c,δ,ε)for the monad (t,μ,η). (2) The following assertions are equivalent: (i) There exists a comonad ((c, ψ), δ, ε) in Mnd π (K). That is, there exists a 1-cell t (c,ψ) t in EM w (K), satisfying cψ ψc tδ = ccμ cψt ψct ηcct δt ψ; (5.3) tε = εt ψ. (5.4) (ii) There is a comonad (t (c,ψ) t,cψ ψc ηcc δ,η ε) in EM w (K). (iii) There are a comonad (J w (t) c J w (t), δ, ε)and a split epi 2-cell cv π vc in K such that δ is a weak π-lifting of δ and ε is a weak π-lifting of ε. If these equivalent statements hold, then we say shortly that the comonad (c, δ, ε)is a weak π-lifting of the comonad (c,δ,ε)for the monad (t,μ,η). (3) The following assertions are equivalent: (i) There exists a 1-cell t (c,ψ) t in EM w (K), satisfying cψ ψc tδ = δt ψ; (5.5) tε = εt ψ. (5.6) (ii) There are a comonad (J w (t) c J w (t), δ,ε) and a split epi-mono pair of 2-cells cv π vc ι cv in K such that δ is both a weak ι-lifting and a weak π-lifting of δ and ε is both a weak ι-lifting and a weak π-lifting of ε. Note that by Lemmas 1.2 and 1.3, in parts (1) and (2) of Corollary 5.1, assertions (i) and (ii) are equivalent in case of an arbitrary 2-category K. Let us stress the (tiny) difference between a 2-cell tc ψ ct in K occurring in Corollary 5.1(3)(i), and a mixed distributive law. A 2-cell ψ in Corollary 5.1(3)(i) satisfies three of the identities defining a mixed distributive law: compatibility with the multiplication of the monad (as (c, ψ) is a 1-cell in EM w (K)), compatibility with the comultiplication of the comonad (by (5.5)) and compatibility with the counit of the comonad (by (5.6)). However, the fourth condition on a mixed distributive law, compatibility ψ ηc = cη with the unit of the monad, does not appear in Corollary 5.1(3)(i) it plays no role in a weak lifting.

221 G. Böhm / Advances in Mathematics 225 (2010) Example 5.2. Generalizing a mixed distributive law of a monad and a comonad (in particular in the bicategory BIM), weak entwining structures were introduced by Caenepeel and De Groot in [9]. The axioms are obtained by weakening the compatibility conditions of a mixed distributive law with the unit of the monad and the counit of the comonad. Precisely, a weak entwining structure in an arbitrary 2-category K consists of a monad (k t k,μ,η), a comonad (k c k, δ,ε) and a 2-cell tc ψ ct subject to the following conditions: ψ μc = cμ ψt tψ; (5.7) δt ψ = cψ ψc tδ; (5.8) ψ ηc = cεt cψ cηc δ; (5.9) εt ψ = μ tεt tψ tηc. (5.10) We claim that under these assumptions ((c, ψ), δt ψ ηc,εt ψ ηc) is a comonad in EM w (K). For that, we need to show that axioms (5.7) (5.10) imply (5.1). Indeed, ccμ cψt ψct tδt tψ tηc = ccμ δtt ψt tψ tηc = δt ψ. The first equality follows by (5.8) and the second one follows by (5.7) and unitality of the monad t. Hence if moreover K admits EM constructions for monads and idempotent 2-cells in K split (hence there exists the pseudo-functor J w ) then, by Corollary 5.1(1), the comonad c has a weak ι-lifting for the monad t. For a commutative, associative and unital ring k, consider a k-algebra A and a k-coalgebra C. Let Ψ : C k A A k C be a k-module map such that the triple (( ) k A,( ) k C, ( ) k Ψ) is a weak entwining structure in CAT. (If we are ready to cope with the more involved situation of a bicategory, we can say simply that (A,C,Ψ)is a weak entwining structure in BIM.) The corresponding weak ι-lifting of the comonad ( ) k C for the monad ( ) k A is studied in [9, Section 2]. Brzeziński showed in [8, Proposition 2.3] that it can be described as a comonad ( ) A C on the category of right A-modules, where the A-coring (i.e. comonad A A in BIM) C is constructed as a k-module retract of A k C. Examples of weak entwining structures, thus examples of weak ι-liftings of comonads for monads, are provided by weak Doi Koppinen data in [3] (see [9]), i.e. by comodule algebras and module coalgebras of weak bialgebras. Further examples are weak comodule algebras of bialgebras in [11, Proposition 2.3]. Example 5.3. Another generalization of a mixed distributive law, motivated by partial coactions of Hopf algebras, is due to Caenepeel and Janssen. Following [10, Proposition 2.6], a partial entwining structure in a 2-category K consists of a monad (k t k,μ,η), a comonad (k c k, δ,ε) and a 2-cell tc ψ ct in K, such that identities (5.4) and (5.7) hold, together with Observe that axiom (5.11) implies (5.3): ccμ cψt cηct δt ψ = cψ ψc tδ. (5.11)

222 28 G. Böhm / Advances in Mathematics 225 (2010) 1 32 ccμ cψt ψct ηcct δt ψ = ccμ ccμt cψtt cηctt δtt ψt ηct ψ = ccμ cψt cηct δt ψ = cψ ψc tδ. The first and last equalities follow by (5.11) and the second equality is obtained using associativity of μ and (3.2). This implies that ((c, ψ), cψ ψc ηcc δ,η ε) is a comonad in EM w (K). Thus if moreover K is a 2-category which admits EM constructions for monads and in which idempotent 2-cells split, then we conclude by Corollary 5.1(2) that a partial entwining structure (t,c,ψ)in K induces a weak π-lifting of the comonad c for the monad t. Consider the particular case when a monad t := ( ) k A in CAT is induced by an algebra A over a commutative, associative and unital ring k, a comonad c := ( ) k C is induced by a k-coalgebra C and a natural transformation tc ψ ct is induced by a k-module map C k A A k C. Then the weak π-lifting of the comonad c for the monad t, induced by a partial entwining ψ, is a comonad ( ) A C on the category of right A-modules. The A-coring C was constructed in [10, Proposition 2.6] as a k-module retract of A k C. Examples of partial entwining structures (hence of weak π-liftings of a comonad for a monad) are provided by partial comodule algebras of bialgebras in [11, Proposition 2.6]. Example 5.4. Yet another way to generalize a mixed distributive law was proposed in [10]. Following [10, Proposition 2.5], a lax entwining structure in a 2-category K consists of a monad (k t k,μ,η), a comonad (k c k,δ,ε) and a 2-cell tc ψ ct in K, such that identities (5.7), (5.10) and (5.11) hold, together with cμ ctεt ctψ ctηc ψc tδ ηc = ψ ηc. As we observed in Example 5.3, (5.11) implies (5.3), and (5.10) is identical to (5.2). However, none of (5.1) and (5.4) seems to hold for an arbitrary lax entwining structure. Still, the axioms of a lax entwining structure allow us to prove that there is a comonad ((c, ψ), cψ ψc ηcc δ,εt ψ ηc) in EM w (K). Therefore, if K admits EM constructions for monads and idempotent 2-cells in K split, then J w takes it to a comonad (J w (t) c J w (t), δ, ε)in K. However, it is neither a weak ι-lifting nor a weak π-lifting of the comonad c, it is of a mixed nature. In the particular case when a lax entwining structure in CAT is induced by modules over a commutative associative and unital ring, the comonad (c, δ, ε)is induced by a coring, which was computed in [10, Proposition 2.5]. Examples of lax entwining structures are provided by lax comodule algebras of bialgebras in [11, Proposition 2.5]. A fourth logical possibility, to obtain a comonad structure on a weak lifting for a monad t of a 1-cell c underlying a comonad (c,δ,ε), is to allow the comultiplication to be a weak ι-lifting (c,ψ) t in EM w (K) of δ and the counit to be a weak π-lifting of ε. That is, to require a 1-cell t to satisfy (5.1) and (5.4). By (the proof of) Lemma 1.3, it yields a coassociative and counital comonad (c, δ, ε)in K. For any 2-category K, one may consider the vertically-opposite 2-category K. The 2-category K has the same 0-, 1-, and 2-cells as K, the same horizontal composition and the opposite vertical composition. Obviously, 2-cells in K split if and only if 2-cells in K split. Since monads in K are the same as the comonads in K, the 2-category K admits EM constructions for monads if and only if K admits EM constructions for comonads, cf. [17]. In this case we denote by J w : EMw (K ) K the pseudo-functor in Theorem 3.5.

223 G. Böhm / Advances in Mathematics 225 (2010) Definition 5.5. Consider a 2-category K which admits EM constructions for comonads. We say that a 1-cell V in K possesses a weak lifting V for some comonads c and c, provided that, regarded as 1-cells in K, V is a weak lifting of V for the monads c and c in K. For a 2-cell ω in K,aweak ι-lifting (resp. weak π-lifting) for some comonads c and c in K is defined as a weak π-lifting (resp. weak ι-lifting) of ω, regarded as a 2-cell in K, for the monads c and c in K. The following corollary is obtained by applying Corollary 5.1 to the vertically-opposite of a 2-category. Therein, the symbol denotes the vertical composition in K (not its opposite). Corollary 5.6. Consider a 2-category K which admits EM constructions for comonads and in which idempotent 2-cells split. Let (k t k,μ,η) be a monad and (k c k,δ,ε) be a comonad in K. (1) The following assertions are equivalent: (i) There is a monad ((t, ψ), μ, η) in Mnd ι (K ). That is, there exists a 1-cell c (t,ψ) c in EM w (K ) (i.e. a 2-cell tc ψ ct in K such that δt ψ = cψ ψc tδ), satisfying ψ μc = cεt cψ cμc ψtc tψc ttδ; (5.12) ψ ηc = cεt cψ cηc δ. (5.13) (ii) There is a monad (c (t,ψ) c,εt ψ μc, εt ψ ηc) in EM w (K ). (iii) There are a monad (J w(c) t J w(c), μ, η)and a split epi 2-cell π in K such that μ is a weak π-lifting of μ and η is a weak π-lifting of η. If these equivalent statements hold, then we say shortly that the monad (t, μ, η)is a weak π-lifting of the monad (t,μ,η)for the comonad (c,δ,ε). (2) The following assertions are equivalent: (i) There is a monad ((t, ψ), μ, η) in Mnd π (K ). That is, there exists a 1-cell c (t,ψ) c in EM w (K ), satisfying cμ ψt tψ = ψ μc εttc ψtc tψc ttδ; (5.14) cη = ψ ηc. (5.15) (ii) There is a monad (c (t,ψ) c,μ εtt ψt tψ,η ε) in EM w (K ). (iii) There are a monad (J w(c) t J w(c), μ, η) and a split monic 2-cell ι in K such that μ is a weak ι-lifting of μ and η is a weak ι-lifting of η. If these equivalent statements hold, then we say shortly that the monad (t, μ, η)is a weak ι-lifting of the monad (t,μ,η)for the comonad (c,δ,ε). (3) The following assertions are equivalent: (i) There exists a 1-cell t (c,ψ) t in EM w (K ), satisfying cμ ψt tψ = ψ μc; (5.16) cη = ψ ηc. (5.17)

224 30 G. Böhm / Advances in Mathematics 225 (2010) 1 32 (ii) There are a monad (J w(c) t J w (c), μ, η) and a split epi-mono pair (π, ι) of 2-cells in K such that μ is both a weak ι-lifting and a weak π-lifting of μ and η is both a weak ι-lifting and a weak π-lifting of η. A 2-cell ψ in Corollary 5.6(3)(i) differs from a mixed distributive law by the compatibility condition with the counit of the comonad. In a 2-category K which admits EM constructions for both monads and comonads and in which idempotent 2-cells split, one can say more about weak entwining structures than it was said in Example 5.2. Proposition 5.7. Consider a 2-category K which admits EM constructions for both monads and comonads and in which idempotent 2-cells split. For a monad (k t k,μ,η), a comonad (k c k, δ,ε), and a 2-cell tc ψ ct in K (with a chosen splitting (π, ι) of the associated idempotent in Lemma 3.2), the following assertions are equivalent: (i) The triple (t,c,ψ)is a weak entwining structure, that is, axioms (5.7) (5.10) are satisfied. (ii) The 2-cell ψ induces both a weak ι-lifting of the comonad c for the monad t and a weak π-lifting of the monad t for the comonad c. That is to say, the assertions in Corollary 5.1(1) and Corollary 5.6(1) hold. Proof. We have seen in Example 5.2 that axioms (5.7) (5.10) imply (5.1). Similarly, they can be seen to imply (5.12) as well, applying first (5.7) and next (5.8). Proposition 5.7 is the basis of a construction in [4] of a 2-category of weak entwining structures in any 2-category. In that paper, for a weak entwining structure in a 2-category K which admits EM constructions for both monads and comonads and in which idempotent 2-cells split, it is proven that the weakly lifted monad, and the weakly lifted comonad, occurring in part (ii) of Proposition 5.7, possess equivalent Eilenberg Moore objects. The characterization of weak entwining structures in Proposition 5.7 can be used, in particular, to describe weak bialgebras [6] in terms of weak liftings. Recall that a weak bialgebra H over a commutative, associative and unital ring k,isak-module which possesses both a k-algebra and a k-coalgebra structure, subject to the following compatibility conditions. Denote the multiplication H k H H in H by juxtaposition of elements. Write 1 for the unit element of the algebra H and write ε : H k for the counit. For the comultiplication H H k H, use a Sweedler type index notation h h 1 k h 2. With these notations, the axioms ( hh ) 1 ( k hh ) 2 = h 1 h 1 k h 2 h 2 ; (5.18) = 1 1 k 1 2 k 1 3 = 1 1 k k 1 2 ; (5.19) ε(h11 )ε ( 1 2 h ) = ε ( hh ) = ε(h1 2 )ε ( 1 1 h ) (5.20) are required to hold, for all elements h and h of H. However, the comultiplication is not required to preserve the unit, i.e. 1 1 k 1 2 is not required to be equal to 1 1 and the counit is not required to be multiplicative, i.e. ε(hh ) is not required to be equal to ε(h)ε(h ), for elements h, h H.

225 G. Böhm / Advances in Mathematics 225 (2010) In the following proposition we deal with the (co)monads H k ( ) and ( ) k H, induced by a k-(co)algebra H on the category of modules over a commutative, associative and unital ring k. Proposition 5.8. For a commutative, associative and unital ring k, consider a k-module H which possesses both a k-algebra and a k-coalgebra structure. Using the notations introduced above the proposition, the following assertions are equivalent: (i) The algebra and coalgebra structures of H constitute a weak bialgebra; (ii) The k-module map Ψ R : H k H H k H, h k h h 1 k hh 2 (5.21) induces a weak ι-lifting of the comonad ( ) k H for the monad ( ) k H and a weak π-lifting of the monad ( ) k H for the comonad ( ) k H, and the k-module map Ψ L : H k H H k H, h k h h 1 h k h 2 (5.22) induces a weak ι-lifting of the comonad H k ( ) for the monad H k ( ) and a weak π-lifting of the monad H k ( ) for the comonad H k ( ). That is to say, ( ( ) k H,( ) k Ψ R ) and ( H k ( ), Ψ L k ( ) ) are comonads in Mnd ι (CAT), via the comultiplication and counit induced by the coalgebra H, and they are monads in Mnd ι (CAT ), via the multiplication and unit induced by the algebra H. Proof. Note first that assertion (ii) implies (5.18). Indeed, (5.21) determines a 1-cell (( ) k H, ( ) k Ψ R ) in EM w (CAT) if and only if ( h h ) 1 k h ( h h ) 2 = h 1 h 1 k hh 2 h 2, for any elements h, h and h of H. Putting h = 1 we obtain (5.18). By Proposition 5.7, assertion (ii) is equivalent to saying that (( ) k H,( ) k H,( ) k Ψ R ) and (H k ( ), H k ( ), Ψ L k ( )) are weak entwining structures in CAT (or, in the terminology of [9], (H,H,Ψ R ) is a right right weak entwining structure and (H,H,Ψ L ) is a left left weak entwining structure in BIM). This statement was proven to be equivalent to (i) in [9, Theorem 4.7]. Acknowledgments The author s work is supported by the Hungarian Scientific Research Fund OTKA F She would like to thank the referee for valuable and helpful comments.

226 32 G. Böhm / Advances in Mathematics 225 (2010) 1 32 References [1] J. Beck, Distributive laws, in: V.B. Eckmann (Ed.), Seminar on Triples and Categorical Homology Theory, in: LNM, vol. 80, Springer, 1969, pp [2] R. Blattner, M. Cohen, S. Montgomery, Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 (1986) [3] G. Böhm, Doi Hopf modules over weak Hopf algebras, Comm. Algebra 28 (2000) [4] G. Böhm, The 2-category of weak entwining structures, arxiv: v2, [5] G. Böhm, T. Brzeziński, Cleft extensions of Hopf algebroids, Appl. Categ. Structures 14 (2006) ; Corrigendum, Appl. Categ. Structures 17 (2009) [6] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf algebras I: Integral theory and C -structure, J. Algebra 221 (1999) [7] T. Brzeziński, Crossed products by a coalgebra, Comm. Algebra 25 (1997) [8] T. Brzeziński, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Algebr. Represent. Theory 5 (2002) [9] S. Caenepeel, E. De Groot, Modules over weak entwining structures, Contemp. Math. 267 (2000) [10] S. Caenepeel, K. Janssen, Partial entwining structures, available (on 24/02/2010) at ~scaenepe/quotient10.pdf. [11] S. Caenepeel, K. Janssen, Partial (co)actions of Hopf algebras and partial Hopf Galois theory, Comm. Algebra 36 (2008) [12] Y. Doi, M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Algebra 14 (1986) [13] J.M. Fernández Vilaboa, R. González Rodríguez, A.B. Rodríguez Raposo, Preunits and weak crossed products, J. Pure Appl. Algebra 213 (2009) [14] J. Gómez-Torrecillas, Comonads and Galois corings, Appl. Categ. Structures 14 (2006) [15] G.M. Kelly, R. Street, Review of the elements of 2-categories, in: G.M. Kelly (Ed.), Category Sem. Proc., Sydney, 1972/1973, in: LNM, vol. 420, Springer, 1974, pp [16] S. Lack, R. Street, The formal theory of monads II, J. Pure Appl. Algebra 175 (2002) [17] J. Power, H. Watanabe, Combining a monad and a comonad, Theoret. Comput. Sci. 280 (2002) [18] A.B. Rodríguez Raposo, Crossed products for weak Hopf algebras, Comm. Algebra 37 (2009) [19] R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972) [20] R. Street, private communication, [21] M.E. Sweedler, Cohomology of algebras over Hopf algebras, Trans. Amer. Math. Soc. 133 (1968) [22] R. Wisbauer, Algebras versus coalgebras, Appl. Categ. Structures 16 (2008)

227 7. fejezet Weak bimonads and weak Hopf monads Böhm, G.; Lack, S.; Street, R. J. Algebra in press. doi: /j.jalgebra

228 FEJEZET. WEAK BIMONADS AND WEAK HOPF MONADS

229 JID:YJABR AID:13051 /FLA [m1g; v 1.45; Prn:10/08/2010; 8:28] P.1 (1-30) JournalofAlgebra ( ) Contents lists available at ScienceDirect Journal of Algebra Weak bimonads and weak Hopf monads Gabriella Böhm a,,stephenlack b,c, Ross Street c a Research Institute for Particle and Nuclear Physics, Budapest, H-1525 Budapest 114, P.O.B. 49, Hungary b School of Computing and Mathematics University of Western Sydney, Locked Bag 1797, Penrith South DC NSW 1797, Australia c Mathematics Department, Macquarie University, NSW 2109, Australia article info abstract Article history: Received 24 February 2010 Available online xxxx Communicated by Nicolás Andruskiewitsch MSC: 18D10 18C15 16T10 16T05 Keywords: Monoidal categories Monads Weak liftings Weak bimonads Weak Hopf monads We define a weak bimonad as a monad T on a monoidal category M with the property that the Eilenberg Moore category M T is monoidal and the forgetful functor M T M is separable Frobenius. Whenever M is also Cauchy complete, a simple set of axioms is provided, that characterizes the monoidal structure of M T as a weak lifting of the monoidal structure of M. The relation to bimonads, and the relation to weak bimonoids in a braided monoidal category are revealed. We also discuss antipodes, obtaining the notion of weak Hopf monad Published by Elsevier Inc. Introduction Bialgebras (say, over a field) have several equivalent characterizations. One of the most elegant is due to Pareigis, who proved that an algebra A over a field K is a bialgebra if and only if the category of (left or right) A-modules is monoidal and the forgetful functor from the category of A-modules to the category of K -vector spaces is strict monoidal. This fact extends to bialgebras in any braided monoidal category [12]. Pareigis characterization of a bialgebra was the starting point of Moerdijk s generalization in [14] of bialgebras to monoidal categories possibly without a braiding. He defined a bimonad (originally * Corresponding author. addresses: G.Bohm@rmki.kfki.hu (G. Böhm), s.lack@uws.edu.au, steve.lack@mq.edu.au (S. Lack), ross.street@mq.edu.au (R. Street) /$ see front matter 2010 Published by Elsevier Inc. doi: /j.jalgebra

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