Galois H-objects with a normal basis in closed categories : a cohomological interpretation

Publicacions Ma emá iques, Vol 37 (1993), 271-284 . GALOIS H-OBJECTS WITH A NORMAL BASIS IN CLOSEDCATEGORIES . A COHOMOLOGICAL INTERPRETATION A bs ac such ha J . N . ALONSO ALVAREZ AND J . M . FERNÁNDEZ VILABOA In his pape , o a cocommu a i e Hop algeb a H in a symme ic closed ca ego y C wi h basic objec K, we ge an isomo phism be ween he g oup o isomo phism classes o Galois H-objec s wi h a no mal basis and he second cohomology g oup H 2 (H, K) o H wi h coe icien s in K . Using his esul , we ob ain a di ec sum decomposi ion o he B aue g oup o H-module Azumaya monoids wi h inne ac ion : BMinn(C, H) -B (C) ® 12 (H, K) In pa icula , i C is he symme ic closed ca ego y o K-mo- dules wi h K a ield, H 2 (H, K) is he second cohomology g oup in oduced by Sweedle in [21] . Mo eo e , i H is a ini ely gen- e a ed p ojec i e, commu a i e and cocommu a i e Hop algeb a o e a commu a i e ing wi h uni K, hen he abo e decompo- si ion heo em is he one ob ained by Bea ie [5] o he B aue g oup o H-module algeb as . P elimina y A monoidal ca ego y (C, ®, K) consis s o a ca ego y C wi h a bi unc o - ® - : CxC -> C and a basic objec K, and wi h na u al isomo phisms : aABC :A®(B®C)=(A(9 B)®C l A :K®A-A A :A®K-A (A (9 aBCD) o aA(B(DC)D 0 (aABC ® D) = aAB(C®D) ° a(A®B)CD (A(9 1B)oaAKB= A®B 272  J . N . ALONSO ALVAREZ, J . M . FERNÁNDEZ VILABOA I he e is a na u al isomo phism TB : A®B=B®A such ha TAB o TB =A ® B, TB®c = (B (9 7-C) o (TB ® C), hen C is called a symme ic monoidal ca ego y . A closed ca ego y is a symme ic monoidal ca ego y in which each unc o -®A : C , C has a speci ied igh adjoin [A, -] : C -> C ([12], [18]) . Examples : 1) The ca ego y o se s and mappings . 2) The ca ego y o R-modules o e a commu a i e ing R . 3) The ca ego y o chain complexes o R-modules and mo phisms o deg ee 0, wi h R a commu a i e ing . 4) The ca ego y o shea es o B-modules o e a opological space, wi h 0 a shea o commu a i e ings . 5) The ca ego y o cohe en shea es o modules o e a scheme . 6) The ca ego y o all R-g aded modules wi h mo phisms o deg ee 0 (R is a commu a i e g aded ing) . 7) (R, u)-Mod, wi h R a commu a i e ing and o, an idempo en ke nel unc o in R-Mod . In wha ollows, C deno es asymme ic closed ca ego y wi h equalize s, co-equalize s and p ojec i e basic objec K . We deno e by am and OM he uni and he co-uni , espec i ely, o he C-adjun ion M ®- -1 [M, -] C ---> C which exis s o each objec M o C . 1 .  An objec M o C is called p o ini e in C i he mo phism [M  Om(K) ®M] o am(M ®M) : M ®M , [M, M] = E(M) ¡san isomo phism, whe e M = [M, K] .  I , mo eo e , he ac o iza ion o Qm(K) : M ® 1V1 ---> K h ough he co-equalize o he mo phisms Qm(M) ® M and M ® ([M, (3m (K) o (Om(M) ® M)] o am (E(M) ® M)) M ® E(M) ® M -> M ® M is an isomo phism, we say ha M is a p ogene a o in C . 2 .  A monoid in C is a iple A = (A, 7A, PA) whe e A is an objec in C and MA : A®A -> A, 7A : K -~ A a e mo phisms in C such ha MA o(A (9 7A) = A= UA o (?1A (9 A), MA 0 (UA ® A) = pA -(A (9 PA) - I MA o Tá = PA, hen we will say ha A is a commu a i e monoid . Gi en wo monoids A = (A, 97A, PA) and B = (B, ?7B, [LB) in C, : A -> B is a monoid mo phism i MB o ( ® ) = o PA and o 7A = ?7B . A comonoid (cocommu a i e), D = (D, ED, 5 D ) is an objec D in C oge he wi h wo mo phisms ED : D --> K, 5D : D --> D ® D, such ha (6 D ® D) o5 D = (D ® 6D) o6 D and (ED ® D) o 6D = 1 D= (D ED) o 6D(TD o 6D = 6D) .  I D= (D, ED, 6D) and E = (E, EE,SE) a e comonoids, : D ---> E is a comonoid mo phism i ( ® ) o S D = 5E o and EE 0 = ED . NORMAL BASIS IN CLOSED CATEGORIES AND COHOMOLOGY  273 3 . Fo a monoid A = (A, T7A, PA) and a comonoid D = (D, ED,SD) in C, we deno e by Reg(D, A) he g oup o in e ible elemen s in C(D, A) (mo phisms in C om D o A) wi h he ope a ion "con olu ion" gi en by : * g = PA o ( ® g) o S D . The uni elemen is ED ® ?1A . Obse e ha Reg(D, A) is aü abelian g oup when D is cocommu a i e and A is commu a i e . 4 . De ini ion . Le II = (C, qc, pc) be a monoidand C = (C, EC, SC) a comonoid in C and le A : C ---> C be a mo phism . Then H = (C = (C, EC, 6C), II = (C, 77c, 7¿C), C , A) is a Hop algeb a in C wi h espec o he comonoid C i Ec and Sc a e monoid mo phisms (equi alen ly, 7c and pc a e comonoid mo phisms) and A is he in e se o 1c : C ---> C in Reg(C, C) . We say ha H is a ini e Hop algeb a i C is p o ini e in C . 5 . De ini ion . (A, WA) = (A, ?7A, PA ; WA) is a le H-module monoid i) A = (A, ?7A, PA) is a monoid in C . ii) (A, c0A) is a le H-module (WA o (C ® WA) = WA o(pc ® A), cO A o ( 7c ® A) = A) . iii) 7A, PA a e mo phisms o le H-modules (WA0(C077A) = 7A0Ec and EPA o(C ® PA) = PA 0IPA®A, whe e (PA®A = (WA (9 WA) o(C , A 0A)o(Sc®A®A)) . We say ha he ac ion WA o H in A is inne i he e exis s a mo phism in Reg(C,A) such ha WA = PA 0(A ® GIA o - A )) 0 ( ® -1 ® A) o (Sc ® A) : C ® A S A, whe e -1 is he con olu ion in e se o . 6 . De ini ion . I H is a cocommu a i e Hop algeb a and (A, WA) is a commu a i e H-module monoid, hen, we say ha a mo phism u in Reg(C ® C, A) is a 2-cocycle i al (Q) * 03 ( ) = 02 (o,) * 04 (o , ), whe e 191 (U) = EPA 0 (00 Q), 192 (U) = Q o (PC ® C), a3(O ) = o, o (C ® pc) and a4(Q) = Q® EC . Two 2-cocycles o, and y a e said o be cohomologous, w i en - -y, i he e exis s a mo phism E Reg(C, A) such ha o, * a 2 ( ) = al ( ) * 03 ( )*y, whe e a l ( ) = WA0(C® ), a2 ( ) = opc and 03 ( ) = 0ec . T i ially, "-" is an equi alence ela ion . The se o equi alence clases shall be called he second cohomology g oup o he cocommu a i e Hop algeb a H wi h he coe icien s in he le H-module monoid (A, WA), and will be deno ed by H 2 (H, A) . I o, is a 2-cocycle in Reg(C ® C, A), hen he mo phism = u * a l (7 ) * a 2 (7 -1 ) *a 3 (7 ) is a 2-cocycle in Reg(C®C, A) cohomologous wi h Q such ha Qo ( 7c ® C) = Ec ® 7A = Q o (C ® 7c), whe e i = o -1 0 (C ® 7 7C) is a mo phism in Reg(C, A) wi h in e se 7 -1 = u o(C ® lc) . Mo eo e , i 274  J . N . ALONSO ALVAREZ, J . M . FERNÁNDEZ VILABOA ,y is cohomologous wi h u, hen he e exis s a mo phism í9 E Reg (C, A) such ha i9 o )c = 91A and Q * c92 (~) = al (19) * o93(19) * 5 . Rema k . Le C he ca ego y o K-modules o e a ield . In his case, H 2 (H, A) is he second cohomology g oup o he Sweedle 's complex q {Reg(®C, A) ; Aq}q>0 Reg (K, A) °o . Reg(C, A) °1 .. . . ~' Reg(®C, A) whe e A q:= al * 82 I *  . . * aq+2)4+1 and o each E Reg(®C, A), aI( ) =WA-(C® ) az( )= o(C®i-20C®u c (9 C®q - i+l0 C) Oq+2 ( ) = ® EC A ~ Reg( q ® 1 C, A) ~~ . . . ([ 21 1) . 7 . De ini ion . (B,PB) = (B,''IB,MB ; PB) is a igh H-comodule monoid i : i) B= (B, nB, MB) is a monoid in C ii) (B, PB) is a igh H-comodule ((PB®C)OPB = (B(9SC)OPB ; (B® -c) o PB = B) . iii) PB : B --> B ®C is a monoid mo phism om (B, ?7B, N-B) o he p oduc monoid BH = (B ® C, ?7B ® ?1c, (AB ® pc) o (B ®TB ® C)) ( ha is, PB o ?1B = 71B ® ? 1c and PB o P, B = (MB (9 ¡ c) o (B ® TB C) o (PB (9 PB)) F om now en we assume ha H is a ini e cocommu a i e and com- mu a i e Hop algeb a . 8 . De ini ion . A igh H-comodulemonoid (B, PB) is said o be a Galois H-objec i and only i : i) The mo phism -YB := (MB (9 C) o (B (9 PB) : B (9 B -> B ®C is an isomo phism . ii) B is a p ogene a o in C . Fo example, in he case o (R, u)-Mod, a commu a i e H-comodule monoid is a couple (B, PB), whe e B is acommu a i e (R, u)-algeb a and PB : B -, B L H := Q, (B (9 R H) is a mo phism o algebas and i de ines a igh H-comodule s uc u e o e B . whe e NORMAL BASIS IN CLOSED CATEGORIES AND COHOMOLOGY  275 (B, PB) is a Galois H-objec i and only i B is a (R, u)-p ogene a o and he mapping pB : B#H -> Hom(B, B) a ising om he le B#H- 0 module s uc u e on B is an isomo phism ([15, (1 .3 .17)]) . I a Galois H-objec is isomo phic o H as an H-comodule hen we say ha i has a no mal basis . I B 1 and B2 a e Galois H-objec s, : B I -> B2 is a mo phism o Galois H-objec s i i is a mo phism o H-comodules (PB,, o = ( ®C) o PB l ) and o monoids . I (A, PA) and (B, PB) a e H-comodule monoids, hen A o B, de ined by he ollowing equalize diag am al A B AoB  -> A®B i :i A®B®C 02 AB aAB = (A (9 TB) o (PA (9 B), and aAB =A® PB is an H-comodulemonoid o be deno ed by (A o B,pAB) . I mo eo e (A, PA) and (B, PB) a e Galois H-objec s, hen (A o B, pAB) is also a Galois H-objec , whe e pAB is he ac o iza ion o he mo phism aAB o iAB (o 02 AB o iAB) h ough he equalize iAB ® C . The se o isomo phism classes o Galois H-objec s (wi h a no mal basis), wi h he ope a ion induced by he one gi en abo e, is an abelian g oup o be deno ed by Galc(H)(Nc(H)) . The uni elemen is he class o (II, SC) and he opposi e o [(B, PB)] is [(B °P , (B (9 A) o PB)] whe e B'P = (B, 7B, MB o 7-B )_ Rema k . In he case o a ini ely gene a ed p ojec i e, commu- a i e and cocommu a i e Hop algeb a H o e a commu a i e ing R, Galc(H) is he g oup o Galois H-objec s in he sense o S . Chase and M . Sweedle in [9] . 9 . P oposi ion . [(B,PB)] E NC(H), hen, he e is a 2-cocycle o, in Reg(C ® C, K) sa is ying a o ( 7C ® C) = EC = Q o(C ® ?1C) . P oo . Le (B, PB) a Galois H-objec wi h a no mal basis . Thenwe ha e an isomo phism - YB : B®B -> B® C and an H-comodule isomo phism : C ---~ B . The e o e he mo phism o H-comodules = (EC ® B) o ( -1 ® ) o (?7B (9 C) : C -> B is in Reg(C, B) wi h in e so .Í -1 =MBo(B®EC®B)o(B® -1 ® )o( - yB l (S ? 7C)o(?7B®C) 276  J . N . ALONSO ALVAREZ, J . M . FERNÁNDEZ VILABOA and sa is ying o IC = ?7B . Indeed : * -1=MBo (eco B0 ECO B)o( -1®B® -1® )o °(?IB0-YB1®7Ic)o( 0C)-SC= =MBo(Ec®B®ec0B)o( -1®B® -1® )o o(B®[-YB1°'YB]®C)°(1]B®?]B0B®97c)o = = (ec ®B® ec) o ( -1 ® ® C) ° ([PB ° nB] ® C) = EC 0 77B -1* =lLB°(B®[Eco -1]0 B)o(-yBl0 )0(?7B®6c)= =MBo(B®[sco -1]® )o(B0PB)°-YB1°(17B®C)- =MB°(B®Ec® )o(B06c)o(B® -1)°-YB10(77B® C)= =(B®ec)°-YB°7B10(?7B®C)= = EC ® ?IB because is an H-comodule isomo phism and he equali ies : (ec ® B) o ( -1 ® ) o (?IB ® ?)c) = (Ec 0 B) o ( -1 0 ) 0 PB ° 71B = ?7B (yBloC)o(B06c)=(B®PB)0-YB1 T i ially, is a mo phism o H-comodules and o ?1c = 77B- The mo phism Q = (P,B o ( ® )) * ( -1 ® P,C) : C® C -> B ac o s h ough he equalize K'IBB ~B®C B077C because H is a Hop algeb a, (B, PB) an H-comodule monoid, a mo - phism o H-comodules and he equali y PB o -1 = ( -1 ® A) o - C o 6C ([14, (2 .3)]) . Mo eo e , he ac o iza ion, á , o u is in Reg(C®C, K) wi h in e se he ac o iza ion o he mo phism u 1 = ( °P-C) * (PB °TB ° ( -1® -1) ) C 0C - B h ough he equalize ?7B . NORMAL BASIS IN CLOSED CATEGORIES AND COHOMOLOGY  27 7 The mo phism Q : C ® C ---> K is a 2-cocycle . Indeed : 77B 0 (191(5~ ) * 0307)) = =( * -1)o(a ®Pc)o(c®TC0C)o(6c®6c)o o(c®ñ~ 0PC)o(cocoTC 0c)o(c®6c®6c)= =PBo(B® -1)°PBo o(& ®PC)o(c®TC®C)o(6c®6c)o o(c(9 ®Pc)o(cocoTC 0c)o(c®6c®6C)= =P-Bo(B® -1)0PB0PB-( ® )o(c®Q 0PC)0 o(coC®TC0c)0(c®6c(9 6c)= =P-Bo(B® -1)0PBOPB0(PB0C)o( ® (9 )= =PBo(B® -1)°PBo o(15~ ®PC)o(c®TC®c)o(6c®6c)o o(ú ®PC®C)o(C®TC®C®C)o(6c®6c®C)= =( * -1)0(ú ®Pc)o(c®TC0C)0(6C06c)o o(a 0PC0c)o(c®TC 0C0c)o(6c®6c(9 c)= _ ?7B o (a2(5~ ) * a4(5~ )) because is a mo phism o H-comodules, H a cocommu a i e Hop algeb a and he equali y o(a (9 Pc)o(c®TC (9 C)o(6c(9 6C)=PB-( ® ) and hen, since 77B is a monomo phism, ú is a 2-cocycle . T i ially, Q o (77C ® C) = EC = o (C ® 77C) . Rema k . I [(B1, PB, )] = [(B2, PB2)] E Nc(H) hen he e is an iso- mo phism o H-comodule monoids h : B1 - B2 . Clea ly, Q , = Qho , . (No ice ha h o 1 E Reg(C, B2) wi h in e se h o i 1) . Mo eo e , a l - Z57 z . Indeed : The mo phism e = (ho l)* a 1 : C --> B2 ac o s h ough he equalize ?7B2 : PB2Oe=(PBZ®PC)o»o l)®T 2(9 ín)o(6c® 21®C)o o(C0(TCo6c))o6c=(B2® )C)0e and hen, he e exis s a mo phism é : C -> K such ha 7B2 o é = e . Clea ly, é o i1c = K . Mo eo e , é is in Reg(C, K) wi h in e se é-1, he ac o iza ion o e-1 = 2 * (h o i 1) h ough he equalize 17B2 . 278  J . N . ALONSO ALVAREZ, J . M . FERNÁNDEZ VILABOA We also ha e ha : l B 2 ° ( j j * a2 (e)) _ - Y"B2 ° ( 1B2 ® 7B2) ° (-j b0 * a2(e)) _ = MB 2 ° (MB2 ® MB2) ° [(h ° i) ® (h ° l) ® ((h ° l 1 ) * (h ° l))® ® 2 ' ] - ( C OCOSC)o(CoCopc)°(COT80C)°(5c0SC)= _ MB2 ° (iíBZ ® B2) ° (PB2 ® (1IB2 °TBZ) 282)° °((ho l)O(h° l)0( 2 1 * 2)0 2 1 O 20 2 1 )° °(cOc0C0b c oli c )o(CococoTcoc)° *(COCob c ob c )o(COTCOC)o(5C05 C )= - Y"B2 ° (MBZ ® B2) ° ((nB2 0 é) ® (77B2 O ~) ® ( 7B2 ° Q 2))° o(coTC0C)°(6C0bC)= = /BZ ° (al (e) * 1 93 (e) * Z 7 2 ) and hen, since 7B 2 is a monomo phism, -5 - - 5 2 10 .  P oposi ion . I o, is a 2-cocycle in Reg(C o C, K) such ha o, ° (71c ® C) = ec = o (C ® 7c), hen (C, = (C, i7c, wc o = (o- ® pc) ° (C ® Tc ® C) ° (S c ® bc)) ; Sc) is a Galois H-objec wi h a no mal basis . P oo . T i ially, (C Q , Sc) is an H-comodule monoid . The mo phism , yco = (loco o C) o (C ® bC) : C ® C -> C®C is an isomo phism wi h in e se ycó = (l-zcoc)°(Cou-1OA®C)°(COPCOCOSC)°(scOaOCOC)o o(Cob c oC)o(Co6C) Indeed : lyc o ° ^ Yco = =(wcoC)°(PC(S o  -1 OaoC)0(C0C0PC0C0bc)o o(CocoPC(S aocoC)o(COTCocobcoC)° °(u0b c 0b c 06 C )o(COTC0C0C)°(6 C OS C OC)o(CO6 C )= =(PC0C)o(PC0o -1 OA0C)0(CoC0PCo5COC)o °(COTC0[mc°(COA)°6 C lo6 C )0(COCOS C OC)o " (o-(S s c 0S C )0(COTc0C)0(6 C 0b c )= NORMAL BASIS IN CLOSED CATEGORIES AND COHOMOLOGY  27 9 =(PC0C)o(PC0a -1 ®a® c)o(C®TC®[TCo5 C ]®C)o o(ao6C®Co6C)o(c®TC®6C)o(6 C 05 C )= = (Pc®C) o (C®[PC o (COA) o 6 C ]®C)o o(u0C0C®o, -1 ®C)o(C®C0C®TC®C®C)o o(c0C0bc05c0C)0(c07506 C )o(6 C 05 C )= =(0'®c®u - 1 ®C)o(COCO [TCosc](96 C )o 0(c®TC®c)o(sc(9 sc)= =([0'*0,-1]oCoc)o(C®Tc (9 c)o(6 C (9 5C)= =C®c ¡YCo o i'Co = = (01®PC®C) o (PC(9TC®C®C) o (C®C®PC®5C®C)o o (C07C®c®sc)o(scou-1®sc®C)o(C®MC®C®a®C)o o(c®C®a®C®5 C )o(5 C ®5 C oC)o(co5 C )= (U®PC®C) o (C®TC®C®C) o (PC®PC®[TC oa C ]®C)o o (C (9Tc ®C ® S c ) o (SC ® Q -1 ® [(A ® . ) o TC oS c ] ® C)o " (co ic®C®C® C)o(5 C0A®C(9 6 C )o " (cobcoc)o(Cobc)= =(0'®C(9 C)o(c®TC(SC)0(PC®Nic(9 6C)o o(c®TC®[PCo(A(9 c)o6 C ]0 c) 0 (6C®0,-1®a(9 6C)o o (C®4 C ®C®6 C ) o (sC®a(gCOC) o (C®6 C ®C) o (C®6C) _ =(o ®C0C)o(lic®Tc0C)o(C®Tco6C)o o([TCo6 C]®01- 1®a® C) 0 (C0 PC ®C®C(9 C)o o(c(9 C®a®C®C® C)0(6 C 0b c (9 C0C)o o(c®[TCo6 C ]0 C)o(C05C)= =(c®o,®C)o(c®C0U - '05 C )0(c®[b C 0P C ]®CoC)o o(c®C®a®C® C)o(b C (9 sc0C)0(c0bc)= =(c®o ®o - 1(9 C)o(c®C(9 TC®CoC)o o(c®s c ®s c oC)o(c®N-C(9 6C)0(6C®a®C)0(c06C)= =(c®[0'*u-1](9 C)o(C(SPC05C)o(5C®a®c)o(c06C)= =C®c and hus (C o , S c ) is a Galois H-objec wi h a no mal basis . Rema k . I o, and -y a e wo 2-cocycles such in P oposi ion 10 and cohomologous, hen (é®C) o5C : (C o , SC) -> (C,,, S c ) is an isomo phism