Duality and products in algebraic (co)homology theories

DUALITY AND PR ODUCTS IN ALGEBRAIC (CO)HO MOLOGY THEORIES NIELS KO W ALZIG AND ULRICH KR ¨ AHMER Abstra ct. The origin and interpla y of prod ucts and du alities in alge- braic (co)homology theories is ascribed to a × A -Hopf algebra structure on the relev an t universal env eloping algebra. This provides a unified treatment for example of results by V an den Bergh about Ho chsc hild (co)homology and by Huebschmann ab out Lie-Rinehart ( co)h omology . 1. Introduction Most classical (co)homolog y theories of algebraic ob jects suc h as groups or Lie, Lie-Rinehart or asso ciativ e algebras can b e realised as (1) H • ( X, M ) := Ext • U ( A, M ) , H • ( X, N ) := T or U • ( N , A ) for an augmen ted rin g X = ( U, A ) (a r ing with a d istin gu ish ed left mo dule) that is functorially attac hed to a giv en ob ject. The cohomology coefficients are left U -mo dules M and those in homology are righ t U -mo d ules N . Our aim here is to clarify the origin and int erplay of multi plicativ e struc- tures and d ualities b et w een suc h (co) homology grou p s, and to pr ovide a unified treat ment of results b y V an den Bergh on Hoc hschild (co)homology [26] and by Hu eb sc hmann on Lie-Rinehart (co)homology [8]. The k ey con- cept inv olved is that of a × A -Hopf algebra introdu ced b y S c hauen bur g [22]. The main results can b e summarised as follo ws: theorem 1 . F or any A -bipr oje ctive × A -Hopf algebr a U ther e is a functor ⊗ : U - Mo d × U op - Mo d → U op - Mo d that induc e s f or M ∈ U - Mo d , N ∈ U op - Mo d and m, n ≥ 0 natur al pr o ducts a : Ext m U ( A, M ) × T or U n ( N , A ) → T or U n − m ( M ⊗ N , A ) . If A ∈ U - Mo d admits a finitely gener ate d pr oje ctive r esolution of finite length and ther e exists d ≥ 0 with Ext m U ( A, U ) = 0 for m 6 = d , then ther e is a c anonic al e lement [ ω ] ∈ T or U d ( A ∗ , A ) , A ∗ := Ext d U ( A, U ) such that for m ≥ 0 and M ∈ U - Mo d with T or A q ( M , A ∗ ) = 0 f or q > 0 · a [ ω ] : Ext m U ( A, M ) → T or U d − m ( M ⊗ A ∗ , A ) is an isomorphism. As we will recall b elo w, × A -bialgebras and × A -Hopf algebras generalise bialgebras and Hopf algebras to w ards p ossibly noncommuta tiv e base al- gebras A . Besides Hopf algebras, b oth the unive rsal en ve loping algebra 1 2 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER U ( A, L ) of a Lie-Rinehart algebra ( A, L ) and the env eloping algebra A e = A ⊗ k A op of an asso ciativ e algebra A are × A -Hopf algebras, see Section 2.5. F or an y × A -bialgebra U , the base algebra A carries a left U -action and the catego ry U - Mo d of left U -mo dules is monoidal with unit ob ject A . But only for × A -Hopf algebras one has a canonical op eration ⊗ as in Theorem 1 whic h turns U op - Mo d in to a mo dule cat egory o v er ( U - Mo d , ⊗ , A ) (Lemma 16). An y × A -Hopf algebra carries t wo left an d tw o righ t actions of the base algebra, all commuting with eac h other. The bipr o j ectivit y assu med in The- orem 1 r efers to the pro jectivit y of t wo of these, see Section 2.1. Under this condition, we can u se the elegan t formalism of s u sp ended monoidal cate- gories f r om [24] to d efine f or M , N ∈ U - Mo d and P ∈ U op - Mo d pro ducts ` : H m ( X, M ) × H n ( X, N ) → H m + n ( X, M ⊗ N ) , a : H n ( X, N ) × H p ( X, P ) → H p − n ( X, N ⊗ P ) , where w e again use the abbreviations from (1) (cf. S ections 3.2 and 3.5). In the last p art of Theorem 1, A ∗ = H d ( X, U ) = Ext d U ( A, U ) is a right U -mo dule via right m ultiplication in U , and if we define th e f unctor ˆ : U - Mo d → U op - Mo d , M 7→ ˆ M := M ⊗ A ∗ , then the statemen t can b e r ewritten as an isomorphism H m ( X, M ) ≃ H dim( X ) − m ( X, ˆ M ) , dim( X ) := pr o j . d im U ( A ) that is giv en as in top ology b y the ca p pro d uct with the fundamental class [ ω ] ∈ H dim( X ) ( ˆ A ) whic h corresp onds u nder the d ualit y to id A ∈ H 0 ( A ) = Hom U ( A, A ). F or M = A this simply means that the H • ( A )-mod ule H • ( A ∗ ) is fr ee with generator [ ω ]. Theorem 1 is w ell-kno wn in group and Lie algebra (co)homolo gy . F or U = A ⊗ k A op it red u ces to V an den Bergh’s r esult [26] that stimulated a lot of recen t research, see e.g. [2, 4, 5, 14]. Note that w e do not need V an den Bergh’s in ve rtibilit y assu mption ab out A ∗ , w h ic h sa ys that ˆ is an equiv- alence. Ho w ev er, it is satisfied f or many well- b eh a v ed algebras [ibid.] and implies the condition T or A q ( M , A ∗ ) = 0 for arbitrary A -bimodu les M (since in v ertible b im o d ules are fi nitely generated pro jectiv e as one-sided mo d ules from either side). F or Lie-Rinehart algebras ( A, L ), Theorem 1 is due to Huebsc hmann [8], an d w e fin d th e general setting helpful for example to u n- derstand the differen t r oles of left and righ t mo d ules that he has observ ed. As Huebschmann has show ed, the conditions of T h eorem 1 are satisfied whenev er L is fin itely generated pr o j ective o v er A , and A ∗ coincides as an A -mo dule with Λ d A L and is in particular pro jectiv e, so also here w e h a v e T or A q ( M , A ∗ ) = 0 for arbitrary ( A, L )-mo du les M . W e were also motiv ated b y the cur ren t discussion of the numerous bialge- broid generalisations of Hopf algebras, see [1]. Sev eral au th ors h a v e raised the question where L ie-Rinehart algebras fit in . They were sh o wn in [28 , 18] to b e × A -bialgebras, see also [12, 9]. Here w e r emark that they are in fact alw a ys × A -Hopf algebras, b ut not n ecessarily Hopf algebroids in th e sense of B¨ ohm an d Szlac h´ an yi (Example 8, this answers a question of B¨ ohm [1]). So b oth th ese examples and th e applications in homological algebra clearly demonstrate the relev ance of the inte rmediate concept of a × A -Hopf algebra. DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 3 Theorem 1 could b e generalised to differen tially graded × A -Hopf alge- bras, shea v es of suc h, or suitable abstract monoidal categories. One can also drop th e condition Ext n U ( A, U ) = 0 for n 6 = d and the assumption that T or A q ( M , A ∗ ) = 0. Th en one obtai ns for a b oun ded b elo w c hain complex M o v er U - Mo d an isomorphism RHom U ( A, M ) ≃ ( M ⊗ L A RHom U ( A, U )) ⊗ L U A . N.K. is su pp orted by th e NW O through the GQT cluster. U.K. is sup- p orted by the EP S R C fello wship EP/E/043267/1 and partially b y the P olish Go v ernment Grant N201 1770 33. W e thank Andy Bak er, Gabriella B¨ ohm, Ken Bro wn , Hennin g K rause and V alery Lunt s for discus s ions. 2. Preliminaries on × A -Hopf a lgebras 2.1. Some con v en tions. Th roughout this pap er, “rin g” m eans “unital and asso ciativ e ring”, and we fix a commutat ive r in g k . All other algebras, mo dules etc. will ha v e an u n derlying structur e of a k -mo du le. Secondly , w e fix a k -algebra A , i.e., a r ing with a ring homomorphism η A : k → Z ( A ) to its centre. W e denote b y A - Mo d the catego ry of left A - mo dules, by A op the opp osite and by A e := A ⊗ k A op the en v eloping algebra of A . Th us left A e -mo dules are A -bimo dules with symmetric action of k . Our main ob j ect is finally an algebra U o v er A e , w h ere we now refer to the less standard notion of an algebra ov er a p ossibly noncommutati ve base algebra: U is a k -algebra w ith a k -algebra homomorphism η = η U : A e → U . This giv es rise to a forget ful functor U - Mo d → A e - Mo d u s ing wh ic h we consider every U -mo d ule M also as an A -bimo d ule with actions (2) a ✄ m ✁ b := η ( a ⊗ k b ) m, a, b ∈ A, m ∈ M . Similarly , ev ery righ t U -mo d ule N is also an A -bim o d ule via (3) a ◮ m ◭ b := nη ( b ⊗ k a ) , a, b ∈ A, n ∈ N . In particular, U itself carries tw o left and t w o right A -actions all comm uting with eac h other. Usually w e consid er U as an A e -mo dule u sing a ✄ u ✁ b , and otherwise w e wr ite e.g. ◮ U ✁ to denote which actions are consid ered. Since this w ill b e r ep eatedly a necessary tec hnical condition, w e define: definition 2 . F or an A e -algebr a U we c al l M ∈ U - Mo d A -bipr oje ctive if b oth ✄ M ∈ A - Mo d and M ✁ ∈ A op - Mo d ar e pr oje ctive mo dules. 2.2. × A -bialgebras [25] . Consider an A e -algebra U as ab o v e whic h is also a co algebra in the monoidal catego ry A e - Mo d . That is, there are maps ∆ : U → U ⊗ A U, ε : U → A satisfying the usu al coalge bra axioms (see e.g. [1] for the details) , where (4) U ⊗ A U = U ⊗ k U / span k { u ✁ a ⊗ k v − u ⊗ k a ✄ v | a ∈ A, u, v ∈ U } . F or A = k one calls U a bialgebra if ∆ and ε are algebra h omomorphisms, but in general there is n o natural algebra structure on U ⊗ A U . The wa y out of this problem w as foun d by T ak euc hi [25] and inv olve s the emb edding (5) ι : U × A U → U ⊗ A U, 4 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER where U × A U is the cen tre of the A -bimo d u le ◮ U ✁ ⊗ A ✄ U ◭ : U × A U := n X i u i ⊗ A v i ∈ U ⊗ A U | X i a ◮ u i ⊗ A v i = X i u i ⊗ A v i ◭ a o . The pro d uct of U turns this in to an alg ebra o v er A e , with η U × A U : A e → U × A U, a ⊗ k b 7→ η ( a ⊗ k 1) ⊗ A η (1 ⊗ k b ) . Similarly , A is an algebra ov er k , b ut not ov er A e in general. T o handle this one needs the canonica l map (6) π : E n d k ( A ) → A, ϕ 7→ ϕ (1) , and the fact that En d k ( A ) is an alge bra o v er A e , with η End k ( A ) : A e → En d k ( A ) , ( η End k ( A ) ( a ⊗ b ))( c ) := acb. No w it make s sense to requ ire ∆ and ε to factor through ι and π : definition 3 . A (left) × A -bialgebr a is an algebr a U over A e to gether with two homomorphisms ˆ ∆ : U → U × A U and ˆ ε : U → En d k ( A ) of algebr as over A e such tha t U is a c o algebr a in A e - Mo d via ∆ = ι ◦ ˆ ∆ and ε = π ◦ ˆ ε . So one has for example for any × A -bialgebra ∆( a ✄ u ✁ b ) = a ✄ u (1) ⊗ A u (2) ✁ b, ∆( a ◮ u ◭ b ) = u (1) ◭ b ⊗ A a ◮ u (2) , where we started to u se S w eedler’s sh orthand notation u (1) ⊗ A u (2) for ∆( u ). Be a wa re that the four A -actio ns are n ot the only f eature of × A -bialgebras that disapp ears for A = k . Another cru cial one is for example that the counit ε : U → A is not necessarily a ring homomorphism. Note also that many authors write s ( a ) := η ( a ⊗ 1) and t ( a ) := η (1 ⊗ a ) and form ulate th e theory using these so-called source and target maps rather than η . 2.3. The monoidal category U - Mo d [21] . Definition 3 might app ear complicated, but it is the correct concept from sev eral p oin ts of view. An imp ortan t one for us is the follo wing resu lt of Schauen burg [21, Theorem 5.1]: theorem 4 . The × A -bialgebr a structur es on an algebr a η : A e → U over A e c orr esp ond bije ctively to monoidal structur es on U - Mo d for which the for getful functor U - Mo d → A e - Mo d induc e d b y η is strictly monoidal. Giv en a × A -bialgebra structur e on U , the monoidal structur e on U - Mo d is d efined as for bialgebras: one tak es the tensor p ro duct M ⊗ A N of the A -bimo dules underlying M , N ∈ U - Mo d and defines a left U -acti on via ∆, (7) u ( m ⊗ A n ) := u (1) m ⊗ A u (2) n, u ∈ U, m ∈ M , n ∈ N . definition 5 . If U is a × A -bialgebr a and M , N ∈ U - Mo d ar e left U - mo dules, we denote the left U -mo dule M ⊗ A N with U - action (7) b y M ⊗ N . The un it ob ject in U - Mo d is A on whic h U acts via ˆ ε ( u )( a ) = ε ( a ◮ u ) = ε ( u ◭ a ) , where the last equalit y is a consequence of the defi n ition of a × A -bialgebra. There is an analogous notion of r ight × A -bialgebra for w h ic h U op - Mo d is monoidal. How ever, for a left × A -bialgebra there is in general no canonical monoidal structure on U op - Mo d or ev en only righ t action of U on A . DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 5 2.4. × A -Hopf algebras [22] . Let U b e a × A -bialgebra and define (8) β : ◮ U ⊗ A op U ✁ → U ✁ ⊗ A ✄ U, u ⊗ A op v 7→ u (1) ⊗ A u (2) v , the so-called Galois map of U , where ◮ U ⊗ A op U ✁ = U ⊗ k U / span { a ◮ u ⊗ k v − u ⊗ k v ✁ a | u, v ∈ U, a ∈ A } . One could flip the tensor comp onen ts in order to a void taking the tensor pro du ct o ve r A op , but w e found it more conv en ient to keep β in the form whic h is standard f or b ialgebras ov er fields. F or the latter it is easily seen that β is bijectiv e if and only if U is a Hopf algebra with β − 1 ( u ⊗ k v ) := u (1) ⊗ S ( u (2) ) v , where S is the antip o de of U . T h is motiv ates the follo w in g definition due to Sc hauen bu r g [22]: definition 6 . A × A -bialgebr a U is a × A -Hopf algebr a if β is a bije c tion. F ollo wing Sc hauenburg, we adopt a Swee dler-type notation (9) u + ⊗ A op u − := β − 1 ( u ⊗ A 1) for th e so-called translation map β − 1 ( · ⊗ A 1) : U → ◮ U ⊗ A op U ✁ . Since sub s tan tial for the subsequent calculations, w e list some prop erties of β − 1 as pr o v en in [22, Prop osition 3.7]: one h as for all u, v ∈ U , a ∈ A u +(1) ⊗ A u +(2) u − = u ⊗ A 1 ∈ U ✁ ⊗ A ✄ U (10) u (1)+ ⊗ A op u (1) − u (2) = u ⊗ A op 1 ∈ ◮ U ⊗ A op U ✁ (11) u + ⊗ A op u − ∈ U × A op U (12) u + ⊗ A op u − (1) ⊗ A u − (2) = u ++ ⊗ A op u − ⊗ A u + − (13) ( uv ) + ⊗ A op ( uv ) − = u + v + ⊗ A op v − u − (14) η ( a ⊗ b ) + ⊗ A op η ( a ⊗ b ) − = η ( a ⊗ 1) ⊗ A op η ( b ⊗ 1) , (15) where in (12) w e abbreviated U × A op U := n X i u i ⊗ A op v i ∈ ◮ U ⊗ A op U ✁ | X i u i ✁ a ⊗ A op v i = X i u i ⊗ A op a ◮ v i o and in (13) the tensor pro d uct o ver A op links the first and third tensor comp onen t (cf. [22, Equation (3. 7)]). By (1 0) and (12) one can wr ite (16) β − 1 ( u ⊗ A v ) = u + ⊗ A op u − v whic h is easily c hec k ed to b e well -defin ed o ve r A with (14) and (15). 2.5. Examples. Clearly , Hopf algebras ov er k suc h as universal env eloping algebras of L ie algebras or group algebras are × k -Hopf algebras. But also the en v eloping algebra of an asso ciativ e algebra that go v erns Ho chsc h ild (co)homolo gy is an example as p oin ted out by Schauen burg [22]: Example 7 . The en v eloping algebra U := A e of any k -algebra A is a × A - bialgebra with η = id A e and co pr o d uct and counit ∆ : U → U ⊗ U, a ⊗ k b 7→ ( a ⊗ k 1) ⊗ A (1 ⊗ k b ) , ε : U → A, a ⊗ k b 7→ ab. As for the × A -Hopf alg ebra structure, the tensor pro duct in question reads ◮ U ⊗ A op U ✁ = U ⊗ k U / span k { ( a ⊗ k cb ) ⊗ k ( a ′ ⊗ k b ′ ) − ( a ⊗ k b ) ⊗ k ( a ′ ⊗ k b ′ c ) } , 6 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER where cb and b ′ c is understo o d to b e the pr o duct in A . One then easily v erifies that ( a ⊗ k b ) + ⊗ A op ( a ⊗ k b ) − := ( a ⊗ k 1) ⊗ A op ( b ⊗ k 1) yields an inv erse of the Galois map d efined as in (16). Finally w e discuss Lie-Rinehart algebras whic h d efine for example Poisson (co)homolo gy . S ev eral authors [28, 12, 18] ha v e sh own th at their en v eloping algebras are × A -bialgebras, b ut they are in fact also × A -Hopf algebras: Example 8 . Let ( A, L ) b e a Lie-Rinehart algebra ov er k [20, 6]. W e d enote b y ( a, X ) 7→ aX the A -mo d u le stru cture on L and by ( X , a ) 7→ X ( a ) the L -action on A giv en by the anc hor ˆ ε : L → Der k ( A ). I ts univ ersal en v eloping algebra U = U ( A, L ) is the univ ersal k -algebra equipp ed w ith t w o maps ι A : A → U, ι L : L → U of k -algebras and of k -Lie algebras, r esp ectiv ely , and su b ject to the ident ities ι A ( a ) ι L ( X ) = ι L ( aX ) , ι L ( X ) ι A ( a ) − ι A ( a ) ι L ( X ) = ι A ( X ( a )) for a ∈ A, X ∈ L ; confer [20] for the pr ecise construction. The m ap ι A is injectiv e, so w e refr ain from f urther ment ioning it. W e will also merely wr ite X wh en we mean ι L ( X ) (if L is A -pro j ectiv e, then ι L is inj ectiv e as w ell). Recall no w from e.g. [28, 18] that U carries the stru cture of a × A -bialgebra: the maps η ( − ⊗ 1) and η (1 ⊗ − ) are equal and giv en by ι A . Th e prescriptions (17) ∆( X ) = 1 ⊗ A X + X ⊗ A 1 , ∆( a ) = a ⊗ A 1 whic h map X ∈ L and a ∈ A into U × A U can b e extended by the universal prop erty to a copro duct ˆ ∆ : U → U × A U . Th e counit is similarly giv en b y the extension of the anc hor ˆ ε to U . Th e bijectivit y of th e Galois map is seen in th e s ame w a y: the translation map is giv en on generators as (18) a + ⊗ A op a − := a ⊗ A op 1 , X + ⊗ A op X − := X ⊗ A op 1 − 1 ⊗ A op X. These maps sta y in U × A op U w hic h is an algebra through th e pr o duct of U in the fi rst and its opp osite in the second tensor factor. By universalit y we obtain a map U → U × A op U , and then β − 1 is d efined usin g (16). On the other hand, U is not necessarily a Hopf algebroid in the sense of [1] (this also answe rs B¨ ohm ’s question ask ed therein whether any × A -Hopf algebra is a Hopf algebroid). This structure assumes the existence of an an tip o de S : U → U op satisfying certain axioms. As a result, the left U - action on A yields b y comp osition with S also a right U -mo d ule structure. Ho w ev er, there m igh t b e an obstru ction for this. F or example, tak e L = Γ( T 1 , 0 S 2 ), wh ere T 1 , 0 S 2 ⊕ T 0 , 1 S 2 = T S 2 ⊗ C is the decomp osition of the complexified tangen t bun dle of the 2-sph ere S 2 ⊂ R 3 in to th e holomorphic and antiholo morph ic part with resp ect to the s tandard complex structure. This d efines together with A = C ∞ ( S 2 , C ) a Lie-Rinehart algebra, where the action of L on A is the u sual action of a ve ctor field on a smo oth function and the action of A on L is giv en b y fi brewise m ultiplication. W e kno w by w ork of Huebsc hmann [7] that the righ t U -mo dule structures on A corresp ond bijectiv ely to left U -mod ule structures on L itself (in general on its top exterior p o w er o v er A , but here this is L b ecause T 1 , 0 S 2 is only a line b undle). Suc h a left U -action corresp onds precisely to a flat connection DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 7 ∇ on the complex line bund le T 1 , 0 S 2 , with X ∈ L acting on sectio ns of T 1 , 0 S 2 b y the co v arian t d eriv ativ e ∇ X (see [7] f or the details). But th e curv atur e of any connection r epresen ts the first Chern class of the bun dle whic h is nonv anishing s ince T 1 , 0 S 2 is n ot trivial. Therefore, there is no flat connection ak a left U -action on L and hence no righ t U -action on A . 3. Mul tiplica t ive structures 3.1. D − ( U ) as a susp ended monoidal category [24] . F or any r ing U , we denote by D − ( U ) the der ived category of b ound ed ab ov e co chain complexes of left U -mo d u les. As usu al, w e identify an y M ∈ U - Mo d w ith a complex in D − ( U ) concen trated in degree 0, and an y b ounded b elo w chain co mplex P • with a b oun ded ab o v e co c hain complex by pu tting P n := P − n . If U is an A -bip ro jectiv e × A -bialgebra, then an y pr o j ectiv e P ∈ U - Mo d is A -bipro jectiv e. Hence the monoidal structure of U - Mo d exte nd s to a monoidal structure on D − ( U ) with u nit ob ject still giv en by A and pro d - uct b eing the total tensor pr o duct ⊗ L = ⊗ L A (the A -b ip ro jectivit y of U - pro jectiv es is n eeded for example to hav e [27, Lemma 10.6.2]). T ogether with the shift functor T : D − ( U ) → D − ( U ), ( T C ) n = C n +1 , D − ( U ) b ecomes what is called a s usp end ed monoidal catego ry in [24]. This just means that for all C, D ∈ D − ( U ), th e canonical isomorphisms T C ⊗ L D ≃ T ( C ⊗ L D ) ≃ C ⊗ L T D giv en b y the ob vious ren umb ering mak e the d iagrams A ⊗ L T C   / / T C y y t t t t t t t t t t T ( A ⊗ L C ) T C ⊗ L A   / / T C y y t t t t t t t t t t T ( C ⊗ L A ) comm utativ e and the diagram T C ⊗ L T D   / / T ( C ⊗ L T D )   T ( T C ⊗ L D ) / / T 2 ( C ⊗ L D ) an ticomm utativ e (comm utativ e up to a sign − 1). 3.2. ` and ◦ [24] . As a sp ecial case of the constructions from [24], we define for an y A -bipro jectiv e × A -bialgebra U a nd L, M , N ∈ U - Mo d the cup pro duct ` : Ext m U ( A, M ) × Ext n U ( A, N ) → Ext m + n U ( A, M ⊗ N ) and the classical Y oneda pro duct ◦ : Ext m U ( N , M ) × Ext n U ( L, N ) → Ext m + n U ( L, M ) . The lat ter is just the comp osition of morphisms in D − ( U ) if one iden tifies Ext n U ( L, N ) ≃ Hom D − ( U ) ( L, T n N ) , and Ext m U ( N , M ) ≃ Hom D − ( U ) ( N , T m M ) ≃ Hom D − ( U ) ( T n N , T m + n M ) . 8 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER The former is obtained as follo ws: giv en ϕ ∈ Ext m U ( A, M ) ≃ Hom D − ( U ) ( A, T m M ) , ψ ∈ Ext n U ( A, N ) ≃ Hom D − ( U ) ( A, T n N ) , one defines ϕ ` ψ as the comp osition A ≃ A ⊗ A ϕ ⊗ ψ / / T m M ⊗ L T n N ≃ T m ( M ⊗ L T n N ) ≃ T m + n ( M ⊗ L N ) / / T m + n ( M ⊗ N ) , where the last map is the augmentat ion M ⊗ L N → H 0 ( M ⊗ L N ) ≃ T or A 0 ( M , N ) ≃ M ⊗ N , or rather T m + n applied to this morphism in D − ( U ). A straightfo rward extension of Theorem 1.7 from [24] no w giv es: theorem 9 . If U is an A -bipr oje ctive × A -bialgebr a, then we have ψ ◦ ϕ = ϕ ` ψ = ( − 1) mn ψ ` ϕ, ϕ ∈ Ext m U ( A, A ) , ψ ∈ Ext n U ( A, M ) as elements of Ext m + n U ( A, M ) ≃ Ext m + n U ( A, A ⊗ M ) ≃ Ext m + n U ( A, M ⊗ A ) . In p articular, Ext U ( A, A ) b e c omes thr ough either of the pr o ducts a gr ade d c ommutative algebr a over the c ommutative subring Hom U ( A, A ) . Pro of. This is pr o v en exactly as in [24]. F or the reader’s con ve nience w e include one of the diagrams inv olved. Th e unlab eled arrows are canonical maps co ming from the susp ended monoidal structure. A ϕ   / / A ⊗ A id ⊗ ϕ   ψ ⊗ i d ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q T m A & & N N N N N N N N N N N N N N N N N / / id   A ⊗ T m A   ψ ⊗ i d ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q T n M ⊗ L A id ⊗ ϕ   T m ( A ⊗ A ) x x p p p p p p p p p p p p p p p p p T m ( ψ ⊗ id )   T n M ⊗ L T m A   v v n n n n n n n n n n n n n n n n n n n T m A T m ( ψ )   T m ( T n M ⊗ L A ) x x q q q q q q q q q q q q q q q q q   T n ( M ⊗ L T m A ) v v n n n n n n n n n n n n n n n n n n n T m + n M T m + n ( M ⊗ L A ) o o The morphism ψ ◦ ϕ ∈ Hom D − ( U ) ( A, T m + n M ) is the path going straigh t do wn f rom A to T m + n M , and ψ ` ϕ is th e one wh ic h go es clo c kwise round the wh ole diagram. All faces of the d iagram commute except the lo w er righ t square whic h in tro du ces a sign ( − 1) mn , so we get ψ ◦ ϕ = ( − 1) mn ψ ` ϕ . The ot her iden tit y is sho wn with a similar diagram. ✷ DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 9 3.3. T ensoring pro jectiv es. This paragraph is a sm all excursus ab out the pro jectivit y of th e tensor pr o duct of t w o pr o jectiv e ob jects of a monoidal catego ry . F or example, U ⊗ U ∈ U - Mo d is not necessarily pro jective ev en for a bialgebra U o ve r a field A = k (so the A -pro jectivit y of U or th e exactness of ⊗ do es not h elp). Here is a sim p le example (for a detailed study of examples of categories of Ma c ke y functors see [15]): Example 10 . Consider the bialgebra U = C [ a, b, c ] ov er A = k = C with ∆( a ) = a ⊗ a, ∆( b ) = a ⊗ b + b ⊗ c, ∆( c ) = c ⊗ c, ε ( a ) = 1 , ε ( b ) = 0 , ε ( c ) = 1 . Geometrical ly , this is the co ordinate rin g of the complex algebraic semigroup G of upp er triangular 2 × 2-matrices, and ∆ and ε are dual to the semigroup la w G × G → G and the em b edding of the iden tit y matrix into G . W e pr o v e that U ⊗ U ∈ U - Mo d is not pr o j ective b y considering the fibres of the semigroup la w G × G → G . Th e fibre ov er a generic an d hence in v ertible element is 3-dimensional, but o v er 0 it is 4-dimensional, and this will imply our claim. W e can use for example [16, Theorem 19 on p. 79]: theorem 11 . L et U ⊂ V b e a flat extension of c ommutative No etherian rings, p ⊂ V b e a prime i de al and q := U ∩ p . Then dim( V p ) = dim( U q ) + dim( V p ⊗ U U ( q )) , wher e dim denotes the Krul l dimension of a ring, V p is the lo c alisation of V at p and U ( q ) := U q / q U q is the r esidue field of the lo c alisation U q . Apply this to our example U ≃ ∆ ( U ) ⊂ V := U ⊗ U : let p b e the ideal of V generated b y a ⊗ C 1, 1 ⊗ C a , b ⊗ C 1, 1 ⊗ C b , c ⊗ C 1, 1 ⊗ C c . Geometrical ly , V is the co ordinate ring of C 6 and V p is the local ring in 0, so dim( V p ) = 6. Since 1 / ∈ p , q = U ∩ p is prop er, and it cont ains the ideal generated b y ∆( a ) = a ⊗ C a , ∆( b ) = a ⊗ C b + b ⊗ C c , ∆( c ) = c ⊗ C c which is maximal in U , so q ⊂ U is the ideal generated b y a, b, c , and U q is the local ring of C 3 at 0 with dim ( U q ) = 3. Th e fi eld U ( q ) is obviously C , and we can write V p ⊗ U U ( q ) also as V p / ∆( q ) V p . Sin ce ∆( q ) V p is conta ined in the ideal r ge nerated in V p b y the elemen ts a ⊗ C 1 , 1 ⊗ C c , w e ha ve dim( V p / ∆( q ) V p ) ≥ dim( V p / r ). No w V p / r is the lo cal ring of C 4 ⊂ C 6 at 0 and hence dim( V p / r ) = 4. In total, we get the strict inequalit y 3 + dim( V p / ∆( q ) V p ) ≥ 3 + 4 = 7 > 6, and hence V is n ot flat o v er U and in particular not pro j ectiv e. F or × A -Hopf algebras the situation is, ho w ev er, muc h simp ler: notice that ◮ U ⊗ A op M ✁ := U ⊗ k M / span { a ◮ u ⊗ k m − u ⊗ k m ✁ a | u ∈ U, a ∈ A, m ∈ M } is for an y × A -bialgebra U and M ∈ U - Mo d a left U -mo dule b y left m ulti- plication on the first factor. Just as for M = U , there is a Galois map β M : ◮ U ⊗ A op M ✁ → U ⊗ M , u ⊗ A op m 7→ u (1) ⊗ A u (2) m, and w e h av e: lemma 12 . F or any × A -bialgebr a U , the gener alise d Galo is map β M is a morphism of U - mo dules. If U is a × A -Hopf algebr a, then β M is bije ctive. 10 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER Pro of. The U -linearity of β M follo ws immediately from the fact that ˆ ∆ : U → U × A U is a homomorphism of algebras o v er A e . F urth ermore, if β is a bij ection, then β M is so as well since we can id en tify β M with β ⊗ U id M , and then the in v erse is simp ly giv en b y β − 1 M ( u ⊗ A m ) = u + ⊗ A op u − m . ✷ Using this one no w gets: theorem 13 . If U is a × A -Hopf algebr a and U ✁ ∈ A op - Mo d is pr oje ctive, then P ⊗ Q ∈ U - Mo d is pr oje ctive for al l pr oje ctives P , Q ∈ U - Mo d . Pro of. By assumption, any pro jectiv e mo d ule o v er U is also p ro jectiv e o v er A op , and if ϕ : R → S is any ring map, then S ⊗ R · : R - Mo d → S - Mo d maps pr o j ectiv es to pro jectiv es. This shows that ◮ U ⊗ A op U ✁ and hence (Lemma 12) U ⊗ U is pro j ective . Since ⊗ = ⊗ A comm utes with arbitrary direct sums, P ⊗ Q is pr o jectiv e for all pro jective s P , Q . ✷ corollar y 14 . If U is as in The or em 13 and P ∈ D − ( U ) is a pr oje ctive r esolution of A ∈ U - Mo d , then so i s P ⊗ P := T ot( P • ⊗ P • ) = P ⊗ L P . This leads to the traditional constr u ction of ` giv en for A = k in [3, Chapter XI]: one fixes a pro jectiv e resolution P of A , and b y the ab o v e, Ext U ( A, M ⊗ N ) is the tot al (co)homology of the double (co chain) complex C 2 mn := Hom U ( P m ⊗ P n , M ⊗ N ) . Then ` is giv en as the comp osition of the canonical map M m + n = p Ext m U ( A, M ) ⊗ k Ext n U ( A, N ) ≃ M m + n = p H m (Hom A ( P • , M )) ⊗ k H n (Hom A ( P • , N )) → H p ( M m + n = • Hom A ( P m , M ) ⊗ k Hom A ( P n , N )) = H p (T ot( C 1 •• )) where C 1 mn := Hom U ( P m , M ) ⊗ k Hom U ( P n , N ), with the map H (T ot( C 1 •• )) → H (T ot( C 2 •• )) ≃ Ext U ( A, M ⊗ N ) that is induced b y the morph ism of double complexes C 1 mn ∋ ϕ ⊗ k ψ 7→ { x ⊗ y 7→ ϕ ( x ) ⊗ ψ ( y ) } ∈ C 2 mn . F or th e sake of completeness let u s finally remark th at as for A = k one can in particular use the bar constr u ction to obtain a canonical resolution: lemma 15 . F or any × A -bialgebr a U , the c omplex of left U -mo dules C bar n := ( ◮ U ✁ ) ⊗ A op n +1 , u ( v 0 ⊗ A op · · · ⊗ A op v n ) := uv 0 ⊗ A op · · · ⊗ A op v n whose b oundar y map is given by b ′ : u 0 ⊗ A op · · · ⊗ A op u n 7→ n − 1 X i =0 ( − 1) i u 0 ⊗ A op · · · ⊗ A op u i u i +1 ⊗ A op · · · ⊗ A op u n +( − 1) n u 0 ⊗ A op · · · ⊗ A op ε ( u n ) ◮ u n − 1 is a c ontr actible r esolution of A ∈ U - Mo d with augmentation ε : C bar 0 = U → A = : C bar − 1 , DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 11 and if U ✁ ∈ A op - Mo d is pr oje ctive, then C bar n ∈ U - Mo d is pr oje ctive. Pro of. All claims are straigh tforw ard: there is a co ntrac ting h omotop y s : C bar n → C bar n +1 , u 0 ⊗ A op · · · ⊗ A op u n 7→ 1 ⊗ A op u 0 ⊗ A op · · · ⊗ A op u n , n ≥ 0 , s : A = C bar − 1 → U = C bar 0 , a 7→ η ( a ⊗ 1) , and the pro jectivit y of C bar n follo ws as in the pro of of Theorem 13. ✷ 3.4. The functor ⊗ : U -Mo d × U op -Mo d → U op -Mo d. No w we in tro du ce the fu nctor ⊗ men tioned in Th eorem 1. lemma 16 . L et U b e a × A -Hopf algebr a and M ∈ U - Mo d , P ∈ U op - Mo d b e left and right U -mo dules, r esp e ctively. Then the formula (19) ( m ⊗ A p ) u := u − m ⊗ A pu + , u ∈ U, m ∈ M , p ∈ P , defines a right U - mo dule structur e on the tensor pr o duct (20) M ⊗ A P := M ⊗ k P / span { m ✁ a ⊗ k p − m ⊗ k a ◮ p | a ∈ A } . If N is any other (left) U -mo dule, then the c anonic al isomorph ism (21) ( M ⊗ N ) ⊗ A P ≃ M ⊗ A ( N ⊗ A P ) of A -bimo dules is also an isomorph ism in U op - Mo d . Final ly, the tensor flip ( M ⊗ A P ) ⊗ U N → P ⊗ U ( N ⊗ A M ) , m ⊗ A p ⊗ U n 7→ p ⊗ U n ⊗ A m is an isomorphism of k -mo dules. Pro of. T o sh o w fi rstly that (19) is w ell-defined o v er A , w e compu te  m ⊗ A ( a ◮ p )  u = u − m ⊗ A pη (1 ⊗ a ) u + = u − m ⊗ A p ( u + ✁ a ) = ( a ◮ u − ) m ⊗ A pu + = u −  η (1 ⊗ a ) m  ⊗ A pu + =  ( m ✁ a ) ⊗ A p  u, where (12) and the action p r op erties w ere used. T ogether with (20) this also pro v es the well-definedness with resp ect to the presen tation of u + ⊗ A op u − . With the help of (14) one sees immediately that for u, v ∈ U we hav e  m ⊗ A p  ( uv ) = ( uv ) − m ⊗ A p ( uv ) + = v − u − m ⊗ A pu + v + =  ( m ⊗ A p ) u  v , since P and M w ere righ t and left U -mo d ules, resp ectiv ely . As a conclu- sion, M ⊗ A P ∈ U op - Mo d . Equ ation (21) is a d irect consequence of the asso ciativit y of the tensor pro duct of A -bimo dules and of (13). F or the last part one has to chec k that the fl ip is w ell-defined: we h a v e η (1 ⊗ a ) m ⊗ A p ⊗ U n 7→ p ⊗ U n ⊗ A η (1 ⊗ a ) m = p ⊗ U η (1 ⊗ a )( n ⊗ A m ) = pη (1 ⊗ a ) ⊗ U ( n ⊗ A m ) , whic h is wh at m ⊗ A pη (1 ⊗ a ) ⊗ U n gets mapp ed to. And seco ndly , we ha ve m ⊗ A p ⊗ U un 7→ p ⊗ U un ⊗ A m = p ⊗ U ( u + ) (1) n ⊗ A ( u + ) (2) u − m = p ⊗ U u + ( n ⊗ A u − m ) = pu + ⊗ U n ⊗ A u − m, whic h is what u − m ⊗ A pu + ⊗ U n = ( m ⊗ A p ) u ⊗ U n ge ts mapp ed to . ✷ definition 17 . We denote the ab ove c onstructe d U op -mo dule by M ⊗ P . 12 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER Th us an unadorned ⊗ refers from no w on either to the monoidal pr o duct on U - Mo d or to the ju st defin ed action of U - Mo d on U op - Mo d . F or example, (21) w ould no w simply b e wr itten as ( M ⊗ N ) ⊗ P ≃ M ⊗ ( N ⊗ P ). Example 18 . Let ( A, L ) b e a Lie-Rinehart algebra and M b e a left and N a right U ( A, L )-modu le, resp ectiv ely (or, in th e terminology of [6, 8], left and right ( A, L )-modu les). Using (18), one gets the right U ( A, L )-mod ule structure on M ⊗ A N fr om form ula (2. 4) in [8, p . 112]: ( m ⊗ A n ) X = m ⊗ A nX − X m ⊗ A n, m ∈ M , n ∈ N , X ∈ L. If w e assume again that U is A -bipro jectiv e, then the ab o v e results extend directly to the deriv ed ca tegory D − ( U op ): we obtain a fu nctor ⊗ L = ⊗ L A : D − ( U ) × D − ( U op ) → D − ( U op ) and w e h av e for all M , N ∈ D − ( U ), P ∈ D − ( U op ) canonical isomorphisms (22) ( M ⊗ L N ) ⊗ L P ≃ M ⊗ L ( N ⊗ L P ) , ( M ⊗ L P ) ⊗ L U N ≃ P ⊗ L U ( N ⊗ L M ) . 3.5. a and • . T hese pr o ducts are dual to ` and ◦ . Th e fir st one is • : Ext m U ( L, M ) × T or U n ( N , L ) → T or U n − m ( N , M ) whic h exists for a r ing U and L, M ∈ U - Mo d , N ∈ U op - Mo d : an elemen t ϕ ∈ Ext m U ( L, M ) ≃ Hom D − ( U ) ( L, T m M ) defines a morphism in D − ( Z ) N ⊗ L U L → N ⊗ L U T m M , x ⊗ U y 7→ x ⊗ U ϕ ( y ) , and ϕ • · is the indu ced map in (co)homology T or U n ( N , L ) ≃ H − n ( N ⊗ L U L ) H − n (id ⊗ ϕ ) / / H − n ( N ⊗ L U T m M ) ≃ H m − n ( N ⊗ L U M ) ≃ T or U n − m ( N , M ) . F or M ∈ U - Mo d , N ∈ U op - Mo d as b efore, the cap pro duct a : Ext m U ( A, M ) × T or U n ( N , A ) → T or U n − m ( M ⊗ N , A ) in v olv es the fu nctor ⊗ from the previous paragraph, so f or this w e wan t U to b e an A -bipro jectiv e × A -Hopf algebra again. S imilarly as for • , ϕ ∈ Ext m U ( A, M ) ≃ Hom D − ( U ) ( A, T m M ) defines a morphism in D − ( k ) N ⊗ L U A ≃ N ⊗ L U ( A ⊗ A ) id ⊗ id ⊗ ϕ / / N ⊗ L U ( A ⊗ L T m M ) ≃ N ⊗ L U ( T m A ⊗ L M ) ≃ ( M ⊗ L N ) ⊗ L U T m A / / ( M ⊗ N ) ⊗ L U T m A, where the last ≃ in the second line is indu ced by the tensor flip as in the deriv ed v ersion (22) of Lemma 16, and the morphism from the s econd to the th ir d line is similarly as in the defi nition of ` indu ced by the morph ism M ⊗ L N → M ⊗ N in D − ( U op ) that tak es zeroth cohomolo gy . P assing no w to cohomology we get ϕ a · : T or U n ( N , A ) → T or n − m ( M ⊗ N , A ). DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 13 More explicitly , if P ∈ D − ( U ) is a p ro jectiv e resolution of A , then a is induced by the morphism B 1 ij ∋ n ⊗ U ( x ⊗ A y ) 7→ { ϕ 7→ ( ϕ ( y ) ⊗ A n ) ⊗ U x } ∈ B 2 ij from the double complex B 1 ij := N ⊗ U ( P i ⊗ A P j ) whose total homology is T or U ( N , A ) to the double complex B 2 ij := Hom k (Hom U ( P j , M ) , ( M ⊗ N ) ⊗ U P i ) whose h omology has a n atural map to Hom k (Ext U ( A, M ) , T or U ( M ⊗ N , A )). In dir ect analogy with T heorem 9 w e get: theorem 19 . If U is an A -bipr oje ctive × A -Hopf algebr a, then we have ϕ • ( x ⊗ U y ) = ϕ a ( x ⊗ U y ) , ϕ ∈ Ext m U ( A, A ) , x ⊗ U y ∈ N ⊗ L U A as elements of N ⊗ L U A ≃ ( A ⊗ N ) ⊗ L U A. 4. Duality and the proof of Theorem 1 4.1. The underiv ed case. In the sp ecial case that A is finitely gener- ated pr o jective itself, Theorem 1 reduces to stand ard linear alg ebra. W e go through this case fi rst since it is b oth instru ctiv e and us ed in the pr o of of the general case. F or the reader’s con v enience w e include fu ll pro ofs. lemma 20 . L et U b e a ring, A ∈ U - Mo d b e finitely gener ate d pr oje ctive, and A ∗ b e Hom U ( A, U ) with its c anonic al U op -mo dule structur e. 1. A ∗ is finitely g ener ate d pr oje ctive, and if e 1 , . . . , e n ar e gener ators of A , then ther e exist gener ators e 1 , . . . , e n ∈ A ∗ with X i e i ( a ) e i = a, X i e i α ( e i ) = α for al l a ∈ A and α ∈ A ∗ . The element ω := X i e i ⊗ e i ∈ A ∗ ⊗ U A is indep endent of the choic e of the gener ators e i , e j . 2. F or al l U op -mo dules M , the assignment δ ( m ⊗ a )( α ) := mα ( a ) , m ∈ M , a ∈ A, α ∈ A ∗ extends uniquely to an isomorph ism of ab elian gr oups δ : M ⊗ U A → Hom U op ( A ∗ , M ) . 3. One has ( A ∗ ) ∗ ≃ A and A ∗ ⊗ U M ≃ Hom U ( A, M ) for M ∈ U - Mo d . Pro of. Since A is pro jectiv e, there is a splitting ι : A → U n of π : U n → A, ( u 1 , . . . , u n ) 7→ X i u i e i . Hence U n ≃ A ⊕ A ⊥ for some A ⊥ ∈ U - Mo d . Dually this giv es A ∗ ⊕ ( A ⊥ ) ∗ = ( U n ) ∗ ≃ U n , whence A ∗ is finitely generated pr o j ectiv e. The e i 14 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER can b e defined as the comp osition of ι with the pr o jection of U n on th e i -th summand. This pro v es the fi rst p arts of 1. F or 2. just note that Hom U op ( A ∗ , M ) ∋ ϕ 7→ X i ϕ ( e i ) ⊗ e i ∈ M ⊗ U A in v erts δ . S ince ω = δ − 1 (id A ∗ ), it do es ind eed not d ep end on the c hoice of generators. 3. no w follo ws fr om 1. and 2. ✷ As in the in tro duction, let us abbreviate in the situ ation of this theorem H 0 ( M ) := Hom U ( A, M ) , H 0 ( N ) := N ⊗ U A for M ∈ U - Mo d , N ∈ U op - Mo d , and call ω ∈ H 0 ( A ∗ ) the fun damen tal class of ( U, A ). Then 3. sa ys for M = A that we ha ve an isomorphism (23) · • ω : H 0 ( A ) → H 0 ( A ∗ ) , ϕ 7→ X i e i ⊗ ϕ ( e i ) . Using Lemma 16 w e can upgrade th is to the und eriv ed ca se of Theorem 1: lemma 21 . L et U b e a × A -Hopf algebr a and assume A is finitely gener ate d pr oje ctive as a U -mo dule. Then the c ap pr o duct with the fu ndamental class ω ∈ H 0 ( A ∗ ) = A ∗ ⊗ U A defines for al l M ∈ U - Mo d an isomorphism · a ω : H 0 ( M ) → H 0 ( M ⊗ A ∗ ) . Pro of. W e hav e ϕ a ω = P i ( ϕ (1) ⊗ A e i ) ⊗ U e i , and Lemma 16 identifies H 0 ( M ⊗ A ∗ ) = ( M ⊗ A ∗ ) ⊗ U A ≃ A ∗ ⊗ U ( A ⊗ M ) ≃ A ∗ ⊗ U M . In this c hain of identificatio ns, ϕ a ω is mapp ed to ϕ a ω 7→ X i e i ⊗ U ( e i ⊗ A ϕ (1)) 7→ X i e i ⊗ U ( e i ϕ (1)) = X i e i ⊗ U ϕ ( e i ) whic h is ident ified with ϕ under the isomorphism Hom U ( A, M ) ≃ A ∗ ⊗ U M giv en b y ϕ 7→ P i e i ⊗ U ϕ ( e i ) as in (2 3). The claim follo ws. ✷ 4.2. The derived case. It remains to throw in some h omologic al algebra to obtain Theorem 1 in general. T o shorten the presen tation, w e define: definition 22 . A mo dule A over a ring U i s p erfe ct if it admits a finite r esolution by finitely gener ate d pr oje ctives. We c al l such a mo dule a duality mo dule if ther e exists d ≥ 0 suc h that Ext n U ( A, U ) = 0 for al l n 6 = d . We abbr eviate i n this c ase A ∗ := Ext d U ( A, U ) and c al l d the dimension of A . The main r emainin g step is to pro v e a d eriv ed ve rsion of Lemma 20. One could use a r esult of Neeman by whic h A ∈ U - Mo d is p erfect if and only if Hom U ( A, · ) commutes with direct sum s [11, 19], or the Ischebeck sp ectral sequence which degenerate s at E 2 if A is a dualit y mo d ule [10, 13, 23]. Ho w ev er, we include a more elemen tary and self-cont ained pr o of. theorem 23 . L et A ∈ U - Mo d b e a duality mo dule of dimension d . 1. The pr oje c tiv e dimension of A ∈ U - Mo d is d . 2. A ∗ is a duality mo dule of the same dimension d . DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 15 3. If P • → A is a finitely gener ate d pr oje ctive r esolution of length d , then P ∗ d −• = Hom U ( P d −• , U ) i s a finitely gener ate d pr oje ctive r eso- lution of A ∗ and the c anonic al isomorph ism δ : M ⊗ U P i → Hom U ( P ∗ i , M ) , m ⊗ U p 7→ { α 7→ mα ( p ) } induc es for al l U op -mo dules M a c anonic al isomorphism T or U i ( M , A ) → Ext d − i U op ( A ∗ , M ) . 4. Ther e is a c anonic al isomorphism ( A ∗ ) ∗ ≃ A . Pro of. Let P • → A b e a finitely generated pro jectiv e resolution of finite length m ≥ 0 (which exists since A is p erfect). Then the (co)homology of 0 → P ∗ 0 → . . . → P ∗ m → 0 , P ∗ n = Hom U ( P n , U ) is Ext • U ( A, U ), so by assumption we ha v e m ≥ d and the ab o v e complex is exact except at P ∗ d where the homology is A ∗ = Ext d U ( A, U ). F u rthermore, all the P ∗ n are fin itely generated pro jectiv e since the P n are so (Lemma 20). Let π i b e the map P ∗ i → P ∗ i +1 and put K := k er π d +1 . By construction, (24) 0 → K → P ∗ d +1 → . . . → P ∗ m → 0 is exact. If one compares this exact sequence w ith the sequence . . . → 0 → 0 → P ∗ m → P ∗ m → 0 using Sc han uel’s lemma (see [17, 7.1.2]), one obtains that K is p ro jectiv e. The exa ctness of P ∗ • at P ∗ d +1 giv es K = im π d , and by the pr o j ectivit y of K , th e map π d : P ∗ d → K ⊂ P ∗ d +1 splits so that P ∗ d ≃ K ⊕ K ⊥ , K ⊥ := k er π d . In particular, b oth K and K ⊥ are finitely generated. It follo ws fr om all this that the complex (25) 0 → P ∗ 0 → . . . → P ∗ d − 1 → K ⊥ → 0 is a finitely generated pro jectiv e resolution of A ∗ : since im π d − 1 ⊂ P ∗ d is con tained in k er π d = K ⊥ , the ab o v e complex is s till exact at P ∗ d − 1 , and the homology at K ⊥ is the homology of P ∗ • at P ∗ d , that is, A ∗ . Since (24) is a finitely generated pro j ectiv e r esolution of 0 and P ∗ d −• is as a complex a direct sum of (25) and (a shift of ) (24) we also kno w that Ext • U op ( A ∗ , M ) is for any M ∈ U op - Mo d the (co)homology of Hom U ( P ∗ d −• , M ). By Lemma 20, this is isomorphic as a c hain complex to M ⊗ U P d −• via the isomorphism giv en in 3., and the homology of this complex is T or U d −• ( M , A ). This p r o v es 3. The sp ecial case M = U implies the r emaining claims. ✷ Assume finally that in the situation of the ab ov e theorem, U is an A - bipro jectiv e × A -Hopf algebra. Since P is a pro jective resolution, w e h a v e M ⊗ U P ≃ M ⊗ L U P and Hom U ( P ∗ , M ) ≃ RHom U ( P ∗ , M ), and δ giv es an isomorphism b et w een them. Th e fundamenta l class is defined to b e ω := δ − 1 (id A ∗ ) ∈ A ∗ ⊗ L U A ≃ P ∗ ⊗ U A ≃ A ∗ ⊗ U P , and Theorem 23 giv es immed iately: 16 NIELS KO W ALZIG AND ULRICH KR ¨ AHMER corollar y 24 . If e 1 , . . . , e n and ˜ e 1 , . . . ˜ e n ar e gener ators of A and of A ∗ , r esp e ctively, then ther e ar e e 1 , . . . , e n ∈ P ∗ 0 and ˜ e 1 , . . . , ˜ e n ∈ P d such that ω = X i e i ⊗ U e i = X i ˜ e i ⊗ U ˜ e i , and δ is given by the Y one da pr o duct · • ω . Theorem 1 follo w s no w as in the un deriv ed case (Lemma 21) w orking with RHom U ( A, M ) and ( M ⊗ L A ∗ ) ⊗ L U A instead of H 0 ( M ) = Hom U ( A, M ) and H 0 ( M ⊗ A ∗ ) = ( M ⊗ A ∗ ) ⊗ U A : usin g Theorem 19 and (2 2) one gets ( M ⊗ L A ∗ ) ⊗ L U A ≃ A ∗ ⊗ L U ( A ⊗ L M ) ≃ A ∗ ⊗ L U M ≃ P ∗ ⊗ L U M ≃ RHom U ( P , M ) ≃ RHom U ( A, M ) , where we hide the reindexing of the complexes for the sak e of b etter read- abilit y (so P ∗ stands for P ∗ d −• , and RHom U ( P , M ) and RHom U ( A, M ) are reindexed in the same wa y). This leads to a con ve rgent sp ectral sequen ce T or U p (T or A q ( M , A ∗ ) , A ) ⇒ Ext d − p − q U ( A, M ) , and u nder the last assump tion of Theorem 1 (T or A q ( M , A ∗ ) = 0 for q > 0) this s p ectral sequence degenerates to the claimed isomorphism. Referen ces [1] Gabriella B¨ ohm, Hopf al gebr oids , (2008), preprint arXiv:0805. 3806 v1 . [2] Kenneth A. Brown and James J. Zh ang, Dualising c omplexes and twiste d Ho chschild (c o)homolo gy for No etherian Hopf al gebr as , J. Algebra 320 (2008), no. 5, 1814–1850. [3] Henri Cartan and Samuel Eilen b erg, Homolo gic al algebr a , Princeton Un ive rsity Press, Princeton, N . J., 1956. [4] V asiliy Dolgushev, The Van den Ber gh duality and the mo dular symmetry of a Poisson variety , ( 2006), prep rin t arXiv:math /0612288 . [5] Victor Ginzbu rg, Calabi -Yau algebr as , (2006), preprint arXiv:ma th/0612139 . [6] Johannes H u ebsc hmann, Poisson c ohomolo gy and quantization , J. Reine An gew. Math. 408 (1990), 57–113. [7] , Lie-Rinehart algebr as, Gerstenhab er algebr as and Batalin-Vil kovisky alge- br as , A nn. I n st. F ourier (Grenoble) 48 (1998), no. 2, 425–440 . [8] , Duali ty for Lie-Rinehart algebr as and the m od ular class , J. Reine An gew. Math. 510 (1999), 103–159. [9] , The universal Hopf algebr a asso ci ate d with a Hopf -Lie-Rinehart algebr a , (2008), preprint arXiv:0802 .3836 . [10] F riedric h Ischeb ec k, Ei ne Dualit¨ at zwischen den Funktor en Ext und Tor , J. Algebra 11 ( 1969), 510–531. [11] Bernhard Keller, On di ffer ential gr ade d c ate gories , International Congress of Mathe- maticians. Vol. I I, Eur. Math. So c., Z ¨ urich, 2006, pp. 151–190. [12] Masoud Khalkhali and Bahram Rangip our, Cycli c c ohomolo gy of (extende d) Hopf algebr as , N oncomm utative geometry and q uantum groups (Warsaw , 2001), Banach Cen ter Publ., vol. 61, Pol ish Acad. Sci., W arsa w, 2003, p p. 59–89. [13] Ulric h Kr¨ ahmer, Poinc ar´ e duality in Ho chschild (c o)homolo gy , New tec hniques in Hopf algebras and graded ring theory , K. Vlaam. Acad. Belgie W et. Kunsten (KV AB), Brussels, 2007, pp. 117–125. [14] St ´ ep hane L au n ois and Lionel R ic hard, Twiste d Poinc ar´ e duality f or some quadr atic Poisson algebr as , Lett. Math. Phys. 79 (2007), n o. 2, 161–174. [15] L. Gaun ce Lewis, Jr., When pr oje ctive do es not i mply flat, and other homolo gic al anomalies , Theory Appl. Categ. 5 (1999), No. 9, 202–250 (electronic). DUALITY A ND PR ODUCTS IN ALGEBRAIC (CO)HOMOLOGY THEORIES 17 [16] Hideyuki Matsumura, Commutative algebr a , second ed., Mathematics Lecture N ote Series, vol. 56, Benjamin/Cummings Pu blishing Co., In c., Reading, Mass., 1980. [17] John C. McConnell and James C. R ob son, Nonc ommutative No etherian rings , revised ed., Graduate Studies in Mathematics, vol. 30, A merican Mathematical So ciety , Pro v- idence, RI, 2001, With th e co operation of L. W. Small. [18] Iek e Mo erd ijk and Janez Mrˇ cun, On the universal enveloping algebr a of a Lie-Rinehart algebr a , ( 2008), prep rin t arXiv:0801 .3929 . [19] Amnon N eeman, T riangulate d c ate gories , An nals of Mathematics S tudies, vol . 148, Princeton Universit y Press, Princeton, NJ, 2001. [20] George S . R inehart, Differ ential forms on gener al c ommutative algebr as , T rans. A mer. Math. S oc. 108 (1963), 195–222. [21] P eter Sc hauenburg, Bialgebr as over nonc ommutative rings and a structur e the or em for Hopf bimo dules , Appl. Categ. St ructures 6 (1998), no. 2, 193–222. [22] , Duals and doubles of quantum gr oup oids ( × R -Hopf algebr as) , New trends in Hopf algebra theory ( La F alda, 1999), Contemp. Math., vol. 267, Amer. Math. So c., Pro vidence, RI, 2000, pp. 273–299. [23] Evgenij G. Sklyarenko , Poinc ar ´ e duality and r elations b etwe en the functors Ext and Tor , Mat. Zametk i 28 (1980), no. 5, 769–776, 803. [24] Maria no Suarez-A lv arez, The Hi lton-He ckmann ar gument f or the anti-c omm utativity of cup pr o ducts , Pro c. Amer. Math. So c. 132 (2004), no. 8, 2241–2246 (electronic). [25] Mitsuhiro T akeuchi, Gr oups of al gebr as over A ⊗ A , J. Math. Soc. Japan 29 (1977), no. 3, 459–492. [26] Mic hel V an d en Bergh, A r elation b etwe en Ho chschild homolo gy and c ohomolo gy for Gor enstein rings , Pro c. Amer. Math. Soc. 126 (1998), no. 5, 1345–1348 , Erratum: Proc. A mer. Math. So c. 130 , no. 9, 2809–28 10 (electronic) (2002). [27] Charles A . W eib el, An i ntr o duction to homolo gic al algebr a , Cam bridge Studies in Adv anced Mathematics, vol. 38, Cambridge Un ivers ity Press, Cambridge, 1994. [28] Ping Xu, Quantum gr oup oids , Comm. Math. Phys. 216 (2001), no. 3, 539–581. N.K.: Utrecht University, Dep ar tment of Ma thema tics, P.O. Box 80.010, 3508T A Utrecht, The Ne therlands E-mail addr ess : nkowalzi@s cience.uva.nl U.K.: Un iversity of Glasgow , Dep ar tme nt of Ma thema ti cs, Universi ty Gar- dens, G lasgo w G 12 8QW, Scotland E-mail addr ess : ukraehmer@ maths.gla.ac.uk