Resolving by a free action linear category and applications to Hochschild-Mitchell (co)homology
aa r X i v : . [ m a t h . K T ] S e p Resolving by a free action linear category andapplications to Hochschild-Mitchell (co)homology
Claude Cibils and Eduardo N. Marcos ∗ Abstract
Let G be a group acting on a small category C over a field k , that is C is a G - k -category. We first obtain that C is resolvable by a categorywhich is G - k -equivalent to it, on which G acts freely on objects.This resolvent category enables to show that if the coinvariants andthe invariants functors are exact, then the coinvariants and invariants ofthe Hochschild-Mitchell (co)homology of C are isomorphic to the trivialcomponent of the Hochschild-Mitchell (co)homology of the skew category C [ G ] . Otherwise the corresponding spectral sequence can be settled.If the action of G is free on objects, there is a canonical decompositionof the Hochschild-Mitchell (co)homology of the quotient category C /G along the conjugacy classes of G . This way we provide a general framefor monomorphisms which have been described previously in low degrees. Index Introduction The resolving category with free action Hochschild-Mitchell homology G - k -categories and graded k -categories . . . . . . . . . . . . . G - k -categories . . . . . . . . . . . . . . . . . . . . . . . ∗ The authors were partially supported by the project USP-Cofecub. The second namedauthor was partially supported by the Projeto tem´atico FAPESP 2014/09310-5 Hochschild-Mitchell cohomology Hochschild cohomology of skew group algebras Introduction
Let k be a field. A k -category C is a small category enhanced over the categoryof k -vector spaces. B. Mitchell in [26] called those categories “algebras withseveral objects”. Indeed, a k -category C with only a finite set of objets providesa k -algebra a ( C ) through the direct sum of all its morphisms. B. Mitchellintroduced Hochschild-Mitchell (co)homology of C , see also [8, 14, 21, 24]. Ifthe number of objects of C is finite, then the Hochschild-Mitchell (co)homologyof C is isomorphic to the Hochschild (co)homology of a ( C ) , see for instance [8].Let G be a group. A G - k -category is a k -category with an action of G on C . More precisely there is a group homomorphism G → Aut k C , where Aut k C isthe group of k -functors C → C which are isomorphisms. In this situation thereexists a skew category C [ G ] . If the number of objects of C is finite, then a ( C [ G ]) is isomorphic to the usual skew group algebra a ( C )[ G ] , see [7].The action of G on C is called free if the action on the objects of C is free.In that case the quotient k -category C /G exists, see for instance [8]. Moreoverthe functor C → C /G is a Galois covering. This construction has several usesin representation theory, see for instance [4, 29, 13, 1, 20].A central result obtained in [7] is that if the action of G is free, then C /G and C [ G ] are equivalent k -categories. However C [ G ] exists for any action, whilea free action is essential for defining C/G . Hence for a non free action C [ G ] isa substitute of C /G .In the same vein, in this paper we first introduce in Section 2 a resolvingmain result. Namely let C be a G - k category. There exists a G - k -category M G ( C ) - called the resolving category - which has a free action of G and whichis G - k -equivalent to C . The resolving category is related to the infinite matrixalgebra considered by J. Cornick in [10], which in turn is linked with Cohen-Montgomery duality in [6], see also [9].In Sections 3 and 4 we will use the resolving category for Hochschild-Mitchellhomology and cohomology. Indeed, for a G - k -category C the resolving categoryenables to relate the Hochschild-Mitchell (co)homologies of C , of C /G , andof C [ G ] . In doing so, we underline that we do not change the bimodules ofcoefficients, in contrast with the Cartan-Leray type spectral sequence obtainedin [8], see also [18] and [27]. n Subsection 3.1 we show that there is a direct sum decomposition ofthe Hochschild-Mitchell homology of a G -graded k -category along the set ofconjugacy classes of G , as in the case of G -graded algebras, see [10, 23, 31].For instance the homology of the graded category C [ G ] decomposes asabove. Suppose that the action is free and the coinvariants functor is exact.In Subsection 3.2 we prove that there is an isomorphism between the trivialconjugacy class direct summand of the homology of C [ G ] , and the coinvariantsof the homology of C . If the coinvariants functor is not exact, then the ad-hocspectral sequence can be settled.The resolving category and results of E. Herscovich in [17] enables then toprove that the isomorphism quoted above also holds when the action is not free.In Subsection 3.3 we infer the analog results for a Galois covering.For Hochschild-Mitchell cohomology, the defining cochains are direct prod-ucts of vector spaces while for homology the chains are direct sums. Neverthe-less, for a graded k -category, we get to show in Subsection 4.1 that the complexof cochains is also a direct product along the conjugacy classes of G . More-over, the subcomplex associated to the trivial conjugacy class is a subdifferentialgraded algebra of it.If the action is free and if the invariants functor is exact, we show that thereis an isomorphism of algebras between the trivial component of the cohomologyof C [ G ] and the invariants of the cohomology of C . Note that the proof is quitedifferent than the one for homology. The resolving category enables then toextend the result for a non free action.in Subsection 4.3 we translate these results into the Galois covering setting.An immediate consequence is that the invariants of the cohomology of C are acanonical direct summand of the cohomology of C/G . We recover this way themonomorphism obtained in [25] which is made explicit in low degrees in [16].In Section 5 we restrict our results to the case of a k -algebra with a finitegroup acting by automorphism, and its Hochschild cohomology. We underlinethat the proof relies in an essential way on the existence of the resolving category.When specialising to algebras, the results of this work are related withexplicit computations made for instance in [12, 15, 30, 28] for Hochschild(co)homology of specific skew group algebras, in particular for the symmet-ric algebra over a finite dimensional vector space V over a field k , with G afinite subgroup of GL ( V ) which order is invertible in k. The resolving category with free action
Let k be a field. A k -category is a small category C enriched over the categoryof k -vector spaces. In other words the objects of C are a set denoted C , forany pair of objects x, y ∈ C the morphisms y C x from x to y have a structure ofvector space, the composition in C is k -bilinear, and the image of the canonicalinclusion k ֒ → x C x is central in x C x for all x ∈ C . In particular x C x is a k -algebra or any x ∈ C .We often write y f x for a morphism f from x to y , that is belonging to y C x . Definition 2.1
Let G be a group. A G - k -category is a k -category C with anaction of G by k -isomorphisms of C . Remark 2.2
Equivalently, a G - k -category C is a k category with firstly anaction of G on the set of objects C . Then for all s ∈ G and y f x ∈ y C x thereis given sy ( sf ) sx ∈ sy C sx . Moreover, the map y C x → sy C sx given by f sf is k -linear. For all s, t ∈ G and for any morphism f we have t ( sf ) = ( ts ) f .Finally for any object x and any s ∈ G , we have s ( x x ) = sx sx . Example 2.3
Let Λ be a k -algebra and let Λ be the single object k -categorywhere the endomorphism algebra of the object is Λ . Let G be a group actingby algebra automorphisms of Λ . Then Λ is a G - k -category. Definition 2.4 A G - k category C is a category with a free action of G if theaction of G on its objects is free. That is if s ∈ G and x ∈ C , then sx = x only holds if s = 1 . Remark 2.5
Except if G is trivial, the G - k -category Λ of Example 2.3 is notwith a free action of G . Recall that for a finite object k -category C , its k -algebra is a ( C ) = M x,y ∈C y C x with product given by matrix multiplication combined with the composition of C . The identity element of a ( C ) is the sum of the identities of the objects. Remark 2.6
Notice that if C and D are finite object k -categories, a k -functor F : C → D do not provides in general a multiplicative k -morphism between a ( C ) and a ( D ) . In particular the Galois coverings of Subsection 3.3 are functorswhich do not provide in general algebra maps. The following result is a straightforward generalisation of Example 2.3.
Lemma 2.7
Let C be a G - k -category with a finite set of objets. The group G acts on a ( C ) by automorphisms of the algebra. Next we will show that a G - k -category C is resolvable by a G - k categorywhich is G - k -equivalent to C , on which G acts freely. efinition 2.8 (see also [10], [11]) Let C be a G - k -category. The objects ofthe resolving k -category M G ( C ) are G × C . The k -vector space of morphismsof M G ( C ) from ( s, x ) to ( t, y ) is ( t,y ) ( M G ( C )) ( s,x ) = y C x . The composition is given in the evident way by the composition of C . The actionof G on M G ( C ) is defined as follows: for r ∈ G , let r ( s, x ) = ( rs, rx ) , and for f ∈ ( t,y ) ( M G ( C )) ( s,x ) = y C x let rf ∈ ry C rx = ( rt,ry ) ( M G ( C )) ( rs,rx ) . Notice that the above action of G is free on the objects of M G ( C ) . Example 2.9
Let Λ be a k -algebra with a group acting by automorphisms, andlet Λ be its single object G - k -category. • The resolving category M G (Λ ) has set of objects G . • Each vector space of morphisms of M G (Λ ) is a copy of Λ . • The composition of M G (Λ ) is given by the product of Λ . • The action of G on the objects is the product of G . On morphisms it isgiven by the action on objects followed by the action on Λ . Next we consider the resolving category through tensor products of cate-gories.
Definition 2.10
Let C and D be k -categories. Their tensor product C ⊗ D hasset of objects C × D . Its morphisms are given by: ( c ′ ,d ′ ) ( C ⊗ D ) ( c,d ) = c ′ C c ⊗ d ′ D d with the obvious composition. If C and D are G - k -categories then C ⊗D is a G - k -category through the diagonalaction of G . Proposition 2.11
Let C be a G - k -category. Let k be the G - k -trivial category,with one object with endomorphisms k and trivial G -action. M G ( C ) = M G ( k ) ⊗ C . Theorem 2.12
Let C be a G - k -category. There is an equivalence of G - k -categories L : M G ( C ) → C . roof. Let L : M G ( C ) → C be the functor defined on the objects by L ( s, x ) = x , while on morphisms L is given by the suitable identity maps. Hence L is afully faithful G -functor which is surjective on the objects, so it is an equivalenceof G - k -categories. ⋄ We end this section by describing the algebra associated to a resolvingcategory.
Proposition 2.13
Let Λ be a k -algebra with an action of a finite group G byautomorphisms of Λ . Let M G (Λ ) be the resolving category of Λ .The k -algebra a ( M G (Λ ) is isomorphic to the matrix algebra M G (Λ) withcolumns and rows indexed by G . The action of G on a matrix is the combinationof the action of G on Λ with the action of G on the indices of the rows and ofthe columns. The proof relies on the description given at example 2.9.
Remark 2.14
Let E be the set of diagonal idempotents of the algebra above M G (Λ) , each one is given by ∈ Λ on a certain spot of the diagonal and zeroselsewhere. The number of elements of E is | G | .The group G acts freely on E . Hochschild-Mitchell homology G - k -categories and graded k -categories We begin this section by recalling the definition of the Hochschild-Mitchell ho-mology of a k -category, see for instance [26, 21, 24, 8]. Next we give propertiesof it for G - k -categories.Then we provide the decomposition of the homology of a graded categoryalong the conjugacy classes of G . This will be used in Subsection 3.2 for skewcategories. Definition 3.1
Let C be a k -category and let C • ( C ) be the chain complex givenby: C ( C ) = M x ∈C x C x ,C n ( C ) = M x ,x ,...,x n ∈C x C x n ⊗ · · · ⊗ x C x ⊗ x C x , with boundary map d given by the usual formulas used to compute the Hochschildhomology of an algebra, see for instance [22, 33, 5].The Hochschild-Mitchell homology HH ∗ ( C ) of C is the homology of thecomplex above. .1.1 G - k -categories Proposition 3.2
Let C be a G - k -category. The group G acts on the chaincomplex C • ( C ) by automorphims. Proof.
For s ∈ G and ( f n ⊗ · · · ⊗ f ⊗ f ) ∈ C n ( C ) we define sf = ( sf n ⊗ · · · ⊗ sf ⊗ sf ) . For n = 2 we have d ( sf ) = sx ( sf sf ) sx ⊗ sx ( sf ) sx − sx ( sf ) sx ⊗ sx ( sf sf ) sx + sx ( sf sf ) sx ⊗ sx ( sf ) sx and sd ( f ) = sx ( s ( f f )) sx ⊗ sx ( sf ) sx − sx ( sf ) sx ⊗ sx ( s ( f f )) sx + sx ( s ( f f )) sx ⊗ sx f x . ⋄ Corollary 3.3
Let C be a G - k -category. The Hochschild-Mitchell homology HH ∗ ( C ) is a kG -module. Recall that for a kG -module M , the kG -module of coinvariants of M is M G = M/ < sm − m > where the denominator is the sub kG -module of M generated by { sm − m | m ∈ M, s ∈ G } . The module of coinvariants is the largest quotient of M withtrivial action of G . Considering M as a kG -bimodule with trivial action on theright, we have M G = kG ⊗ kG ⊗ ( kG ) op M. If G is of finite order invertible in k , then the coinvariants functor is exact.Moreover, M G is canonically isomorphic to the invariants M G = { m ∈ M | sm = m for all s ∈ G } through the morphism m | G | P s ∈ G sm . Graded categories
Definition 3.4
Let G be a group. A k -category B is G -graded if for all x, y ∈B there is a direct sum decomposition of vector spaces y B x = M s ∈ G y B sx such that z B ty y B sx ⊂ z B tsx for all objects x, y, z ∈ B and for every s, t ∈ G. A morphism f ∈ y B sx is called homogeneous of degree s from x to y , we oftenwrite it y f sx instead of f . ext we provide a decomposition of the chain complex of a graded category.This corresponds to M. Lorenz [23] decomposition for the Hochschild homologyof a G -graded k -algebra (see also [32]). Proposition 3.5
Let B be a G -graded k -category. Let D be a conjugacy classof G. The following is a subcomplex of C • ( B ) : C Dn ( B ) = M s n ...s s ∈ Dx n ,...,x ,x ∈B x B s n x n ⊗ · · · ⊗ x B s x ⊗ x B s x . Let Cl G be the set of conjugacy classes of G. There is a decomposition C • ( B ) = M D ∈ Cl G C D • ( B ) . Proof.
The verification for n = 2 provides the way for proving the result forany n . It has the advantage of avoiding long and useless technical computations.Let x f s x ⊗ x f s x ⊗ x f s x ∈ C D ( B ) , with s s s ∈ D. We have d ( f ⊗ f ⊗ f ) = f f ⊗ f − f ⊗ f f + f f ⊗ f ∈ (cid:0) x B s s x ⊗ x B s x (cid:1) ⊕ (cid:0) x B s x ⊗ x B s s x (cid:1) ⊕ (cid:0) x B s s x ⊗ x B s x (cid:1) . Note that for the last summand s s s = s ( s s s ) s − ∈ D. ⋄ Theorem 3.6
Let B be a G -graded k -category and let HH D ∗ ( B ) = H ∗ ( C D • ( B )) .There is a decomposition HH ∗ ( B ) = M D ∈ Cl G HH D ∗ ( B ) . Skew categories
In this section we compare the coinvariants of the Hochschild-Mitchell homol-ogy of a G - k -category C with the trivial component of the homology of theskew category C [ G ] . Indeed C [ G ] (defined below) is graded, hence we will useTheorem 3.6. Definition 3.7 [7] Let C be a G - k -category. The skew category C [ G ] has thesame set of objects than C . Let y C [ G ] xs = y C sx . The morphisms of C [ G ] from x to y are y C [ G ] x = M s ∈ G y C [ G ] xs . (3.1) The composition is defined through adjusting the first morphism in order tomake possible to compose it in C with the second one, as follows. If y f sx ∈ y C sx ⊆ y C [ G ] x and z g ty ∈ z C ty ⊆ z C [ G ] y , then z g ty )( y f sx ) = z ( g ◦ tf ) tsx ∈ z C [ G ] x , where ◦ denotes the composition of C . Remark 3.8
1. By definition, the direct summands of (3.1) are in one to one correspon-dence with elements of G . However some of the summands may be zero.2. If C is a single object G - k -category with endomorphism algebra Λ , it isshown in [7] that the single object k -category C [ G ] has endomorphismalgebra the usual skew group algebra. Namely as a vector space, Λ[ G ] =Λ ⊗ kG . For a, b ∈ Λ and s, t ∈ G , the product is given by ( a ⊗ s )( b ⊗ t ) = as ( b ) ⊗ st. Lemma 3.9
The category C [ G ] is graded. Proof.
We have z C [ G ] yt y C [ G ] xs ⊂ z C [ G ] xts . (3.2) ⋄ Hence the decomposition of Theorem 3.6 holds for the homology of C [ G ] . Free actionTo prove the next result we first need to restrict ourselves to a free action. Thenwe will be able to consider the general case by using the resolving category ofSection 2.
Theorem 3.10
Let C be a G - k -category with free action of G . Let { } be thetrivial conjugacy class of G . There is an isomorphism HH { }∗ ( C [ G ]) ≃ H ∗ (( C • ( C ) G ) . If the coinvariants functor is exact then HH { }∗ ( C [ G ]) ≃ H ∗ ( C ) G . Remark 3.11
If the covariants functor is not exact, it follows by standard ar-guments that there is a spectral sequence for computing HH { }∗ ( C [ G ]) . We provide next properties of a skew category that we will need for prov-ing Theorem 3.10. The following result is proved in [7], we give a proof forcompleteness. emma 3.12 Let C be a G - k -category. Let x and y be objects in the sameorbit of the action of G . They are isomorphic in C [ G ] . Proof.
Let t ∈ G such that y = tx . Recall that tx C [ G ] x = L s ∈ G tx C sx . Let a = tx tx ∈ tx C [ G ] tx = tx C tx and b = x x ∈ x C [ G ] t − tx = x C x . Using the composition defined in C [ G ] , we obtain that a and b are mutualinverses in the skew category. ⋄ Definition 3.13
Let G be a group acting on a set E. A transversal T of theaction is a subset of E consisting of exactly one element in each orbit of theaction. Equivalently T ⊂ E is a transversal if for each x ∈ E there exists a unique u ( x ) ∈ T such that there exists some s ∈ G such that x = su ( x ) .Note that the action is free if and only if for each x ∈ E , there exists aunique s ∈ G such that x = su ( x ) . Lemma 3.14
Let C be a G - k -category, let T ⊆ C be a transversal of theaction of G on C , and let C T [ G ] be the full subcategory of C [ G ] with set ofobjects T. For each conjugacy class D of G we have HH D ∗ ( C T [ G ]) = H D ∗ ( C [ G ]) . Proof.
In [17] E. Herscovich (see also [3, 19]) proved that if C and D are k -categories and if F : C → D is a k -equivalence, then F induces a quasi-isomorphism C • ( C ) → C • ( D ) . Moreover, if C and D are G -graded and if F is homogeneous, then the induced quasi-isomorphism clearly preserves thedecomposition along conjugacy classes of G . Since T is a transversal, theabove Lemma 3.12 shows that the inclusion functor C T [ G ] ⊂ C [ G ] is dense.Moreover it corresponds to a full subcategory, hence it is full and faithful, inaddition of being homogeneous. The induced quasi-isomorphism provides thenthe result. ⋄ Proof of Theorem 3.10.
Let T be a transversal of the free action of G on C . In order to define an isomorphism of chain complexes A : C • ( C ) G → C • ( C { } T [ G ]) , let x f x ⊗ x f x ⊗ x f x be a chain of C ( C ) . In ( C ( C )) G , webegin by modifying the chain in order that the starting (and hence the ending)object x belongs to T. More precisely, there exists a unique s ∈ G such that sx = u ∈ T. Then f ⊗ f ⊗ f ≡ s ( f ⊗ f ⊗ f ) = u ( sf ) sx ⊗ sx ( sf ) sx ⊗ sx ( sf ) u . In other words we can assume that the chain is of the form u ( f ) x ⊗ x ( f ) x ⊗ x ( f ) u for u ∈ T. or i = 1 , , let u i = u ( x i ) be the unique element of T which is in the orbit of x i . Moreover let s i be the unique element of G such that x i = s i u i . We define A ( u ( f ) s u ⊗ s u ( f ) s u ⊗ s u ( f ) u ) = u ( f ) s u ⊗ u ( s − f ) s − s u ⊗ u ( s − f ) s − u = u [ f ] s u ⊗ u [ s − f ] s − s u ⊗ u [ s − f ] s − u . This chain belongs to C { } ( C T [ G ]) since s ( s − s ) s − = 1 . For a 3-chain theformula defining A is A ( u f s u ⊗ s u f s u ⊗ s u f s u ⊗ s u f u ) = u [ f ] s u ⊗ u [ s − ( f )] s − s u ⊗ u [( s − f )] s − s u ⊗ u [ s − f ] s − u . Next we verify that A is a chain map. dA ( f ⊗ f ⊗ f ) = [ f ] s [ s − f ] s − s ⊗ [ s − f ] s − − [ f ] s ⊗ [ s − f ] s − s [ s − f ] s − +[ s − f ] s − [ f ] s ⊗ [ s − f ] s − s = [ f f ] s ⊗ [ s − f ] s − − [ f ] s ⊗ [( s − f )( s − f )] s − +[( s − f )( s − f )] s − s ⊗ [ s − f ] s − s . Recall that d ( f ⊗ f ⊗ f ) = u ( f f ) s u ⊗ s u ( f ) u − u ( f ) s u ⊗ s u ( f f ) u + s u ( f f ) s u ⊗ s u ( f ) s u . In order to compute Ad , notice that up to the action, that is in the coinvariants,the last term of the previous sum can be rewrited: s u ( f f ) s u ⊗ s u ( f ) s u ≡ u ( s − ( f f )) s − s u ⊗ s − s u ( s − f ) u . This way the last summand of d ( f ⊗ f ⊗ f ) starts and ends at u ∈ T ,which is required in order to apply A. Hence Ad ( f ⊗ f ⊗ f ) = A (cid:0) u ( f f ) s u ⊗ s u ( f ) u − u ( f ) s u ⊗ s u ( f f ) u + u ( s − ( f f )) s − s u ⊗ s − s u ( s − f ) u (cid:1) = [( f f )] s ⊗ [ s − f ] s − − [ f ] s ⊗ [ s − ( f f )] s − +[ s − ( f f )] s − s ⊗ [( s − s ) − s − f ] ( s − s ) − and this shows Ad = dA. et g ⊗ g ⊗ g ∈ C { } ( C T [ G ]) , that is g ⊗ g ⊗ g = u [ g ] s u ⊗ u [ g ] s u ⊗ u [ g ] s u = u g s u ⊗ u g s u ⊗ u g s u with s s s = 1 . Let B : C { }• ( C T [ G ]) → ( C • ( C )) G be defined by B ( g ⊗ g ⊗ g ) = u ( g ) s u ⊗ s u ( s g ) s s u ⊗ s s u ( s s g ) s s s u . We observe that since s s s = 1 , we have that s s s u = u . Next we willshow that A and B are mutual inverses. This will imply that B is a chain map,since A is a chain map. AB ( g ⊗ g ⊗ g ) = A ( u g s u ⊗ s u ( s g ) s s u ⊗ s s u ( s s g ) u ) = u g s u ⊗ u ( s − s g ) s u ⊗ u (( s s ) − ( s s ) g ) ( s s ) − u . Since s s s = 1 , we obtain u g s u ⊗ u g s u ⊗ u g s u = u [ g ] s u ⊗ u [ g ] s u ⊗ u [ g ] s u .BA ( u f s u ⊗ s u f s u ⊗ s u f u ) = B (cid:18) u [ f ] s u ⊗ u [ s − f ] s − s u ⊗ u [ s − f )] s − u (cid:19) = u f s u ⊗ s u ( s s − f ) s u ⊗ s s − s u (( s s − s ) s − f ) ( s s − s ) s − u = u f s u ⊗ s u f s u ⊗ s u f u . ⋄ General caseThe next Lemma enables to generalise Theorem 3.10 to a G - k -category wherethe action of G is non necessarily free. Recall that M G ( C ) is the resolvingcategory of a G - k -category, see Definition 2.8. Lemma 3.15
Let C be a G - k -category. The chain complexes C • ( M G ( C )) and C • ( C ) are kG -quasi-isomorphic. Proof.
By Theorem 2.12, there is an equivalence of categories L : M G ( C ) →C . As for Lemma 3.14, we infer from [17] that there is an induced quasi-isomorphism C • ( M G ( C )) → C • ( C ) . ⋄ heorem 3.16 Let C be a G - k -category. HH { }∗ ( C [ G ]) = H ∗ ( C • ( C ) G ) . If the coinvariants functor is exact HH { }∗ ( C [ G ]) = HH ∗ ( C ) G . Proof.
Due to Theorem 3.10 the result holds for the G - k -category M G ( C ) . The equivalence of G - k -categories L : M G ( C ) → C provides an homogeneousequivalence of G -graded k -categories L [ G ] : M G ( C )[ G ] → C [ G ] which gives a quasi-isomorphism C • ( M G ( C )[ G ]) → C • ( C [ G ]) preserving the decomposition of chain complexes along the conjugacy classes of G . Hence HH { }∗ ( M G ( C )[ G ]) = HH { }∗ ( C [ G ]) . By the above proposition H ∗ ( C • ( M G ( C )) G ) = H ∗ ( C • ( C ) G ) . If the coinvariants functor is exact then HH ∗ ( M G ( C )) G = ( HH ∗ ( C )) G . ⋄ Galois coverings
In this section we reformulate Theorem 3.10 in terms of Galois coverings. Firstwe recall the definition of a quotient category.
Definition 3.17 (see [4, 29] and also [7]) Let C be a G - k -category with a freeaction of G . The quotient category C /G has set of objects the set of orbits C /G. Let α and β be orbits. The vector space of morphisms from α to β is β ( C /G ) α = M x ∈ αy ∈ β y C x G . Let γ, β, α be orbits. Let g ∈ z C y ′ and f ∈ y C x , here z ∈ γ , y and y ′ ∈ β , and x ∈ α. Let s be the unique element of G suchthat sy = y ′ , then f ≡ sf in the coinvariants. The composition gf in C /G is gf = z g y ′ sy sf sx ∈ γ ( C /G ) α . There is no difficulty in verifying that this is a well defined associative compo-sition.
Definition 3.18
A Galois covering of k -categories is a functor C → C /G, where C is a G - k -category with free action, and where the functor is the canonicalprojection functor. Let
C → C /G be a Galois covering and let T be a transversal of the actionof G on C . For each orbit α , let u α ∈ T be the unique element of T whichbelongs to α. It is shown in Lemma 2.2 of [7] that through a canonical identification wehave β ( C /G ) α = M s ∈ G u β C su α . This provides that C /G is graded by G . Indeed let β ( C /G ) sα = u β C su α , andnotice that ( u γ g tu β )( u β f su α ) = (cid:0) u γ g tu β )( tu β tf tsu α (cid:1) = u γ ( g ( tf )) tsu α ∈ γ ( C /G ) tsα . The following result can be deduced from [7]. We provide a proof for complete-ness.
Proposition 3.19
Let
C → C /G be a Galois covering. Let T ⊂ C be atransversal of the free action of G , and consider the G -grading of C /G deter-mined by T. The graded categories C /G and C T [ G ] are isomorphic by an homogeneousfunctor, and the graded categories C /G and C [ G ] are equivalent by an homo-geneous functor. Proof.
There is a bijection between the objects of C /G and those of C T [ G ] .The previous considerations shows that the morphisms of both categories aresubsequently identified in an homogeneous manner. Moreover we have alreadyused that the inclusion C T [ G ] ⊂ C [ G ] provides an equivalence of categories,and this equivalence is homogeneous. ⋄ The above analysis provides the translation of Theorem 3.10 in terms ofGalois coverings, as follows.
Theorem 3.20
Let
C → C /G be a Galois covering. HH { }∗ ( C /G ) = H ∗ ( C • ( C ) G ) . If the coinvariants functors is exact HH { }∗ ( C /G ) = HH ∗ ( C ) G . Hochschild-Mitchell cohomology
Graded categories
As mentioned in the Introduction, Hochschild-Mitchell cohomology is more in-tricate than homology since it makes use of direct products. We begin thissection by recalling its definition. If the category is graded, we provide thedecomposition along conjugacy classes.
Definition 4.1
Let C be a k -category. Let C • ( C ) be the complex of cochainsgiven by: C ( C ) = Y x ∈C x C x , C n ( C ) = Y x n +1 ,...,x ∈C C x n +1 ,...,x for n > where C x n +1 ,...,x = Hom k ( x n +1 C x n ⊗ · · · ⊗ x C x , x n +1 C x ) The coboundary d is given by the formulas below which are the usual ones forcomputing Hochschild cohomology, see for instance [5, 33].Let ϕ be a cochain of degree n , that is a family of k morphisms ϕ = { ϕ ( x n +1 ,...,x ) } . Its coboundary dϕ is the family { ( dϕ ) ( x n +2 ,...,x ) } given by ( dϕ ) ( x n +2 ,...,x ) (cid:0) x n +2 ( f n +1 ) x n +1 ⊗ · · · ⊗ x ( f ) x (cid:1) =( − n +1 f n +1 ϕ ( x n +1 ,...,x ) ( f n ⊗ · · · ⊗ f )+ P ni =1 ( − i +1 ϕ ( x n +1 ,...,x i +2 ,x i ,...x ) ( f n +1 ⊗ · · · ⊗ f i +1 f i ⊗ · · · ⊗ f )+ ϕ ( x n +2 ,...x ) ( f n +1 ⊗ · · · ⊗ f ) f . (4.1) Note that d is well defined since the cochains are direct products. The Hochschild-Mitchell cohomology of C is HH ∗ ( C ) = H ∗ ( C • ( C )) . In degree zero we set HH ( C ) = { ( x f x ) x ∈C | y g x x f x = y f y y g x for all y g x ∈ y C x } . As for Hochschild cohomology of algebras, the cup product is defined at thecochain level as follows. Let ϕ ∈ C x n +1 ,...,x and ψ ∈ C y m +1 ,...,y . If x n +1 = y the cup product ψ ⌣ ϕ is zero. Otherwise the cup product ψ ⌣ ϕ ∈C y m +1 ,...,y ,x n ,...,x is ( ψ ⌣ ϕ )( f n + m ⊗ · · · ⊗ f ) = ψ ( f n + m ⊗ · · · ⊗ f n +1 ) ϕ ( f n ⊗ · · · ⊗ f ) . The cup product verifies the graded Leibniz rule, and it provides a graded com-mutative k -algebra structure on HH ∗ ( C ) . In particular HH ( C ) is a commuta-tive k -algebra which is the center of the category. roposition 4.2 Let B be a G -graded category, and let Cl ( G ) be the set ofconjugacy classes of G. There is a decomposition HH ∗ ( B ) = Y D ∈ Cl ( G ) HH ∗ D ( B ) where HH ∗{ } ( B ) is a subalgebra. Proof.
For D ∈ Cl ( G ) we provide a subcomplex of cochains C • D ( B ) of C • ( B ) as follows. Let ϕ be a cochain of degree n . We say that ϕ is homogeneous oftype ( s n , . . . , s , s ) if:1. Each component of ϕ has its image contained in the homogeneous mor-phisms of degree s .2. For ( s ′ n , . . . , s ′ ) = ( s n , . . . , s ) , each component of ϕ restricted to tensorsof homogeneous morphisms degree ( s ′ n , . . . , s ′ ) is zero.The formula (4.1) which defines the coboundary d has n +2 summands. Let d n +1 be the first one, let d be the last one, and let d i denotes the in between sum-mands indexed according to the appearance of the composition “ f i +1 f i ” for i = n, . . . , . Let ϕ be homogeneous of type ( s n , · · · , s , s ) . We observe the following: • d n +1 ϕ is a sum of homogeneous cochains of types ( s, s n , . . . , s , ss ) for s ∈ G. • d i ϕ is a sum of homogeneous cochains of types ( s n , . . . , s i +1 , s ′′ , s ′ , s i − , . . . , s , s ) for s ′′ , s ′ ∈ G with s ′′ s ′ = s i . • d ϕ is a sum of homogeneous cochains of types ( s n , . . . , s , s, s s ) for s ∈ G. Let the class of ( s n , . . . , s , s ) be the product s n . . . s s − ∈ G. The aboveconsiderations show that if ϕ is homogeneous of type ( s n , . . . , s , s ) , hence ofclass c = s n . . . s s − , then dϕ is a sum of homogeneuos cochains, possibly ofdifferent types but whose classes are conjugated to c. Let D be a conjugacy class and let C • D ( C ) be the homogeneous cochainswhich classes of types are in D. We have showed that C • D ( C ) is a cochainssubcomplex of C • ( C ) . Moreover C • ( C ) = Y D ∈ Cl ( G ) C • D ( C ) . Clearly, if ϕ and ψ are homogeneous cochains which classes of types are both1, then ψ ⌣ ϕ is also of class type 1. ⋄ .2 Skew categories
We recall that the skew category is graded, so the results of the Subsection 4.1are in force. As for homology, at first glance we are only able to provide a resultif the action is free. Then the resolving category of Section 2 enables to extendthe result for any G - k -category.Firstly we provide some properties of the cohomology of a G - k -category thatwe need.If C is a G - k -category then C n ( C ) is a kG -module as follows. Let ϕ = { ϕ ( x n +1 ,...,x ) } ∈ C n ( C ) be a cochain, where ϕ ( x n +1 ,...,x ) : x n +1 C x n ⊗ · · · ⊗ x C x → x n +1 C x . Let s ∈ G and let s. [ ϕ ( x n +1 ,...,x ) ] : sx n +1 C sx n ⊗ · · · ⊗ sx C sx → sx n +1 C sx be defined by s. [ ϕ ( x n +1 ,...,x ) ]( f n ⊗ · · · ⊗ f ) = s [ ϕ ( x n +1 ,...,x ) ( s − f n ⊗ · · · ⊗ s − f )] . Finally we set s.ϕ = { s. [ ϕ x n +1 ,...,x ] } . Remark 4.3
Let ( ) G be the invariants functor. Then ϕ ∈ ( C n ( C )) G if andonly if for all s ∈ G and for any sequence of objects x n +1 , . . . , x we have that ϕ ( sx n +1 ,...,sx ) ( sf n ⊗ · · · ⊗ sf ) = s (cid:2) ϕ ( x n +1 ,...,x ) ( f n ⊗ · · · ⊗ f ) (cid:3) . Clearly the action of G commutes with the coboundary of C • ( C ) . Moreoverthe action of G is by automorphisms of the cup product. In other words C • ( C ) is a differential graded algebra with an action of G by automorphisms of itsstructure.In particular ( C • ( C )) G is a graded differential algebra. Moreover the inferredaction of G on HH ∗ ( C ) is by automorphisms of the algebra. If the invariantsfunctor is exact, then ( HH ∗ ( C )) G = H ∗ ( C • ( C ) G ) as k -algebras. Free action
Theorem 4.4
Let C be a G - k -category with a free action of G , and let C [ G ] be the G -graded skew category. There is an isomorphism of k -algebras HH ∗{ } ( C [ G ]) ≃ H ∗ (cid:16) ( C • ( C ) G (cid:17) . If the invariants functor is exact, we infer an isomorphism of k -algebras HH ∗{ } ( C [ G ]) ≃ HH ∗ ( C ) G . f the invariants functor is not exact, the standard considerations provide aspectral sequence. Proof.
Let T be a transversal of the action of G on C and let C T [ G ] be thefull subcategory of C [ G ] with set of objets T. Let D be a conjugacy class of G. We assert that HH ∗ D ( C T [ G ]) = HH ∗ D ( C [ G ]) . Indeed the equivalence of categories given by the inclusion C T [ G ] ⊆ C [ G ] in-duces a quasi-isomorphism of the complexes of cochains which preserves thedecomposition along the conjugacy classes of G. Moreover for the trivial conjugacy class the quasi-isomorphism is a morphismof differential graded algebras. Hence HH ∗{ } ( C T [ G ]) = HH ∗{ } ( C [ G ]) as k -algebras.In what follows we will prove that there are morphisms of graded differentialalgebras ( C • ( C )) G A ⇄ B C •{ } ( C T [ G ]) which are inverses one of each other.Let ψ ∈ (cid:0) C ( C ) (cid:1) G and let u , u , u , u ∈ T. We will define ( Aψ ) ( u ,u ,u ,u ) on each homogeneous component.Let f ⊗ f ⊗ f ∈ u C T [ G ] s u ⊗ u C T [ G ] s u ⊗ u C T [ G ] s u . Recall that by the definition of the morphisms of C [ G ] we have that f i ∈ u i +1 C t i u i for i = 1 , , . Let ( Aψ ) ( u ,u ,u ,u ) ( f ⊗ f ⊗ f ) = ψ ( u ,s u ,s s u ,s s s u ) ( f ⊗ s f ⊗ s s f ) . We observe that this definition makes sense since f ⊗ s f ⊗ s s f ∈ u C s u ⊗ s u C s s u ⊗ s s u C s s s u . Moreover Aψ ( f ⊗ f ⊗ f ) ∈ u C s s s u = u C T [ G ] s s s u , that is we have indeed defined an homogeneous cochain of type ( s , s , s , s s s ) ,which is of class { } . The verification that dA = Ad is straightforward, it uses in a crucial waythat ψ is an invariant; the formulas defining the composition in C [ G ] are requiredas well . Analogously, it is easy to verify that A ( ψ ′ ⌣ ψ ) = A ( ψ ′ ) ⌣ A ( ψ ) .Let ϕ ∈ C { } ( C T [ G ]) . In order to define ( Bϕ ) ( x ,x ,x ,x ) we first observethat since the action of G on C is free, there exist s , s , s , s ∈ G which areunique such that x i = s i u i for i = 1 , , , . et g ⊗ g ⊗ g ∈ s u C s u ⊗ s u C s u ⊗ s u C s u . We define ( Bϕ ) ( x ,x ,x ,x ) as follows: ( Bϕ )( g ⊗ g ⊗ g ) = s ϕ ( u ,u ,u ,u ) ( s − g ⊗ s − g ⊗ s − g ) . In order to verify that this is well defined, note first that s − g ⊗ s − g ⊗ s − g ∈ u C s − s u ⊗ u C s − s u ⊗ u C s − s u = u C [ G ] s − s u ⊗ u C [ G ] s − s u ⊗ u C [ G ] s − s u . Secondly, using that ϕ is a cochain for the trivial conjugacy class, we obtain ϕ ( u ,u ,u ,u ) ( s − g , s − g , s − g ) ∈ u C [ G ] s − s s − s s − s u = u C [ G ] s − s u = u C s − s u . Hence ( Bϕ )( g ⊗ g ⊗ g ) ∈ su C su , therefore Bϕ ∈ C ( C ) . Next we checkthat Bϕ is an invariant cochain. Let t ∈ G , we assert that t ( Bϕ ) ( s u ,s u ,s u ,s u ) ( g ⊗ g ⊗ g ) = Bϕ ( ts u ,ts u ,ts u ,ts u ) ( tg ⊗ tg ⊗ tg ) . Indeed, the second term is by definition ts ϕ ( u ,u ,u ,u ) (cid:0) ( ts ) − tg ⊗ ( ts ) − tg ⊗ ( ts ) − tg (cid:1) , which equals the first term.Let ψ ∈ C ( C ) G , we assert that BAψ = ψ. Recall that if f ⊗ f ⊗ f ∈ u C [ G ] t u ⊗ u C [ G ] t u ⊗ u C [ G ] t u , then ( Aψ ) u ,u ,u ,u ( f ⊗ f ⊗ f ) = ψ ( f ⊗ t f ⊗ t t f ) . Let g ⊗ g ⊗ g ∈ s u C s u ⊗ s u C s u ⊗ s u C s u . Then
BAψ ( g ⊗ g ⊗ g ) = s Aψ ( s − g ⊗ s − g ⊗ s − g ) where s − g ⊗ s − g ⊗ s − g ∈ u C [ G ] s − s u ⊗ u C [ G ] s − s u ⊗ u C [ G ] s − s u . Hence
BAψ ( g ⊗ g ⊗ g ) = s ψ ( s − g ⊗ ( s − s ) s − ) g ⊗ ( s − s s − s ) s − g ) = s ψ ( s − g ⊗ s − g ⊗ s − g ) . Since ψ is invariant, the later equals ψ ( g ⊗ g ⊗ g ) . et ϕ ∈ C { } ( C T [ G ]) , next we will show ABϕ = ϕ. Consider g ⊗ g ⊗ g ∈ t u C t u ⊗ t u C t u ⊗ t u C t u . We have Bϕ ( g ⊗ g ⊗ g ) = t ϕ ( t − g ⊗ t − g ⊗ t − g ) . Let f ⊗ f ⊗ f ∈ u C [ G ] s u ⊗ u C [ G ] s u ⊗ u C [ G ] s u . Then
ABϕ ( f ⊗ f ⊗ f ) = ( Bϕ )( f ⊗ s f ⊗ s s f ) where f ⊗ s f ⊗ s s f ∈ u C s u ⊗ s u C s s u ⊗ s s u C s s s u Hence
ABϕ ( f ⊗ f ⊗ f ) = ϕ ( f ⊗ s − s f ⊗ ( s s ) − s s f ) = ϕ ( f ⊗ f ⊗ f ) . ⋄ General caseOur next aim is to show that the isomorphism of Theorem 4.4 remains validwhen the action of the group is not necessarily free. The following result hasbeen proved in [3, 17], see also [18].
Proposition 4.5
Let C and D be k -categories and let F : C → D be anequivalence of k -categories. There is an induced map C • F : C • ( D ) → C • ( C ) which is a quasi-isomorphism. Remark 4.6
In the following the explicit definition of C • ( F ) will be useful, itis as follows. Let ϕ = (cid:0) ϕ y n +1 ,...,y (cid:1) ∈ C n ( D ) where ϕ y n +1 ,...,y : y n +1 D y n ⊗ · · · ⊗ y D y −→ y n +1 D y is a k -morphism. The component ( x n +1 , · · · , x ) of ( C • F )( ϕ ) is given asfollows. Let f n +1 ⊗ · · · ⊗ f ∈ x n +1 C x n ⊗ · · · ⊗ x C x . Then [( C • F )( ϕ )] x n +1 ,...,x ( f n +1 ⊗ · · · ⊗ f ) =20 x n +1 F x ) − (cid:16) ϕ F ( xn +1) ,...,F ( x ( F ( f n +1 ) ⊗ · · · ⊗ F ( f )) (cid:17) where x n +1 F x : x n +1 C x → F ( x n +1 ) D F ( x ) is the k isomophism provided by the equivalence F. Observe that in [17] the above Proposition is obtained in a more generalsetting, that is for a D -bimodule of coefficients N . In our case N = D . Therestricted C -bimodule of coefficients is denoted F N in [17], observe that F D is isomorphic to C via F. This later isomorphism explains that in our setting x n +1 F − x is required in the above formula while in [17] it is not needed since thebimodule of coefficients there is F D , not C . Theorem 4.7
Let C and D be G - k -categories and let F : C → D be a G - k -equivalence of categories. Then F induces an isomorphism of G - k -algebras HH • ( D ) → HH • ( C ) . Proof.
The explicit description of C • F given above enables to check withoutdifficulty that it is multiplicative with respect to the cup product. Moreover, C • F commutes with the actions of G on C • C and C • D , that is C • F is a kG -morphism. Therefore the induced map in cohomology is an isomorphism of G - k -algebras. ⋄ We recall that if C is a G - k -category, then M G ( C ) is a G - k -category where theaction of G on the objects of M G ( C ) is free, see Definition 2.8. Moreover thereis a a G - k -functor L : M G ( C ) → C which is an equivalence of categories. Theorem 4.8
Let C be a G - k -category. Let C [ G ] be the graded skew category,and let { } be the trivial conjugacy class of G . There is an isomorphism of k -algebras HH ∗{ } ( C [ G ]) ≃ H ∗ ( C • ( C ) G ) . If the invariants functor is exact, we have an isomorphism of k -algebras HH ∗{ } ( C [ G ]) ≃ HH ∗ ( C ) G . Proof.
Let L [ G ] : M G ( C )[ G ] → C [ G ] be the homogeneous equivalence of G -graded k -categories obtained from the G - k -equivalence of categories L : M G ( C ) → C of Theorem 2.12.We observe that if B and D are G -graded categories and K : B → D is anhomogeneous equivalence, then the quasi-isomorphism C • ( K ) : C • ( B ) → C • ( D ) described in Remark 4.6 preserves the decomposition along the conjugacy classesof G. Hence HH ∗{ } ( B ) and HH ∗{ } ( D ) are isomorphic k -algebras. ⋄ .3 Galois coverings
In this Subsection we will translate the results we have obtained for the coho-mology of skew categories to a Galois coverings
C → C /G . Then we will providea canonical monomorphism from the invariants of the cohomology of C /G tothe cohomology of C . This corresponds to the monomorphism of [25]. In lowdegrees it is described in [16].The proof of the following result is along the same lines than the proof ofTheorem 3.20. Theorem 4.9
Let
C → C /G be a Galois covering. HH ∗{ } ( C /G ) = H ∗ ( C • ( C ) G ) . If the invariants functors is exact HH ∗{ } ( C /G ) = HH ∗ ( C ) G . Corollary 4.10
Let
C → C /G be a Galois covering. If the invariants functor isexact, there is a canonical injective morphism HH ∗ ( C ) G ֒ → HH ∗ ( C /G ) which splits canonically. Proof.
The cohomology of C /G has a direct sum decomposition along the con-jugacy classes of G . The direct summand corresponding to the trivial conjugacyclass is isomorphic to the invariants of the cohomology of G . ⋄ Hochschild cohomology of skew group algebras
In this section we will specialise Theorem 4.8 for k -algebras. Note that theproof of Theorem 4.8 requires the resolving category. We do not know a proofof Theorem 5.1 without using a resolving object which makes the action of thegroup free on a set. See Remark 5.2.Let Λ be a k -algebra, and let G be a group acting by algebra automorphismsof Λ . Let Λ[ G ] be the usual skew group algebra recalled in Remark 3.8. TheHochschild cohomology of a k -algebra Λ is denoted HH ∗ (Λ) . Theorem 5.1
Let G be a finite group whose order is invertible in k . Let Λ be a k -algebra with an action of G by algebra automorphisms. There is anisomorphism of algebras HH ∗{ } (Λ[ G ]) ≃ HH ∗ (Λ) G . roof. Let Λ be the single object G - k -category of Λ considered at Example2.3. As noticed, the action of G is not free on Λ unless G is trivial. ByTheorem 4.8 we have an isomorphism of k -algebras HH ∗{ } (Λ [ G ]) ≃ HH ∗ (Λ ) G . (5.1) Of course a (Λ ) = Λ . Moreover we have that HH ∗ ( C ) = HH ∗ ( a ( C )) , seefor instance [8]. Hence the right hand side of (5.1) is isomorphic to HH ∗ (Λ) G .On the other hand, as quoted in the Introduction, if G is finite and if C isa G - k -category with a finite number of objects, then a ( C [ G ]) = a ( C )[ G ] . Thus the left hand side of (5.1) is HH ∗{ } (Λ [ G ]) = HH ∗{ } ( a (Λ [ G ])) = HH ∗{ } ( a (Λ )[ G ]) = HH ∗{ } (Λ[ G ]) . ⋄ Remark 5.2
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C.C.:IMAG, Univ Montpellier, CNRS, Montpellier, FranceInstitut Montpelli´erain Alexander Grothendieck
[email protected] .N.M.:Departamento de Matem´atica, IME-USP,Rua do Mat˜ao 1010, cx pt 20570, S˜ao Paulo, Brasil. [email protected]@ime.usp.br