Hochschild-Mitchell (co)homology of skew categories and of Galois coverings
aa r X i v : . [ m a t h . K T ] A p r Hochschild-Mitchell (co)homology of skew categoriesand of Galois coverings
Claude Cibils and Eduardo N. Marcos ∗ Abstract
Let C be category over a commutative ring k , its Hochschild-Mitchellhomology and cohomology are denoted respectively HH ∗ ( C ) and HH ∗ ( C ) . Let G be a group acting on C , and C [ G ] be the skew category. We providedecompositions of the (co)homology of C [ G ] along the conjugacy classesof G . For Hochschild homology of a k -algebra, this corresponds to thedecomposition obtained by M. Lorenz.If the coinvariants and invariants functors are exact, we obtain isomor-phisms ( HH ∗ ( C )) G ≃ HH { }∗ ( C [ G ]) and ( HH ∗ ( C )) G ≃ HH ∗{ } ( C [ G ]) , where { } is the trivial conjugacy class of G .We first obtain these isomorphisms in case the action of G is free onthe objects of C . Then we introduce an auxiliary category M G ( C ) withan action of G which is free on its objects, related to the infinite matrixalgebra considered by J. Cornick. This category enables us to show thatthe isomorphisms hold in general, and in particular for the Hochschild(co)homology of a k -algebra with an action of G by automorphisms.We infer that ( HH ∗ ( C )) G is a canonical direct summand of HH ∗ ( C [ G ]) .This provides a frame for monomorphisms obtained previously, and whichhave been described in low degrees. Introduction
Let k be a commutative ring. A k -category C is a small category enhanced overthe category of k -modules. In other words the objects of C are a set denoted C , for any pair of objects x, y ∈ C the set of morphisms from x to y is denoted y C x and has a k -module structure, the composition in C is k -bilinear, and theimage of the canonical inclusion k ֒ → x C x is central in x C x for all x ∈ C . Notethat in particular x C x is a k -algebra for any x ∈ C .Mitchell in [2] called those categories “algebras with several objects”. In-deed, to a k -algebra Λ we associate a k -category with a single object x with ∗ 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 C x = Λ . Moreover a k -category C with a finite set of objets provides a k -algebra a ( C ) = ⊕ x,y ∈C y C x with product given by the composition of C com-bined with the matrix multiplication. However we point out that if C and D are finite object k -categories, a k -functor F : C → D provides a multiplicative k -morphism a ( F ) : a ( C ) → a ( D ) , but in general a ( F )(1) = 1 . In particularthe Galois coverings that we consider in this work are functors which are notalgebra morphisms.Hochschild-Mitchell homology and cohomology of a k -category C are theo-ries introduced by Mitchell [2], see also [8, 14, 21, 24]. When the number ofobjects of C is finite, they coincide with Hochschild cohomology and homologyof 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 , that is with a group homomorphism G → Aut k C where Aut k C is the groupof k -functors C → C which are isomorphisms. In particular a G - k -algebra Λ isa k -algebra with an action of G by k -automorphisms of Λ .Let C be a G - k -category. The skew category C [ G ] has been considered in[7]. It has the same set of objects than C while the morphism between twoobjects x, y ∈ C are direct sums, y C [ G ] x = ⊕ s ∈ G y C sx , with composition givenby ( z g ty )( y f sx ) = g ◦ tf . If the G - k -category C has a finite number of objectsand if G is finite, then a ( C )[ G ] = a ( C [ G ]) , where Λ[ G ] denotes the usual skewalgebra of a G - k -algebra Λ .The action of G on C is called free if for x ∈ C , the equality sx = x onlyholds for s = 1 . In this case the quotient k -category C /G exists, see for instance[8], and C → C /G is a Galois covering. This construction has several uses inrepresentation theory, see for instance [4, 28, 13, 1, 20].A main result obtained in [7] is the following: if the action of G is free onthe objects of C , then C /G and C [ G ] are equivalent k -categories. Consequently,if the action is not free on the objects, the skew category is a substitute to thequotient category.In this work we consider a G - k -category C and we relate the Hochschild-Mitchell (co)homology of C and of C /G , or of C [ G ] .Hochschild-Mitchell (co)homology theories of k -categories can also be de-fined with bimodules of coefficients. We underline that the comparison resultsthat we obtain do not rely on a change of the bimodules of coefficients. Moreprecisely the (co)homology of k -categories that we consider in this paper are al-ways with coefficients in themselves. In contrast the Cartan-Leray type spectralsequence obtained in [8] makes use of an adequate change in the bimodule ofcoefficients, see also [18] and [26].If Λ is a G -graded algebra its Hochschild homology decomposes HH ∗ (Λ) = M D ∈ Cl ( G ) HH D ∗ (Λ) where Cl ( G ) is the set of conjugacy classes of G , see [10, 23, 30]. This decom-position also holds for the Hochschild-Mitchell homology of a G -graded category , for instance for C [ G ] where C is a G - k -category. If the action of G is freeon C , and if the coinvariants functor ( ) G is exact, we prove that there is anisomorphism HH { }∗ ( C [ G ]) = ( HH ∗ ( C )) G , where { } is the trivial conjugacy class. If ( ) G is not exact, then the ad-hocspectral sequence can be settled.In order to extend the above result to a non free action, we introduce anauxiliary G - k -category M G ( C ) . It has a natural free action of G on its objects,and there is a G - k -equivalence of categories M G ( C ) → C . The category M G ( C ) is related to the infinite matrix algebra considered by J. Cornick in [10] for agraded algebra, which in turn is linked with Cohen-Montgomery duality in [6],see also [9].In [17] E. Herscovich proved that equivalent k -categories have isomorphicHochschild-Mitchell co(homologies), see also [3], note that if the k -categoriesare isomorphic the result is obvious. Beyond, in [19] it is proven that Hochschild-Mitchell (co)homology is a derived invariant. In case of equivalent k -categories,the quasi-isomorphism is induced by the given equivalence, hence from the G - k -equivalence M G ( C ) → C we infer a kG -isomorphism in homology. Thisenables us to prove that the above isomorphism also holds for Hochschild-Mitchell homology, without assuming freeness of the action on the objects.If the coinvariants functor ( ) G is exact, the invariance of the Hochschild-Mitchell homology for equivalence of k -categories implies that for a Galois cov-ering C → C /G the following holds: HH { }∗ ( C /G ) = ( HH ∗ ( C )) G . Hochschild-Mitchell cohomology is defined via a complex of cochains whichin each degree is the direct product of k -modules, in contrast with Hochschild-Mitchell homology where the chains are direct sums. Nevertheless we first showthat for a G - k -category, the complex is a direct product along the conjugacyclasses of G . Moreover the complex of cochains is a differential graded alge-bra for the cup product. The subcomplex associated to the trivial conjugacyclass is a sub differential graded algebra, the corresponding Hochschild-Mitchellcohomology is denoted HH ∗{ } ( B ) and is a subalgebra of HH ∗ ( B ) . For a G - k -category, the group G acts on the complex of cochains C ∗ ( C ) byautomorphisms of the differential graded algebra, hence HH ∗ ( C ) is a G -algebra.In case the action of G on C is free and the invariants functor ( ) G is exact,we show that there is an isomorphism of algebras ( HH ∗ ( C )) G ≃ HH ∗{ } ( C [ G ]) . Using the isomorphism on Hochschild-Mitchell cohomology between equiv-alent k -categories we infer ( HH ( C )) G ≃ HH ∗{ } ( C /G ) , ence ( HH ∗ ( C )) G is a canonical direct summand of HH ∗ ( C /G ) . This way werecover the monomorphism obtained in [25] which is made explicit in low degreesin [16].Through the same mentioned auxiliary category, we extend the result in casethe action of G on C is not necessarily free. We establish that if the invariantsfunctor is exact, the above isomorphism of algebras holds in general. If theinvariants functor is not exact a spectral sequence can be considered instead.In particular, for Λ a G - k -algebra, if the invariants functor is exact, there isan algebra isomorphism ( HH (Λ)) G ≃ HH ∗{ } (Λ[ G ]) . In other words ( HH (Λ)) G is a subalgebra of HH ∗ (Λ[ G ]) , and there is a canon-ical two-sided ideal complementing it, provided by the non trivial conjugacyclasses of G . This result is related with explicit computations made for instancein [12, 15, 29, 27] for Hochschild (co)homology of specific skew group algebras,in particular for the symmetric algebra over a finite dimensional vector space V over a field k , with G a finite subgroup of GL ( V ) which order is invertible in k. Graded and skew categories
Let k be a commutative ring. A k -category is a small category C enriched overthe category of k -modules, its set of objects is denoted C . For x, y ∈ C , weusually write y f x for an element f of the k -module of morphisms y C x from x to y .Let G be a group. A G - k -category is a k -category C with an action of G by k -isomorphisms of C . That is firstly G acts on C and secondly for s ∈ G and y f x ∈ y C x there is sy ( sf ) sx ∈ sy C sx such that the map given by y C x → sy C sx , f sf is a k -module morphism, and t ( sf ) = ( ts ) f for s, t ∈ G , as well as s ( y x ) = sy sx for s ∈ G and x ∈ C . Observe that if C is a single object G - k -category, then its k -algebra ofendomorphisms has an action of G by automorphisms of the algebra. Definition 2.1 [7] Let C be a G - k -category. The skew category C [ G ] has thesame set of objects than C . Let y C [ G ] sx = y C sx . The morphisms of C [ G ] from x to y are y C [ G ] x = M s ∈ G y C [ G ] sx . (2.1) 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 . emark 2.2
1. By definition, the direct summands of (2.1) are in one to one correspon-dence with elements of G .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, i.e. Λ[ G ] = Λ ⊗ kG where Λ and kG are subalgebras, and with product determined by the intertwining kG ⊗ Λ → Λ ⊗ kG given by s ⊗ a sa ⊗ s .3. Observe that z C [ G ] ty y C [ G ] sx ⊂ z C [ G ] tsx . (2.2) Definition 2.3
Let G be a group. A k -category B is G -graded if for all x, y ∈B there is a direct sum decomposition by means of k -modules 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 . A morphism y f sx ∈ y C sx = y C [ G ] sx is denoted by y [ f ] sx . Recall that in C [ G ] z [ g ] ty y [ f ] sx = z [( g ) ◦ ( tf )] tsx , (2.3) that is the k -category C [ G ] is G -graded, as observed in (2.2). Homology for free actions and for Galois coverings
We recall the definition of Hochschild-Mitchell homology, see for instance [2,21, 24, 8].
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, 32, 5].The Hochschild-Mitchell homology of C is the homology of the complexabove, that is HH ∗ ( C ) = H ∗ ( C • ( C )) . ext we decompose the above chain complex for a G -graded k -category, asLorenz [23] have done for the complex which computes the Hochschild homologyof a G -graded k -algebra (see also [31]). Proposition 3.2
Let B be a G -graded k -category. Let D be a conjugacy classof G. Then 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 is a subcomplex of C • ( B ) , and there is a decomposition C • ( B ) = M D ∈ Cl G C D • ( B ) where Cl G denotes the set of conjugacy classes of G. Proof.
The following is a verification for n = 2 which provides the wayfor proving the result for any n . It has the advantage of avoiding long anduseless technical computations. For subsequent proofs we will often maintainthis approach of providing the computations for small values of n. 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. ⋄ Corollary 3.3
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 ) . For a G - k -category C our aim is to compare HH ∗ ( C ) and HH ∗ ( C [ G ]) . We firstobserve the following fact.
Proposition 3.4
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 ) . or 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 3.5
Let C be a G - k -category. The Hochschild-Mitchell homology HH ∗ ( C ) is a kG -module. Recall that if M is a kG -module, its kG -module of coinvariants is M G = M/
Let C be a G - k -category and suppose that the action of G on C is free. Let C [ G ] be the skew category, considered with its G -grading (see2.3). Let { } be the trivial conjugacy class of G . There is an isomorphism HH { }∗ ( C [ G ]) ≃ H ∗ (( C • ( C ) G ) . If the coinvariant functor is exact then HH { }∗ ( C [ G ]) ≃ ( H ∗ ( C )) G . Remark 3.8
If the covariants functor is not exact, it follows that there is aspectral sequence for computing HH { }∗ ( C [ G ]) . Before proving the Theorem we provide some properties of the skew categorythat we will need. emma 3.9 Let C be a k − G -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 3.10
Let G be a group acting on a set E. A transversal T of theaction is a subset of E obtained by choosing precisely one element in each orbitof the action. Observe that equivalently T ⊂ E is a transversal if for each x ∈ E there existsa unique u ( x ) ∈ T such that x = su ( x ) for some s ∈ G . Note that the actionis free if for each x ∈ E , there is only one s ∈ G such that x = su ( x ) . Lemma 3.11
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, 19] (see also [3]) it is proven that if C and D are k -categoriesand 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 F is homogeneous,then the induced quasi-isomorphism clearly preserves the decomposition alongconjugacy classes of G . Since T is a transversal, the above Lemma 3.9 showsthat the inclusion functor C T [ G ] ⊂ C [ G ] is dense. Moreover it corresponds to afull subcategory, hence it is full and faithful, in addition of being homogeneous.The induced quasi-isomorphism provides then the result. ⋄ Proof of Theorem 3.7.
Let T be a transversal of the free action of G on C . Inorder 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 ) . Up to the action, that isin ( C ( C )) G , we begin by modifying the chain in order that the starting (andhence 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 . ⋄ We end this section by restating the Theorem above in terms of Galoiscoverings. We recall first the definition of a quotient category.
Definition 3.12 (see [4, 28]) Let C be a G - k -category such that the action of G on C is free. The quotient category C /G has set of objects the set of orbits C /G. Let α and β be orbits. The k -module 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 3.13
A Galois covering of k -categories is a functor C → C /G, where C is a G - k -category with a free action of G on the objects and where the functoris the canonical projection 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 3.14
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. ⋄ Corollary 3.15
Let
C → C /G be a Galois covering. HH { }∗ ( C /G ) = H ∗ ( C • ( C ) G ) . If the coinvariant functors is exact HH { }∗ ( C /G ) = ( HH ∗ ( C )) G . Homology for skew categories
In this section we construct an auxiliary G - k -category affording a free action of G on its objects, in order to prove Theorem 3.7 without the assumption thatthe action of G is free on the objects. Definition 4.1 (see also [10], [11]) Let C be a G - k -category. The objects ofthe k -category M G ( C ) are G × C . The k -module of morphisms of 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,ty ) ( M G ( C )) ( rs,rx ) . Observe that this action is free on the objects of M G ( C ) . Remark 4.2
Let Λ be a G - k -algebra, considered as a single object G - k -category.The category M G (Λ) is the so-called Λ -complete category over G : its set ofobjects is G and each k -module of morphisms is a copy of Λ . The compositionis given by the product in the algebra. The action of G on the objects is theproduct of G, while on morphism it is given by the G -action on Λ combinedwith the action on the objects.If G is finite, the k -algebra a ( M G (Λ)) is a matrix algebra over Λ withcolumns and rows indexed by G. Observe that the action of s ∈ G on a givenmatrix is the action of s on the columns and the rows, combined with the actionof s on the entries of the matrix. We recall next the definition of the tensor product
C ⊗ D of k -categories C and D . The objects are C × D , while the morphisms are given by: ( c ′ ,d ′ ) ( C ⊗ D ) ( c,d ) = c ′ C c ⊗ d ′ D d with the obvious composition. Moreover, if C and D are G - k -categories then C ⊗ D is a G - k -category through the diagonal action of G .Hence M G ( C ) = M G ( k ) ⊗ C , where k is the single objet G - k -category, the object has endomorphisms k , andthe action of G is trivial. Lemma 4.3
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. ⋄ Proposition 4.4
Let C be a G - k -category. The chain complexes C • ( M G ( C )) and C • ( C ) are kG -quasi-isomorphic. Proof.
As before we use the result in [17]. The above equivalence of categories L : M G ( C ) → C induces a quasi-isomorphism C • ( M G ( C )) → C • ( C ) . Since L is a G - k -functor, the quasi-isomorphism is a kG -module chain map, which providesa kG -isomophism in homology. ⋄ Theorem 4.5
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 3.7 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 ]) which preserves the decomposition of chain complexes along the conjugacyclasses 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 . ⋄ Cohomology
Definition 5.1
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 homology, see for instance [5, 32].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 . (5.1) 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. Proposition 5.2
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. roof. 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 (5.1) 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.Let C be a G - k -category. We assert that 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 e 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 5.3
Let ( ) G be the invariant functor. Then ϕ ∈ ( C n ( C )) G if and onlyif 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, in other words HH ∗ ( C ) is a G - k -algebra. If the invariants functor is exact, then ( HH ∗ ( C )) G = H ∗ ( C • ( C ) G ) as k -algebras. Theorem 5.4
Let C be a G - k -category with free action of G on the objects, andlet 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 invariant functor ( ) G is exact, we infer an isomorphism HH ∗{ } ( C [ G ]) ≃ ( HH ∗ ( C )) G of k -algebras. Remark 5.5
1. Based on this Theorem, we will prove that the result also holds if theaction of G on C is not free.2. If the invariant functor is not exact, the standard considerations providea spectral 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. oreover 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 , , , .Let 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 . econdly, 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 ) . Let 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 )18 here 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 ) . ⋄ Our next aim is to show that the isomorphism of Theorem 5.4 remains validwhen the action of the group is not necessarily free on the set of objects of the k -category. The following result have been proved in [3, 17], see also [ ? ]. Proposition 5.6
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 5.7
For future use we make precise the definition of C • ( F ) . 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 ) =( 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 . heorem 5.8 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 easilythat it is multiplicative with respect to the cup product. Moreover, C • F com-mutes 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 4.1. Moreover thereis a a G - k -functor L : M G ( C ) → C which is an equivalence of categories. Theorem 5.9
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 invariant functor ( ) G 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 .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 5.7 preserves the decomposition along the conjugacy classesof G. Hence HH ∗{ } ( B ) and HH ∗{ } ( D ) are isomorphic k -algebras. Remark 5.10
Let Λ be a G - k -algebra, considered as a single object G - k -category with endomorphism algebra Λ . The action of G on the object is trivial,which is not free unless G is trivial. Let k be a field and G be a finite groupwhose order is invertible in k , the invariants functor is exact and the previousTheorem provides the isomorphism HH ∗{ } (Λ[ G ]) ≃ ( HH ∗ (Λ)) G . We review now the proof that we have provided, specified to the above situation.We have considered the matrix G -algebra M G (Λ) , where the action of G is on he indices of the rows and of the columns, and on Λ . Note that the action of G on the set of | G | idempotents of the diagonal is free.The track of the previous categorical proof translates into first decomposingthe cochains of M G (Λ) through the mentioned diagonal idempotents. Thenthe freeness of the action on this set enables to show that the invariants ofthe complex of cochains of M G (Λ) and the homogeneous cochains of class1 of M G (Λ)[ G ] are isomorphic. The final step consists in showing that theHochschild cohomology, as a kG module, remains the same when consideringthe algebra M G (Λ)[ G ] . References [1] Assem, I.; Bustamante, J. C.; Le Meur, P.; Coverings of Laura Algebras:the Standard Case. J. Algebra 323 (2010) 83–120.[2] Bautista, R.; Gabriel, P.; Ro˘ıter, A. V.; Salmer´on, L.; Representation-finitealgebras and multiplicative bases. Invent. Math. 81 (1985), 217–285.[3] Baues H.J.; Wirsching G.; Cohomology of small categories. J. Pure Appl.Algebra 38 (1985), 187–211.[4] Bongartz; K., Gabriel, P.; Covering spaces in representation-theory. Invent.Math. 65 (1982), 331–378.[5] Cartan, H.; Eilenberg S.; Homological Algebra, Princeton, New Jersey,Princeton University press, 1958.[6] Cohen, M.; Montgomery, S.; Group-graded rings, smash products, andgroup actions. Trans. Am. Math. Soc. 282 (1984), 237–258.[7] Cibils, C.; Marcos, E. N.; Skew category, Galois covering and smash productof a k-category. Proc. Amer. Math. Soc. 134 (2006), 39–50.[8] Cibils, C.; Redondo, M. J.; Cartan-Leray spectral sequence for Galois cov-erings of linear categories. J. Algebra 284 (2005), 310–325.[9] Cibils, C.; Solotar, A.; The fundamental group of a Hopf linear category.J. Algebra 462 (2016), 137–162.[10] Cornick, J.; Homological techniques for strongly graded rings: a survey. Ge-ometry and cohomology in group theory (Durham, 1994), 88–107, LondonMath. Soc. Lecture Note Ser., 252, Cambridge Univ. Press, Cambridge,1998.[11] Corti˜nas, G.; Ellis, E.; Isomorphism conjectures with proper coefficients. J.Pure Appl. Algebra 218 (2014), 1224–1263.
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C.C.:IMAG, Univ Montpellier, CNRS, Montpellier, FranceInstitut Montpelli´erain Alexander Grothendieck
E.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