A remark on Hochschild cohomology and Koszul duality
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A remark on Hochschild cohomology and Koszul duality
Bernhard Keller
Dedicated to the Jos´e Antonio de la Pe˜na on the occasion of his sixtieth birthday
Abstract.
Applying recent results by Lowen–Van den Bergh we show that Hochschild coho-mology is preserved under Koszul–Moore duality as a Gerstenhaber algebra. More precisely, thecorresponding Hochschild complexes are linked by a quasi-isomorphism of B -algebras.
1. Introduction
Consider the following statement:Hochschild cohomology is preserved under Koszul duality.This is clearly wrong. Indeed, if V is a non zero finite-dimensional vector space over a field k , thenthe center (=zeroth Hochschild cohomology) of the symmetric algebra SV is SV but the centerof the Koszul dual exterior algebra Λ p V ˚ q is finite-dimensional (for a study of the Hochschildcohomology of Λ p V ˚ q , cf. [ ]). To try and save the statement, let us recall that Λ p V ˚ q is in factthe Yoneda algebra Ext ˚ SV p k, k q and that, with the zero differential, it is even quasi-isomorphic to the derived endomorphism algebra RHom SV p k, k q . Thus, it is natural to endow Λ p V ˚ q with the grading so that V ˚ sits in degree 1and with the zero differential. With this new interpretation of Λ p V ˚ q as a dg (=differential graded)algebra, we find that its zeroth Hochschild cohomology is the completion y SV of the symmetricalgebra at the augmentation ideal. It turns out that in order to get exactly SV , it suffices toreplace the dg algebra Λ p V ˚ q with the k -dual dg coalgebra Λ V . In fact, we then get an algebraisomorphism HH ˚ p SV, SV q „ ÝÑ HH ˚ p Λ V, Λ V q . Our main results are that this isomorphism generalizes from SV to any augmented dg k -algebra(when we replace Λ V with the Koszul–Moore dual dg coalgebra) and that it lifts to an isomorphismbetween the corresponding Hochschild cochain complexes in the homotopy category of B -algebras.In particular, the isomorphism in Hochschild cohomology is an isomorphism of Gerstenhaber alge-bras. The algebra isomorphism for augmented dg algebras follows from the setup of Koszul–Mooreduality, which we recall in section 2. Our first proof of the lift to the B -level is now supersededby recent work of Lowen–Van den Bergh [ ], which we use in the very short proof of Theorem 2.6.Previous results relating the Hochschild cohomologies of Koszul(-Moore) dual algebras can befound in [ ], [ ] [ ], [ ] and [ ] cf. remark 2.7. Mathematics Subject Classification.
Key words and phrases.
Hochschild cohomology, Koszul duality, Gerstenhaber algebra, B -algebra. c (cid:13) Acknowledgments
I am grateful to Jos´e Antonio de la Pe˜na for his friendship throughout the years and for theco-organization of many successful scientific events. I thank the organizers of the ARTA 7, wherethis material was first presented. I am indebted to Wendy Lowen for a talk at Trinity CollegeDublin in May 2019 on the results of [ ]. Many thanks to Vladimir Dotsenko for reminding me ofreferences [ ] and [ ] and to Pedro Tamaroff for pointing out [ ].
2. Reminder on Koszul–Moore duality
We follow Lef`evre–Hasegawa [ ] and Positselski [ ], cf. also [ ] and Appendix A to [ ]. Let k be a field and A a dg k -algebra. Thus, A is a Z -graded associative algebra with 1 A “ à p P Z A p endowed with a homogeneous linear endomorphism d of degree 1, the differential, such that d “ d p ab q “ p da q b ` p´ q p a p db q for all a P A p and all b P A . Let ε : A Ñ k be an augmentation (a morphism of dg algebras). Forexample, if V is a vector space (concentrated in degree 0), we can consider A “ SV (concentratedin degree 0 with d “ C be a dg coalgebra and ε : k Ñ C a co-augmentation. Forexample, we may consider C “ Λ V , where V is concentrated in degree ´ d “
0. Denote by
Hom k p C, A q the graded vector space whose n th component is formed by the homogeneous k -linearmaps f : C Ñ A of degree n . We make Hom k p C, A q into a differential graded algebra by setting d p f q “ d ˝ f ´ p´ q n f ˝ d for f homogeneous of degree n and f ˚ g “ µ ˝ p f b g q ˝ ∆for homogeneous f and g , where µ is the multiplication of A and ∆ the comultiplication of C . A twisting cochain is an element τ P Hom k p C, A q of degree 1 such that ε ˝ τ “ “ τ ˝ η and d p τ q ` τ ˚ τ “ . For example, with the above notations, the composition of the natural projection and inclusionmorphisms Λ V Ñ V Ñ SV is a twisting cochain. We denote by Tw p C, A q the set of twisting cochains. From now on, we assumethat C is cocomplete , i.e. that C “ cok p η q is the union of the kernels of the maps induced by theiterated comultiplications ∆ p n q : C Ñ C b n , n ě . We denote by
Alg the category of augmented dg algebras and by
Coalg the category of cocompleteco-augmented dg coalgebras.
Proposition . a) The functor Tw p ? , A q : Coalg op Ñ Set is representable, i.e. thereis an object BA P Coalg and a functorial bijection Tw p C , A q „ ÝÑ Coalg p C , BA q . b) The functor Tw p C, ? q : Alg Ñ Set is co-representable, i.e. there is an object Ω A P Alg anda functorial bijection Tw p C, A q „ ÝÑ Alg p Ω C, A q . The dg coalgebra BA is known as the bar construction and the dg algebra Ω C as the cobarconstruction . It is not hard to describe BA and Ω C explicitly but we will not need this. Noticethat if we denote by DBA the k -dual dg algebra of BA , then we have a canonical isomorphism DBA „ ÝÑ RHom A p k, k q REMARK ON HOCHSCHILD COHOMOLOGY AND KOSZUL DUALITY 3 and that the latter is quasi-isomorphic to the Koszul dual A ! (with the generators in differentialdegree 1) if A is a Koszul algebra concentrated in degree 0. We denote the category of dg right A -modules by Mod A . Its localization with respect to all quasi-isomorphisms is the derived category D A . A (right) dg comodule M is cocomplete if it is the union of the kernels of the maps M Ñ M b C n ´ , n ě , induced by the iterated comultiplications. We denote by Com C the category of cocomplete rightdg C -comodules. It becomes a Frobenius exact category when endowed with the conflations givenby the exact sequences whose underlying sequences of graded comodules split. The category up tohomotopy is the associated stable category. We denote by by D C the co-derived category , i.e. thelocalization of Com C at the class of all co-quasi-isomorphisms . Here, a morphism s : L Ñ M of dgcomodules is a co-quasi-isomorphism if its cone lies in the smallest triangulated subcategory of thecategory up to homotopy of dg comodules stable under coproducts and containing all totalizationsof short exact sequences 0 Ñ X Ñ Y Ñ Z Ñ ]. One can also characterize the co-quasi-isomorphisms using the cobar construction for dg comodules, cf. [ ] and below. For the comparisonbetween the two, cf. Appendix A of [ ].Let us fix a twisting cochain τ : C Ñ A . For M P Mod A , let M b τ C be the graded comodule M b C endowed with the differential d b ` b d ` d τ , where d τ “ p µ b q ˝ p b τ b q ˝ p b ∆ q . For L P Com C , let L b τ A be the graded A -module L b A endowed with the differential d b ` b d ` d τ , where d τ “ p b µ q ˝ p b τ b q ˝ p ∆ b q . Proposition . The pair p ? b τ A, ? b τ C q is a pair of adjoint functors between Com C and Mod A . It induces a pair of adjoint functors between D C and D A . We define the twisting cochain τ : C Ñ A to be acyclic if the associated adjoint functorsbetween D C and D A are equivalences. In this case, we say that A and C are Koszul–Moore dual to each other.
Theorem ]) . The following are equivalent: i) τ is acyclic; ii) τ induces a quasi-isomorphism Ω C Ñ A ; iii) τ induces a weak equivalence C Ñ BA ( i.e. the induced morphism Ω C Ñ Ω BA is aquasi-isomorphism); iv) The natural morphism A b τ C b τ A Ñ A is a quasi-isomorphism. For example, it follows from part iv) that SV and Λ V (with the above notations) are Koszul–Moore dual to each other. By part iii), we have Koszul–Moore duality between A and BA andtherefore a triangle equivalence D p BA q „ ÝÑ D A. It is remarkable that the dg coalgebra BA determines all of D A whereas the dual dg algebra DBA „ ÝÑ RHom A p k, k q a priori only determines the thick subcategory of D A generated by k .Assume that τ : C Ñ A is an acylic cochain. Put A e “ A b A op and C e “ C b C op and let τ e “ τ b η ` η b τ : C e Ñ A e . Clearly τ e is a twisting cochain. Proposition . The twisting cochain τ e is acyclic and the induced equivalence D p C e q „ ÝÑ D p A e q takes the dg bicomodule C to the dg bimodule A . Thus we have an induced isomorphism of gradedalgebras HH ˚ p C q “ Ext ˚ C e p C, C q „ ÝÑ Ext ˚ A e p A, A q “ HH ˚ p A q . BERNHARD KELLER
Remark . We see that Koszul–Moore duality preserves Hochschild cohomology as a gradedalgebra.
Proof.
Let ϕ A : A b τ C b τ A Ñ A be the canonical morphism. Then the canonical morphism ϕ A e : A e b τ e C e b τ e A e Ñ A e is the composition of the isomorphism p A b A op q b τ e p C b C op q b τ e p A b A op q „ ÝÑ p A b τ C b τ A q b p A op b τ C op b τ A op q with the quasi-isomorphism ϕ A b ϕ A op : p A b τ C b τ A q b p A op b τ C op b τ A op q Ñ A b A op . Thus τ e is an ayclic twisting cochain by Theorem 2.3. The induced equivalence takes C to C b τ e p A b A op q “ A b τ C b τ A which is quasi-isomorphic to A since τ is acyclic. ‘ Theorem . The isomorphism of the proposition lifts to an isomorphism in the homotopycategory of B -algebras between the corresponding Hochschild cochain complexes. In particular, itpreserves the Gerstenhaber brackets. Remark . In [ ] , Buchweitz related the Hochschild cohomology algebras of a Koszul algebraand its Koszul dual. In Theorem 3.5 of [ ] , we showed that for a Koszul algebra A (concentratedin differential degree ), there is a canonical isomorphism in the category of Adams graded B -algebras between the Hochschild complex of A and that of the Koszul dual algebra A ! whose p thgraded piece is put into bidegree p p, ´ p q , where the first component is the differential degree and thesecond component the Adams degree. Here, we get rid of the Koszulity assumption and the Adamsgrading by using dg coalgebras.In [ ] , the F´elix–Menichi–Thomas show that for a simply connected coalgebra C , the Hochschildcohomologies of the dg algebras Ω C and Hom k p C, k q are isomorphic as Gerstenhaber algebras. An-other proof of this, under less stringent connectedness assumptions, is given in section 3 of Briggs–G´elinas’ [ ] . For Koszul A –algebras, an isomorphism of the Hochschild cohomologies as weightgraded A -algebras (but not as Gerstenhaber algebras) is proved by Berglund–B¨orjeson in Theo-rem 3.2 of [ ] . Proof.
The twisting cochain induces a weak equivalence BA Ñ C . By the argument of[ ], this yields an isomorphism in the homotopy category of B -algebras between the Hochschildcomplexes of BA and C (these are sometimes called coHochschild complexes). Thus, we mayassume that C “ BA and τ is the canonical twisting cochain. Let r BA “ A b τ C b τ A . Then r BA has a natural structure of dg coalgebra in the category Mod p A e q of dg A - A -bimodules endowedwith b A . We have a lax monoidal functor F : Mod p A e q Ñ C p k q taking a bimodule M to k b A M b A k . The lax structure is given by the morphism F p L b A M q “ k b A p L b A M qb A k “ k b A p L b A A b A M qb A k Ñ k b A L b A k b A M b A k “ p F L qb k p F M q . The functor F sends the A e -coalgebra r BA to the k -coalgebra BA “ C . It induces a morphismfrom the Hochschild complex of r BA to that of C and this is easily seen to be a quasi-isomorphismcompatible with the cup product and the brace operations. The claim follows by Theorem 5.1 of[ ] which states that there is a canonical isomorphism in the homotopy category of B -algebrasbetween the Hochschild complex of r BA and the Hochschild complex of A . It is not hard to checkthat in homology, it induces the isomorphism of Propostion 2.4. ‘ Example . Suppose that k is of characteristic and g a finite-dimensional Lie algebra over k . Let U g be its enveloping algebra and Λ g the supersymmetric coalgebra on g placed in degree ´ and endowed with the coalgebra differential whose p´ q -component is the bracket Λ g Ñ g . Thus,the underlying complex of Λ g is the Chevalley–Eilenberg complex of g . Then the map τ : Λ g Ñ U g REMARK ON HOCHSCHILD COHOMOLOGY AND KOSZUL DUALITY 5 which is the composition of the projection Λ g Ñ g with the inclusion g Ñ U g is an acyclic twistingcochain and we obtain an isomorphism of Gerstenhaber algebras HH ˚ p Λ g q „ ÝÑ HH ˚ p U g q . It would be interesting to generalize this example to Lie algebroids.
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B. Keller: Universit´e Paris Diderot – Paris 7, UFR de Math´ematiques, CNRS, Institut de Math´ematiquesde Jussieu–Paris Rive Gauche, IMJ-PRG, Bˆatiment Sophie Germain, 75205 Paris Cedex 13, France
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