Derived invariance of the Tamarkin--Tsygan calculus of an algebra
DDERIVED INVARIANCE OF THE TAMARKIN–TSYGANCALCULUS OF AN ALGEBRA
MARCO ANTONIO ARMENTA AND BERNHARD KELLER
Abstract.
We prove that derived equivalent algebras have isomorphicdifferential calculi in the sense of Tamarkin–Tsygan. Introduction
Let k be a commutative ring and A an associative k -algebra projective asa module over k . We write ⊗ for the tensor product over k . We point outthat all the constructions and proofs of this paper extend to small dg cate-gories cofibrant over k . The Hochschild homology HH • ( A ) and cohomology HH • ( A ) are derived invariants of A , see [3, 4, 9, 10, 12]. Moreover, these k -modules come with operations, namely the cup product ∪ : HH n ( A ) ⊗ HH m ( A ) → HH n + m ( A ) , the Gerstenhaber bracket[ − , − ] : HH n ( A ) ⊗ HH m ( A ) → HH n + m − ( A ) , the cap product ∩ : HH n ( A ) ⊗ HH m ( A ) → HH n − m ( A )and Connes’ differential B : HH n ( A ) → HH n +1 ( A ) , such that B = 0 and(1) [ Bi α − ( − | α | i α B, i β ] = i [ α,β ] , where i α ( z ) = ( − | α || z | z ∩ α . This is the first example [2, 11] of a differentialcalculus or a Tamarkin-Tsygan calculus , which is by definition a collection( H • , ∪ , [ − , − ] , H • , ∩ , B ) , such that ( H • , ∪ , [ − , − ]) is a Gerstenhaber algebra, the cap product ∩ en-dows H • with the structure of a graded Lie module over the Lie algebra( H • [1] , ∪ , [ − , − ]) and the map B : H n → H n +1 squares to zero and satisfiesthe equation (1). The Gerstenhaber algebra ( HH • ( A ) , ∪ , [ − , − ]) has beenproved to be a derived invariant [8, 6]. The cap product is also a derivedinvariant [1]. In this work, we use an isomorphism induced from the cyclic Key words and phrases.
Derived category, Hochschild homology, Connes differential. a r X i v : . [ m a t h . K T ] N ov MARCO ANTONIO ARMENTA AND BERNHARD KELLER functor [7] to prove derived invariance of Connes’ differential and of the ISB-sequence. To obtain derived invariance of the differential calculus, we needto prove that this isomorphism equals the isomorphism between Hochschildhomologies used in [1] to prove derived invariance of the cap product.2.
The cyclic functor
Let
Alg be the category whose objects are the associative dg (=differentialgraded) k -algebras cofibrant over k (i.e. ‘closed’ in the sense of section 7.5of [7]) and whose morphisms are morphisms of dg k -algebras which do notnecessarily preserve the unit. Let rep( A, B ) be the full subcategory of thederived category D ( A op ⊗ B ) whose objects are the dg bimodules X suchthat the restriction X B is compact in D ( B ), i.e. lies in the thick subcategorygenerated by the free module B B . Define ALG to be the category whoseobjects are those of
Alg and whose morphisms from A to B are the isomor-phism classes in rep( A, B ). The composition of morphisms in
ALG is givenby the total derived tensor product [7]. The identity of A is the isomorphismclass of the bimodule A A A . There is a canonical functor Alg → ALG thatassociates to a morphism f : A → B the bimodule f B B with underlyingspace f (1) B and A - B -action given by a.f (1) b.b (cid:48) = f ( a ) bb (cid:48) .Let Λ be the dg algebra k [ (cid:15) ] / ( (cid:15) ) where | (cid:15) | = − C : Alg → DMix be the cyclic functor [7], that is, theunderlying dg k -module of C ( A ) is the mapping cone over (1 − t ) viewed asa morphism of complexes ( A ⊗∗ +1 , b (cid:48) ) → ( A ⊗∗ , b ) and the first and seconddifferentials of the mixed complex C ( A ) are (cid:20) b − t − b (cid:48) (cid:21) and (cid:20) N (cid:21) . Clearly, a dg algebra morphism f : A → B (even if it does not preserve theunit) induces a morphism C ( f ) : C ( A ) → C ( B ) of dg Λ-modules. Let X be an object of rep( A, B ). We assume, as we may, that X is cofibrant (i.e.‘closed’ in the sense of section 7.5 of [7]). This implies that X B is cofibrantas a dg B -module and thus that morphism spaces in the derived categorywith source X B are isomorphic to the corresponding morphism spaces in thehomotopy category. Consider the morphisms A α X (cid:47) (cid:47) End B ( B ⊕ X ) B β X (cid:111) (cid:111) where End B ( B ⊕ X ) is the differential graded endomorphism algebra of B ⊕ X , the morphism α X be given by the left action of A on X and β X is ERIVED INVARIANCE OF THE TAMARKIN–TSYGAN CALCULUS 3 induced by the left action of B on B . Note that these morphisms do notpreserve the units. The second author proved in [7] that C ( β X ) is invertiblein DMix and defined C ( X ) = C ( β X ) − ◦ C ( α X ). We recall that C is welldefined on ALG and that this extension of C from Alg to ALG is uniqueby Theorem 2.4 of [7].Let X : A → B be a morphism of ALG where X is cofibrant. Put X ∨ = Hom B ( X, B ). We can choose morphisms u X : A → X L ⊗ B X ∨ and v X : X ∨ L ⊗ B X → B such that the following triangles commute X u X ⊗ (cid:47) (cid:47) = (cid:38) (cid:38) X L ⊗ B X ∨ L ⊗ A X ⊗ v X (cid:15) (cid:15) X X ∨ ⊗ u X (cid:47) (cid:47) = (cid:38) (cid:38) X ∨ L ⊗ A X L ⊗ B X ∨ v X ⊗ (cid:15) (cid:15) X ∨ . Then the functors ? L ⊗ A e ( X ⊗ X ∨ ) : D ( A e ) → D ( B e )and ? L ⊗ B e ( X ∨ ⊗ X ) : D ( B e ) → D ( A e )form an adjoint pair. We will identify X L ⊗ B X ∨ ∼ → ( X ⊗ X ∨ ) L ⊗ B e B and X ∨ L ⊗ A X ∼ → ( X ∨ ⊗ X ) L ⊗ A e A , and still call u X and v X the same mor-phisms when composed with this identification. Since k is a commutativering, the tensor product over k is symmetric. We will denote the symme-try isomorphism by τ . Let D ( k ) denote the derived category of k -modules.We define a functor ψ : Alg → D ( k ) by putting ψ ( A ) = A L ⊗ A e A , and ψ ( f ) = f ⊗ f for a morphism f : A → B . There is a canonical quasi-isomorphism ψ ( A ) → ϕ ( A ) for any algebra A , where ϕ ( A ) is the underlyingcomplex of C ( A ). Therefore, the functors ϕ and ψ take isomorphic valueson objects. We now define ψ on morphisms of ALG as follows: Let X be acofibrant object of rep( A, B ). Define ψ ( X ) to be the composition A L ⊗ A e A → A L ⊗ A e X ⊗ X ∨ L ⊗ B e B ∼ → B L ⊗ B e X ∨ ⊗ X L ⊗ A e A → B L ⊗ B e B. That is, we put ψ ( X ) = (1 ⊗ v X ) ◦ τ ◦ (1 ⊗ u X ). Theorem 2.1.
The assignments A (cid:55)→ ψ ( A ) , X (cid:55)→ ψ ( X ) define a functoron ALG that extends the functor ϕ : Alg → D ( k ) . Corollary 2.2.
The functors ϕ and ψ : ALG → D ( k ) are isomorphic. MARCO ANTONIO ARMENTA AND BERNHARD KELLER
Proof of the Corollary.
This is immediate from Theorem 2.4 of [7] and theremark following it. (cid:3)
Proof of the Theorem.
For ease of notation, we write ⊗ and Hom insteadof L ⊗ and RHom. Let f : A → B be a morphism of Alg . The associatedmorphism in
ALG is X = f B B . Note that X ∨ = B B f . The diagrams A ⊗ A e ( f B ⊗ B B f ) (cid:39) (cid:41) (cid:41) (cid:39) (cid:15) (cid:15) A ⊗ A e ( f B ⊗ B f ) ⊗ B e B (cid:39) (cid:47) (cid:47) A ⊗ A e f B f and A ⊗ A e ( f B ⊗ B f ) ⊗ B e B (cid:39) (cid:47) (cid:47) τ (cid:15) (cid:15) A ⊗ A e f B fτ (cid:15) (cid:15) B ⊗ B e B f ⊗ f B ⊗ A e A (cid:39) (cid:47) (cid:47) f B f ⊗ A e A are commutative. Since f B f ⊗ A e A τ (cid:47) (cid:47) ⊗ f (cid:15) (cid:15) A ⊗ A e f B ff ⊗ (cid:15) (cid:15) B ⊗ B e B τ (cid:47) (cid:47) B ⊗ B e B is also commutative and the bottom morphism equals the identity, we getthat ψ ( f B B ) is the morphism induced by f from A ⊗ A e A to B ⊗ B e B .Therefore ψ ( f B B ) = ϕ ( f B B ). Let X : A → B and Y : B → C be morphismsin ALG . We have canonical isomorphismsHom C ( Y, C ) ⊗ B Hom B ( X, B ) ∼ → Hom B ( X, Hom C ( Y, C )) ∼ → Hom C ( X ⊗ B Y, C ) . Whence the identification( X ⊗ B Y ) ∨ = Y ∨ ⊗ B X ∨ . Put Z = X ⊗ B Y . For u Z , we choose the composition A u X (cid:47) (cid:47) X ⊗ B X ∨ ⊗ u Y ⊗ (cid:47) (cid:47) X ⊗ B Y ⊗ C ⊗ Y ∨ ⊗ B X ∨ and for v Z the composition( Y ∨ ⊗ B X ∨ ) ⊗ A ( X ⊗ B Y ) ⊗ v X ⊗ (cid:47) (cid:47) Y ∨ ⊗ B Y v Y (cid:47) (cid:47) C .
By definition, the composition ψ ( Y ) ◦ ψ ( X ) is the composition of (1 ⊗ v Y ) ◦ τ ◦ (1 ⊗ u Y ) with (1 ⊗ v X ) ◦ τ ◦ (1 ⊗ u X ). We first examine the composition(1 ⊗ u Y ) ◦ (1 ⊗ v X ): B ⊗ B e ( X ∨ ⊗ X ) ⊗ A e A ⊗ v X (cid:47) (cid:47) B ⊗ B e B ⊗ u Y (cid:47) (cid:47) B ⊗ B e ( Y ⊗ Y ∨ ) ⊗ C e C ERIVED INVARIANCE OF THE TAMARKIN–TSYGAN CALCULUS 5
Clearly, the following square is commutative B ⊗ B e ( X ∨ ⊗ X ) ⊗ A e A c (cid:47) (cid:47) ⊗ v X (cid:15) (cid:15) (( X ∨ ⊗ X ) ⊗ A e A ) ⊗ B e B v X ⊗ (cid:15) (cid:15) B ⊗ B e B τ (cid:47) (cid:47) B ⊗ B e B , where c is the obvious cyclic permutation. Notice that τ : B ⊗ B e B → B ⊗ B e B equals the identity. Thus, we have 1 ⊗ u Y = (1 ⊗ u Y ) ◦ τ and(1 ⊗ u Y ) ◦ (1 ⊗ v X ) = (1 ⊗ u Y ) ◦ τ ◦ (1 ⊗ v X ) = (1 ⊗ u Y ) ◦ ( v X ⊗ ◦ c. Let σ (( X ∨ ⊗ X ) ⊗ A e A ) ⊗ B e ( Y ⊗ Y ∨ ) ⊗ C e C ∼ → A ⊗ A e ( X ⊗ B Y ) ⊗ ( Y ∨ ⊗ B X ∨ ) ⊗ C e C be the natural isomorphism given by reordering the factors. Then we have ψ ( Y ) ◦ ψ ( X ) = f ◦ g , where f = σ ◦ (1 ⊗ u Y ) ◦ c ◦ τ ◦ (1 ⊗ u X ) and g =( v Y ⊗ ◦ τ ◦ ( v X ⊗ ◦ σ − . It is not hard to see that f equals 1 ⊗ u Z and g equals (1 ⊗ v Z ) ◦ τ . Intuitively, the reason is that given the available data,there is only one way to go from A ⊗ A e A to A ⊗ A e ( X ⊗ B Y ) ⊗ ( Y ∨ ⊗ B X ∨ ) ⊗ C e C and only one way to go from here to C ⊗ C e C . It follows that ψ ( Y ) ◦ ψ ( X ) = ψ ( Z ). (cid:3) Derived invariance
Let A and B be derived equivalent algebras and X a cofibrant object ofrep( A, B ) such that ? L ⊗ A X : D ( A ) → D ( B ) is an equivalence. Then C ( X )is an isomorphism of DMix and ϕ ( X ) an isomorphism of D ( k ). There is acanonical short exact sequence of dg Λ-modules0 → k [1] → Λ → k → k [1] B (cid:48) (cid:47) (cid:47) Λ I (cid:47) (cid:47) k S (cid:47) (cid:47) k [2] . We apply the isomorphism of functors ? L ⊗ Λ C ( A ) ∼ → ? L ⊗ Λ C ( B ) to thistriangle to get an isomorphism of triangles in D ( k ), where we recall that ϕ ( A ) is the underlying complex of C ( A ) k [1] L ⊗ Λ C ( A ) B (cid:48) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) ϕ ( A ) I (cid:47) (cid:47) ϕ ( X ) (cid:15) (cid:15) k L ⊗ Λ C ( A ) S (cid:47) (cid:47) ∼ = (cid:15) (cid:15) k [2] L ⊗ Λ C ( A ) ∼ = (cid:15) (cid:15) k [1] L ⊗ Λ C ( B ) B (cid:48) (cid:47) (cid:47) ϕ ( B ) I (cid:47) (cid:47) k L ⊗ Λ C ( B ) S (cid:47) (cid:47) k [2] L ⊗ Λ C ( B ) MARCO ANTONIO ARMENTA AND BERNHARD KELLER
Taking homology and identifying H j ( k L ⊗ Λ C ( A )) = HC j ( A ) as in [5], givesan isomorphism of the ISB-sequences of A and B , · · · (cid:47) (cid:47) HC n − ( A ) B (cid:48) n − (cid:47) (cid:47) ∼ = (cid:15) (cid:15) HH n ( A ) I n (cid:47) (cid:47) HH n ( X ) (cid:15) (cid:15) HC n ( A ) S n (cid:47) (cid:47) ∼ = (cid:15) (cid:15) HC n − ( A ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) · · ·· · · (cid:47) (cid:47) HC n − ( B ) B (cid:48) n − (cid:47) (cid:47) HH n ( B ) I n (cid:47) (cid:47) HC n ( B ) S n (cid:47) (cid:47) HC n − ( B ) (cid:47) (cid:47) · · · , where HH n ( X ) is the map induced by ϕ ( X ). In terms of the differentialcalculus, Connes’ differential is the map B n : HH n ( A ) → HH n +1 ( A ) , given by B n = B (cid:48) n I n . This shows that B n is derived invariant via HH n ( X ).By Theorem 2.1, the map HH n ( X ) is equal to the map induced by ψ ( X )used in the proof of the derived invariance of the cap product [1]. Therefore,we get the following Theorem 3.1.
The differential calculus of an algebra is a derived invariant.
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ERIVED INVARIANCE OF THE TAMARKIN–TSYGAN CALCULUS 7
CIMAT A. C. Guanajuato, M´exicoIMAG, Univ Montpellier, CNRS, Montpellier, France.
E-mail address : [email protected] Universit´e Paris Diderot – Paris 7, Sorbonne Universit´e, UFR de Math´ematiques,CNRS, Institut de Math´ematiques de Jussieu–Paris Rive Gauche, IMJ-PRG,Bˆatiment Sophie Germain, 75205 Paris Cedex 13, France
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