DERIVED CATEGORIES: LECTURE 5 References 1. Hochschild Homology and Cohomology of Algebras Let a Be an Associative, Non Necessar

DERIVED CATEGORIES: LECTURE 5

EVGENY SHINDER

References [Ca] Andrei Caldararu, The Mukai pairing, II: The Hochschild-Konstant-Rosenberg isomorphism, arXiv:math/0308080 [math.AG] [Gi] Viktor Ginzburg, Lectures on noncommutative geometry, arXiv:math/0506603 [math.AG] [Ke] B. Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998), no. 1–3, 223–273. [Ka1] D.Kaledin, Homological methods in Non-commutative Geometry, Lectures in Tokyo 2007–08 http://imperium.lenin.ru/∼kaledin/tokyo/ [Ka2] D.Kaledin, Non-commutative Geometry from the homological point of view, Lectures in Seoul, 2009 http://imperium.lenin.ru/∼kaledin/seoul/ [Kuz] A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, arXiv:0904.4330v1 [math.AG] [Ma] Nikita Markarian, The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem arXiv:math/0610553 [math.AG] [We] Weibel, Introduction to homological algebra

1. Hochschild homology and cohomology of algebras Let A be an associative, non necessarily commutative k-algebra, and let Ae = A⊗Aop. Exchanging the factors gives rise to an isomorphism Ae = A ⊗ Aop =∼ Aop ⊗ A = (Ae)op. Left Ae-module is the same thing as a right Ae-module and the same thing as A-bimodule, that is a vector space endowed with commuting left and right actions of A. A can be considered as an Ae-module. Let M be an arbitrary Ae-module. We have the following deﬁnitions: Ae HH∗(M) := T or∗ (A, M) (1.1) ∗ ∗ HH (M) := ExtAe (A, M). To compute these we can use a projective resolution of A as Ae-module. Note that Ae ∼ HH0(M) = T or0 (A, M) = A ⊗Ae M = M/[M,A], in particular

HH0(A) = A/[A, A] and 0 0 HH (M) = ExtAe (A, M) = HomAe (A, M) = ZA(M), in particular HH0(A) = Z(A). 1 1.1. Bar complex. For any two associative k-algebras A and A0 we define their free product A?A0 as a vector space generated by finite strings 0 0 a1 ? a1 ? a2 ? a2 ?... 0 0 where ai ∈ A, ai ∈ A . The multiplication is concatenation of strings and we factor out the ideal generated by relations α ? a = a ? α = αa for α ∈ k and a ∈ A t A0. Let B•(A) be the algebra A ? k[]. B•(A) can be endowed with a structure of dg- algebra, that is a multiplicative grading, and a differential of degree +1 which respects the multiplication according to the Leibnitz rule d(a · b) = d(a) · b + (−1)deg(a)a · d(b). The grading of B•(A) is given as deg() = −1 deg(a) = 0, a ∈ A. The differential is defined on generators by d() = 1 d(a) = 0, a ∈ A and on general element by the Leibnitz rule. • Explicitly elements of B (A) of degree −k have the form a1a2 . . . ak+1 (possibly some of ai = 1). We use the ”bar” notation a1|a2| ... |ak+1 to denote the latter element. It follows that B−k(A) = A⊗k+1 and d : B−k(A) → B−k+1(A) has the form

d(a1|a2| ... |ak+1) = a1a2|a3| ... |ak+1−

− a1|a2a3| ... |ak+1+ ... k+1 + (−1) a1|a2| ... |akak+1. Note that each B−k(A) is in fact an A-bimodule with the action deﬁned as 0 0 (a ⊗ a ) · a1|a2| ... |ak+1 = aa1|a2| ... |ak+1a and d is a morphism of A-bimodules. In fact each B−k(A) is a free A-bimodule for k ≥ 1: −k e ⊗k−1 indeed, as A-bimodules we have B (A) = A ⊗ Vk for k-vector space A . • −k Lemma 1.1. The bar complex B (A) is acyclic, that is the complex (B (A), d)k≥1 is a free Ae-resolution of B0(A) = A. Proof. For any cycle ξ ∈ B•(A) we have ξ = 1 · ξ = d( · ξ) therefore ξ is cohomologous to zero. The bar resolution is convenient to use for Hochschild homology and cohomology computation. 2 Lemma 1.2. Let A be commutative. Then we have the following isomorphisms ∼ HH1(A) = ΩA HH1(A) =∼ Der(A).

Here ΩA is the module of K¨ahlerdiﬀerentials {a · db : a, b ∈ A} ΩA = {a · d(b1b2) = ab2 · d(b1) + ab1 · d(b2)} and Der(A) is the module of derivations

Der(A) = {g ∈ Homk(A, A): g(ab) = ag(b) + bg(a)}. −k e Proof. We use the bar resolution. Recall that for each k ≥ 1 we have B (A) = A ⊗ Vk ⊗k−1 where Vk = A . Therefore for any A-bimodule M we can compute Hochschild homology of M using the complex

−k−1 ⊗k (1.2) Ck(M) := B (A) ⊗Ae M = M ⊗ A , k ≥ 0

· · · → M ⊗ A⊗2 → M ⊗ A → M (when the tensor product is taken over the base field we omit the subscript). In particular we have Ker(d : A ⊗ A → A) HH (A, A) = , 1 Im(d : A ⊗ A ⊗ A → A ⊗ A) and it is easy to see using identification (1.2) that for the differential we have d(a ⊗ b) = ab − ba = 0

d(a ⊗ b ⊗ c) = ab ⊗ c − a ⊗ bc + ca ⊗ b, thus the morphism a ⊗ b 7→ a · db gives an isomorphism HH1(A, A) = ΩA. Similarly we have

k −k−1 ⊗k C (M) := HomAe (B (A),M) = Hom(A ,M), are the terms of the complex computing HH∗(A, M) M → Hom(A, M) → Hom(A⊗2,M) → ... and in particular Ker(d : Hom(A, A) → Hom(A⊗2,A)) HH1(A) = Im(d : A → Hom(A, A)) with diﬀerentials d(b)(a) = ab − ba = 0, a, b ∈ A

d(g)(a1 ⊗ a2) = a1g(a2) − a2g(a1), a1, a2 ∈ A, g ∈ Hom(A, M). 1 Thus HH (A, A) coincides with Der(A). 3 Remark 1.3. If X = Spec(A) is an aﬃne variety, then

1 HH1(A) = ΩA = Γ(X, ΩX )

1 HH (A) = Der(A) = Γ(X,TX ) In the case X is smooth the Hochschild-Konstant-Rosenberg theorem will give further identification of Hochschild homology with differential forms and of Hochschild cohomology with polyvector fields. 1.2. Morita invariance. Two rings A and B are called Morita equivalent if their abelian categories of left modules are equivalent.

Theorem 1.4. Let P be a (B,A)-bimodule, and denote by FP the functor A − mod → B − mod sending M to P ⊗A M. This way we get an equivalence of categories: (B,A) − bimodules → F un(A − mod, B − mod). Here on the right we consider the category of right exact additive functors with morphisms given by natural transformations. In particular, every functor of this kind is isomorpic to some FP . Proof. We only prove essential surjectivity, the fully faithfulness is similar. Let F be a right exact additive functor A − mod → B − mod. Let P := F (A). By functoriality, P is (B,A)-bimodule. The proof goes by constructing a natural transformation FP → F . Since both functors are right exact and the transformation is an isomorphism on free modules by construction it follows that FP is isomorphic to F . Remark 1.5. Theorem of Toëngives an analog for this in dg-context. Corollary 1.6. A and B are Morita equivalent if and only if there exist an (B,A)- ∼ ∼ bimodule P and a (A, B)-bimodule Q such that Q⊗AP = B as B-bimodules and P ⊗BQ = A as A-bimodules. Example 1.7. Let A = k be a field, V be a finite dimensional k-vector space and ∼ ∗ B = End(V )(= Mn(k)). Let P = V considered as (End(V ), k)-bimodule and Q = V considered as (k, End(V ))-bimodule. We have the following isomorphisms of bimodules: ∗ ∼ V ⊗k V = End(V )

∗ ∼ V ⊗End(V ) V = k. Therefore the conditions of the corollary are satisﬁed and k and End(V ) are Morita equivalent. Corollary 1.8. If A and B are Morita equivalent, then mod − A and mod − B are equivalent, A − mod − A and B − mod − B are equivalent, A − modf and B − modf are equivalent. Proposition 1.9. Let A and B be Morita invariant k-algebras. Then their Hochschild homology, resp. cohomology, are isomorphic. 4 Proof. We have an equivalence of categories:

FP,Q :(A, A) − bimodules → (B,B) − bimodules given by the functor M 7→ P ⊗A M ⊗A Q. ∼ We have FP,Q(A) = P ⊗ Q = B as a B-bimodule. Note that FP,Q is a tensor functor. Therefore (A,A)−bimod ∼ (B,B)−bimod T or∗ (A, A) = T or∗ (B,B) and similarly for cohomology. 2. Hochschild homology and cohomology of varieties Let X be a smooth variety of dimension n and let ∆ : X → X × X be the diagonal. We do not assume that X is projective. Definition 2.1. Hochschild homology and cohomology complexes of X are defined as HH• := p1∗ O∆ ⊗ O∆ • HH := p1∗Hom(O∆, O∆). Hochschild homology and cohomology groups of X are −k X×X HHk(X) := RΓ (X, HH•) ' T ork (O∆, O∆) k k • k HH (X) := RΓ (X, HH ) ' ExtX×X (O∆, O∆). (in agreement with (1.1) when X = Spec(A)). Proposition 2.2. We have the following expression for Hochschild homology and cohomology complexes: ∗ HH• ' ∆ ∆∗OX • ∗ ∨ HH ' ∆ ∆∗OX ⊗ ωX [−n] and HH• = p1∗Hom(∆∗OX , ∆∗ωX [n]). Proof. For homology we use the projection formula to compute ∗ ∗ ∗ HH• = p1∗ O∆ ⊗ O∆ ' p1∗∆∗∆ O∆ ' ∆ O∆ = ∆ ∆∗OX . For cohomology we first note that by Lemma below we have ∨ NX/X×X = TX =⇒ det(NX/X×X ) = ωX ∨ ∨ O∆ ' ∆∗ωX [−n] and then compute using the projection formula: • ∨ HH = p1∗Hom(O∆, O∆) ' p1∗ O∆ ⊗ O∆ ' ∗ ∨ ' p1∗∆∗ ∆ O∆ ⊗ ωX [−n]) ' ∗ ∨ ' ∆ ∆∗OX ⊗ ωX [−n]). 5 Lemma 2.3. Let i : Y → X be a closed embedding of smooth projective varieties of codimension c and let F be a coherent sheaf on Y . Then 1. ∨ ∨ (i∗F) ' i∗(F ⊗ det(NY/X )[−c]) ∨ where det(NY/X ) is the the top exterior power of the conormal bundle of Y in X. In particular ∨ (i∗OY ) ' i∗(det(NY/X )[−c]. ∗ k 2. i i∗F has non-zero cohomology sheaves H only in the range [−k, 0] and they are given as k k ∨ H = F ⊗ Λ NY/X . Proof. Let D be the duality operator ∨ DX := Hom(•, ωX [dim(X)]) = (•) ⊗ ωX [dim(X)].

It follows from the Grothendieck-Verdier duality formalism that DX i∗ = i∗DY . We compute ∨ ∨ (i∗F) = DX (i∗F) ⊗ ωX [−dim(X)] = ∨ = i∗DY (F) ⊗ ωX [−dim(X)] = ∨ ∨ = i∗(F ⊗ ωY [dim(Y )]) ⊗ ωX [−dim(X)] = ∨ ∗ ∨ = i∗(F ⊗ ωY ⊗ i ωX [−c]) = ∨ = i∗(F ⊗ det(NY/X )[−c]). Proposition 2.4. 1. Hochschild homology is functorial for Fourier-Mukai transforms. 2. Hochschild cohomology is functorial for Fourier-Mukai equivalences.

Proof. We omit the proof. See Section 6 of [Kuz]. ∼ Theorem 2.5. If X and Y are derived equivalent, then HH∗(X) = HH∗(Y ) and HH∗(X) =∼ HH∗(Y ). Proof. By a theorem of Orlov, any equivalence Db(X) → Db(Y ) is a Fourier-Mukai transform. The result now follows from the previous Proposition.

3. Hochschild-Konstant-Rosenberg’s theorem Theorem 3.1 (HKR). We have the following quasi-isomorphisms of complexes on X: n M i HH• ' ΩX [i] i=0 n • M i HH ' TX [−i] i=0 In this section we give a sketch of the proof of this theorem. We start with the following Lemma: 6 Lemma 3.2. Let Y ⊂ X be a smooth complete intersection of codimension n, or more generally, Y = Z(s) for a regular section s of a vector bundle E of rank n over X. Then we have ∗ ∨ OY ⊗OX OY = Λ (NY/X [1]) Hom (O , O ) = Λ∗(N [−1]) OX Y Y Y/X (both complexes on the right have zero diﬀerentials).

Proof. We have the following locally free Kozsul resolution of OY : • n ∨ n−1 ∨ 2 ∨ ∨ K (s) := [Λ (E ) → Λ (E ) → · · · → Λ (E ) → E → OX ] with diﬀerentials given as contraction with s. Now E Y = NY/X and we have: ∼ • ∗ ∨ ∗ ∨ OY ⊗OX OY = K (s) ⊗OX OY = Λ (E Y [1]) = Λ (NY/X [1]) Hom (O , O ) ∼ Hom (K•(s), O ) = Λ∗(E [−1]) = Λ∗(N [−1]) OX Y Y = OX Y Y Y/X and the diﬀerentials in both complexes are zero since s vanishes on Y ⊂ X. Corollary 3.3. The HKR theorem holds for a regular local k-algebra A.

Proof. Let M ⊂ A be the maximal ideal of A. Let B = (A ⊗ A)m−1(M), this is also a regular local ring, and A is a B-module. One can see that L HH• = A ⊗B A HH• = RHomB(A, A). Since A and B are regular local rings, the ideal I = Ker(B → A) is generated by a regular sequence f1, . . . , fn ∈ B of length n = dim(A) = dim(B) − dim(A) and we ∨ apply the Lemma above. To finish the proof recall that NX/X×X = TX and NX/X×X = 1 ΩX . Proof of HKR theorem. In fact the two statements are equivalent to each other. This follows from Proposition 2.2 and isomorphisms i n−i ΩX = Λ TX ⊗ ωX which are easy to see fiberwise. We now prove the isomorphism for Hochschild homology. b Note that by definition HH• is a commutative algebra object in D (X). This allows us to Ln i define a morphism ψ• : i=0 ΩX [i] → HH• by giving its component ψ1 :ΩX [1] → HH•. Corollary 3.4. ∼ M p q HHk(X) = H (X, ΩX ) p−q=−k k ∼ M p q HH (X) = H (X, Λ TX ) p+q=k Proof. n n −k M i M i−k i M p q HHk(X) = RΓ (X, ΩX [i]) = H (X, ΩX ) = H (X, ΩX ). i=0 i=0 q−p=k 7 n n k k M i M −i+k i M p q HH (X) = RΓ (X, Λ TX [−i]) = H (X, Λ TX ) = H (X, Λ TX ). i=0 i=0 p+q=k ∗ Remark 3.5. Assume that X is a smooth projective over C, so that H (X, C) admits a Hodge decomposition. The Euler characterstic of X is equal to the Euler characteristic of the Hochschild homology: X k χ(X) = (−1) dimHHk(X), in particular χ(X) is derived invariant. It is not known whether the Betti numbers bi(X) are derived invariant.

4. Hochschild homology and cohomology of admissible subcategories Let A ⊂ Db(X) be an admissible subcategory. There are two possible ways of deﬁning Hochschild homology and cohomology of A. On the one hand one can use Bondal-van den Bergh to deduce that A has a strong ∼ b generator T , and then by a theorem of Keller we have A = Dperf (mod − A), where A is the dg-algebra of endomorphisms of T . Then one deﬁnes

HH∗(A) := HH∗(A) HH∗(A) := HH∗(A). On the other hand Kuznetsov [Kuz] deﬁnes Hochschild homology and cohomology of A as follows: let P ∈ Db(X × X) be the Fourier-Mukai kernel giving the projection Db(X) → A. Then HH∗(A) := HomDb(X×X)(P,P ◦ SX ) ∗ HH (A) := HomDb(X×X)(P,P ). Kuznetsov has shown that the two deﬁnitions agree. Moreover we have the following fact ([Kuz], Theorem 7.3): b Proposition 4.1. If D (X) = hA1,..., Ari is a semi-orthogonal decomposition, then there is a canonical decomposition

HH∗(X) = HH∗(A1) ⊕ · · · ⊕ HH∗(Ar).