Homology, Homotopy and Applications, vol.5(1), 2003, pp.407–421

EXTENSIONS OF HOMOGENEOUS COORDINATE RINGS TO A_{∞}-ALGEBRAS

A. POLISHCHUK

(communicated by James Stasheff)

Abstract We study A_{∞}-structures extending the natural algebra structure on the cohomology of ⊕_{n∈ℤ}L^{n}, where L is a very ample line bundle on a projective d-dimensional variety X such that H^{i}(X, L^{n}) = 0 for 0 < i < d and all n ∈ ℤ. We prove that there exists a unique such nontrivial A_{∞}-structure up to a strict A_{∞}-isomorphism (i.e., an A_{∞}-isomorphism with the identity as the first structure map) and rescaling. In the case when X is a curve we also compute the group of strict A_{∞}- automorphisms of this A_{∞}-structure.

1. Introduction

Let X be a projective variety over a field k, L be a very ample line bundle on X. Recall that the graded k-algebra

R_{L} = ⊕_{n⩾0}H^{0}(X, L^{n})

is called the homogeneous coordinate ring corresponding to L. More generally, one can consider the bigraded k-algebra

A_{L} = ⊕_{p,q∈ℤ}H^{q}(X, L^{p}).

We call A_{L} the extended homogeneous coordinate ring corresponding to L.

Since A_{L} can be represented naturally as the cohomology algebra of some dgalgebra (say, using injective resolutions or Čech cohomology with respect to an affine covering), it is equipped with a family of higher operations called Massey products. A better way of recording this additional structure uses the notion of A_{∞}-algebra due to Stasheff. Namely, by the theorem of Kadeishvili the product on A_{L} extends to a canonical (up to A_{∞}-isomorphism) A_{∞}-algebra structure with m_{1} = 0 (see [4] 3.3 and references therein). More precisely, this structure is unique up to a strict A_{∞}-isomorphism, i.e., an A_{∞}-isomorphism with the identity map as the first structure map (see section 2.1 for details). Note that the axioms of A_{∞}- algebra use the cohomological grading on A_{L} (where H^{q}(X, L^{p}) has cohomological degree q), and all the operations (m_{n}) have degree zero with respect to the internal

This work was partially supported by NSF grant DMS-0070967 Received May 24, 2003, revised September 15, 2003; published on October 15, 2003. 2000 Mathematics Subject Classification: 18E30, 55P43. Key words and phrases: A-infinity algebra, A-infinity isomorphism, homogeneous coordinate ring. \begin{matrix}c \\ ◯ \end{matrix} 2003, A. Polishchuk. Permission to copy for private use granted.

grading (where H^{q}(X, L^{p}) has internal degree p). The natural question is whether it is possible to characterize intrinsically this canonical class of A_{∞}-structures on A_{L}. This question is partly motivated by the homological mirror symmetry. Namely, in the case when X is a Calabi-Yau manifold, the A_{∞}-structure on A_{L} is supposed to be A_{∞}-equivalent to an appropriate A_{∞}-algebra arising on a mirror dual symplectic side. An intrinsic characterization of the A_{∞}-isomorphism class of our A_{∞}-structure could be helpful in reducing the problem of constructing such an A_{∞}-equivalence to constructing an isomorphism of the usual associative algebras. More generally, it is conceivable that the algebra A_{L} can appear as cohomology algebra of some other dg-algebras (for example, if there is an equivalence of the derived category of coherent sheaves on X with some other such category), so one might be interested in comparing corresponding A_{∞}-structures on A_{L}.

Thus, we want to study all A_{∞}-structures (m_{n}) on A_{L} (with respect to the cohomological grading), such that m_{1} = 0, m_{2} is the standard double product and all m_{n} have degree 0 with respect to the internal grading. Let us call such an A_{∞}-structure on A_{L} admissible. As we have already mentioned before, there is a canonical strict A_{∞}-isomorphism class of such structures coming from the realization of A_{L} as cohomology of the dg-algebra ⊕_{n}C^{•}(L^{n}) where C^{•}(?) denotes the Čech complex with respect to some open affine covering of X. By definition, an A_{∞}-structure belongs to the canonical class if there exists an A_{∞}-morphism from A_{L} equipped with this A_{∞}-structure to the above dg-algebra inducing identity on the cohomology. The simplest picture one could imagine would be that all admissible A_{∞}-structures are strictly A_{∞}-isomorphic, i.e., that A_{L} is intrinsically formal. It turns out that this is not the case. However, our main theorem below shows that if the cohomology of the structure sheaf on X is concentrated in degrees 0 and dim X then for sufficiently ample L the situation is not too much worse.

We will recall the notion of a homotopy between A_{∞}-morphisms in section 2.1 below. ^{1} Let us say that an A_{∞}-structure is nontrivial if it is not A_{∞}-isomorphic to an A_{∞}-structure with m_{n} = 0 for n ≠ 2. By rescaling of an A_{∞}-structure we mean the change of the products (m_{n}) to (λ^{n−2}m_{n}) for some constant λ ∈ k^{∗}. Our main result gives a classification of admissible A_{∞}-structures on A_{L} up to a strict A_{∞}-isomorphism and rescaling (under certain assumptions).

Theorem 1.1. Let L be a very ample line bundle on a d-dimensional projective variety X such that H^{q}(X, L^{p}) = 0 for q ≠ 0, d and all p ∈ ℤ. Then

(i) up to a strict A_{∞}-isomorphism and rescaling there exists a unique non-trivial admissible A_{∞}-structure on A_{L}; moreover, A_{∞}-structures on A_{L} from the canonical strict A_{∞}-isomorphism class are nontrivial; (ii) for every pair of strict A_{∞}-isomorphisms f, f′ : (m_{i}) → (m′_{i}) between admissible A_{∞}-structures on A_{L} there exists a homotopy from f to f′.

Remarks. 1. One can unify strict A_{∞}-isomorphisms with rescalings by considering A_{∞}-isomorphisms (f_{n}) with the morphism f_{1} of the form f_{1}(a) = λ^{deg(a)} for

1All A_{∞}-morphisms and homotopies between them are assumed to respect the internal grading on A_{L}.

some λ ∈ k^{∗} (where a is a homogeneous element of A_{L}). In particular, part (i) of the theorem implies that all non-trivial admissible A_{∞}-structures on A_{L} are A_{∞}- isomorphic.

2. As we will show in section 2.1, part (ii) of the theorem is equivalent to its particular case when f′ = f. In this case the statement is that every strict A_{∞}- automorphism of f is homotopic to the identity.

3. If one wants to see more explicitly how a canonical A_{∞}-structure on A_{L} looks like, one has to choose one of the natural dg-algebras with cohomology A_{L} (an obvious algebraic choice is the Čech complex; in the case k = ℂ one can also use the Dolbeault complex), choose a projector π from the dg-algebra to some space of representatives for the cohomology such that π = 1 − dQ − Qd for some operator Q, and then apply formulas of [5] for the operations m_{n} (they are given by certain sums over trees).

The above theorem is applicable to every line bundle of sufficiently large degree on a curve. In higher dimensions it can be used for every sufficiently ample line bundle on a d-dimensional projective variety X such that there exists a dualizing sheaf on X and H^{i}(X, O_{X}) = 0 for 0 < i < d. For example, this condition is satisfied for complete intersections in projective spaces. At present we do not know how to extend this theorem to the case when O_{X} has some nontrivial middle cohomology. Note that for a smooth projective variety X over ℂ the natural (up to a strict A_{∞}- isomorphism) A_{∞}-structure on H^{∗}(X, O_{X}) is trivial as follows from the formality theorem of [1]. This suggests that for sufficiently ample line bundle L one could try to characterize the canonical A_{∞}-structure on A_{L} (up to a strict A_{∞}-isomorphism and rescaling) as an admissible A_{∞}-structure whose restriction to H^{∗}(X, O_{X}) is trivial.

In the case when X is a curve we can also compute the group of strict A_{∞}- automorphisms of an A_{∞}-structure on A_{L}. As we will explain below 2.1, strict A_{∞}-isomorphisms on A_{L} form a group HG, which is a subgroup of automorphisms of the free coalgebra Bar(A_{L}) (preserving both gradings). The dual to the degree zero component of Bar(A_{L}) (with respect to both gradings) can be identified with the completed tensor algebra T̂(H^{1}(X, O_{X})^{∗}) = ∏_{n⩾0} T^{n}(H^{1}(X, O_{X})^{∗}). Therefore, we obtain a natural homomorphism from HG to to the group G of continuous automorphisms of T̂(H^{1}(X, O_{X})^{∗}).

Theorem 1.2. Let L be a very ample line bundle on a projective curve X such that H^{1}(X, L) = 0. Let also HG(m) ⊂ HG be the group of strict A_{∞}-automorphisms of an admissible A_{∞}-structure m on A_{L}. Then the above homomorphism HG → G restricts to an isomorphism of HG(m) with the subgroup G_{0} ⊂ G consisting of inner automorphisms of T̂(H^{1}(X, O_{X})^{∗}) by elements in 1 + ∏_{n>0} T^{n}(H^{1}(X, O_{X})^{∗}).

Assume that X is a smooth projective curve. Then there is a canonical noncommutative thickening J̃ of the Jacobian J of X (see [3]). As was shown in [10], a choice of an A_{∞}-structure in the canonical strict A_{∞}-isomorphism class gives rise to a formal system of coordinates on J̃ at zero. More precisely, by this we mean an isomorphism of the formal completion of the local ring of J̃ at zero with T̂(H^{1}(X, O_{X})^{∗}) inducing the identity map on the tangent spaces. Formal coordinates associated with two strictly isomorphic A_{∞}-structures are related by the coordinate change given by the image of the corresponding A_{∞}-isomorphism under the homomorphism HG → G. Now Theorem 1.2 implies that two A_{∞}-structures in the canonical class that induce the same formal coordinate on J̃ can be connected by a unique strict A_{∞}-isomorphism. Indeed, two such isomorphisms would differ by a strict A_{∞}-automorphism inducing the trivial automorphism of T̂(H^{1}(X, O_{X})^{∗}), but such an A_{∞}-automorphism is trivial by Theorem 1.2.

Convention. Throughout the paper we work over a fixed ground field k. The symbol ⊗ without additional subscripts always denotes the tensor product over k.

Acknowledgment. I’d like to thank the referee for helpful remarks and suggestions.

2. Preliminaries

2.1. Strict A_{∞}-isomorphisms and homotopies

We refer to [4] for an introduction to A_{∞}-structures. We restrict ourselves to several remarks about A_{∞}-isomorphisms and homotopies between them.

A strict A_{∞}-isomorphism between two A_{∞}-structures (m) and (m′) on the same graded space A is an A_{∞}-morphism f = (f_{n}) from (A, m) to (A, m′) such that f_{1} = id. The equations connecting f, m and m′ can be interpreted as follows. Recall that m and m′ correspond to coderivations d_{m} and d_{m′} of the bar-construction Bar(A) = ⊕_{n⩾1}T^{n}(A[1]) such that d_{m}^{2} = d_{m′}^{2} = 0. Now every collection f = (f_{n})_{n⩾1}, where f_{n} : A^{⊗n} → A has degree 1 − n, f_{1} = id, defines a coalgebra automorphism α_{f} : Bar(A) → Bar(A), with the component Bar(A) → A[1] given by (f_{n}). The condition that f is an A_{∞}-morphism is equivalent to the equation α_{f} ∘d_{m} = d_{m′} ∘α_{f}. In other words, strict A_{∞}-isomorphisms between A_{∞}-structures precisely correspond to the action of the group of automorphisms of Bar(A) as a coalgebra on the space of coderivations d such that d^{2} = 0. More precisely, we consider only automorphisms of Bar(A) of degree 0 inducing the identity map A → A. Let us denote by HG = HG(A) the group of such automorphisms which we will also call the group of strict A_{∞}-isomorphisms on A. We will denote by m → g ∗ m, where g ∈ HG, the natural action of this group on the set of all A_{∞}-structures on A.

One can define a decreasing filtration (HG_{n}) of HG by normal subgroups by setting

HG_{n} = {f = (f_{i}) | f_{i} = 0, 1 < i ⩽ n}.

Note that f ∈ HG_{n} if and only if the restriction of α_{f} to the sub-coalgebra Bar(A)_{⩽n} = ⊕_{i⩽n}(A[1])^{⊗i} is the identity homomorphism. Furthermore, it is also clear that HG ≃ proj lim_{n} HG/HG_{n}. In particular, an infinite product of strict A_{∞}-isomorphisms … ∗ f(3) ∗ f(2) ∗ f(1) is well-defined as long as f(n) ∈ HG_{i(n)}, where i(n) → ∞ as n → ∞.

The notion of a homotopy between A_{∞}-morphisms is best understood in a more general context of A_{∞}-categories. Namely, for every pair of A_{∞}-categories C, D one can define the A_{∞}-category Fun(C, D) having A_{∞}-functors F : C → D as objects (see [6], [8]). In particular, there is a natural notion of closed morphisms between two A_{∞}-functors F, F′ : C → D. Specializing to the case when C and D are A_{∞}-categories with one object corresponding to A_{∞}-algebras A and B we obtain a notion of a closed morphism between a pair of A_{∞}-morphisms f, f′ : A → B. Following [4] we call such a closed morphism a homotopy between A_{∞}-morphisms f and f′. More explicitly, a homotopy h is given by a collection of maps h_{n} : A^{⊗n} → B of degree −n, where n ⩾ 1, satisfying some equations. These equations are written as follows: there exists a unique linear map H : Bar(A) → Bar(B) of degree −1 with the component Bar(A) → B given by (h_{n}), such that

Δ ∘ H = (α_{f} ⊗ H + H ⊗ α_{f′}) ∘ Δ,  (2.1.1)

where α_{f}, α_{f′} : Bar(A) → Bar(B) are coalgebra homomorphisms corresponding to f and f′, Δ denotes the comultiplication. Then the equation connecting h, f and f′ is

α_{f} − α_{f′} = d_{A} ∘ H + H ∘ d_{B},  (2.1.2)

where d_{A} (resp., d_{B}) is the coderivation of Bar(A) (resp., Bar(B)) corresponding to the A_{∞}-structure on A (resp., B). It is not difficult to check that for a given A_{∞}-morphism f from A to B the equations (2.1.1) and (2.1.2) imply that α_{f′} is a homomorphism of dg-coalgebras, so it defines an A_{∞}-morphism f′ from A to B. Moreover, similarly to the case of strict A_{∞}-isomorphisms we have the following result.

Lemma 2.1. Let A and B be A_{∞}-algebras and f = (f_{n}) be an A_{∞}-morphism from A to B. For every collection (h_{n})_{n⩾1}, where h_{n} : A^{⊗n} → B has degree −n, there exists a unique A_{∞}-morphism f′ from A to B such that h is a homotopy from f to f′.

Proof. It is easy to see that equation (2.1.1) is equivalent to the following formula

H[a_{1}| … |a_{n}] = ∑_{i_{1}<...<i_{k}<m<j_{1}<...<j_{l}=n}
±[f_{i_{1}}(a_{1}, …, a_{i_{1}})| … |f_{i_{k}−i_{k−1}}(a_{i_{k−1}+1}, …, a_{i_{k}})|
h_{m−i_{k}}(a_{i_{k}+1}, … a_{m})|f′_{j_{1}−m}(a_{m+1}, …, a_{j_{1}})| … |f′_{j_{l}−j_{l−1}}(a_{j_{l−1}+1}, …, a_{j_{l}})],

(2.1.3)

where a_{1}, …, a_{n} ∈ A, n ⩾ 1. We are going to construct the maps H|_{Bar(A)_{⩽n}} and α_{f′}|_{Bar(A)_{⩽n}} recursively, so that at each step the equations (2.1.2) and (2.1.3) are satisfied when restricted to Bar(A)_{⩽n}. Of course, we also want H to have (h_{n}) as components. Then such a construction will be unique. Note that H|_{A[1]} is given by h_{1} and α_{f′}|_{A[1]} is given by f′_{1} = f_{1} − m_{1} ∘ h_{1} − h_{1} ∘ m_{1}. Now assume that the restrictions of H and α_{f′} to Bar(A)_{⩽n−1} are already constructed, so that the maps f′_{i} : A^{⊗i} → B are defined for i ⩽ n − 1. Then the formula (2.1.3) defines uniquely the extension of H to Bar(A)_{⩽n} (note that in the RHS of this formula only f′_{i} with i ⩽ n − 1 appear). It remains to apply formula (2.1.2) to define α_{f′}|_{Bar(A)_{⩽n}}.

Let HG be the group of strict A_{∞}-isomorphisms on a given graded space A. In other words, HG is the group of degree 0 coalgebra automorphisms of Bar(A) with the componenent A → A equal to the identity map. This group acts on the set of all A_{∞}-structures on A. The stabilizer subgroup of some A_{∞}-structure m is the group of strict A_{∞}-automorphisms HG(m). We can consider the set of all strict A_{∞}-automorphisms f_{h} ∈ HG(m) such that there exists a homotopy h from the trivial A_{∞}-automorphism f^{tr} to f_{h} (where f_{i}^{tr} = 0 for i > 1). It is easy to see that A_{∞}-automorphisms of the form f_{h} constitute a normal subgroup in HG(m) that we will denote by HG(m)^{0}. Furthermore, for every g ∈ HG we have HG_{g∗m} = gHG(m)^{0}g^{−1}. Also, for a pair of elements g_{1}, g_{2} ∈ HG such that m′ = g_{1} ∗ m = g_{2} ∗ m, there exists a homotopy between g_{1} and g_{2} (where g_{i} are considered as A_{∞}-morphisms from (A, m) to (A, m′)) if and only if g_{1}^{−1}g_{2} ∈ HG(m)^{0}.

2.2. Obstructions

Below we use Hochschild cohomology HH(A) := HH(A, A) for a graded associative algebra A (see [7] for the corresponding sign convention). When considering A = A_{L} as a graded algebra we equip it with the cohomological grading, so in the situation of Theorem 1.1 this grading has only 0-th and d-th non-trivial graded components.

The following lemma is well known and its proof is straightforward.

Lemma 2.2. Let m and m′ be two admissible A_{∞}-structures on A. Assume that m_{i} = m′_{i} for i < n, where n ⩾ 3.

(i) Set c(a_{1}, …, a_{n}) = (m′_{n} − m_{n})(a_{1}, …, a_{n}). Then c is a Hochschild n-cocycle, i.e.,

δc(a_{1}, …, a_{n+1}) = ∑_{j=1}^{n}(−1)^{j}c(a_{1}, …, a_{j}a_{j+1}, …, a_{n+1})+

(−1)^{n deg(a_{1})}a_{1}c(a_{2}, …, a_{n+1}) + (−1)^{n+1}c(a_{1}, …, a_{n})a_{n+1} = 0.

(ii) If m′ = f ∗m for a strict A_{∞}-isomorphism f such that f_{i} = 0 for 1 < i < n−1, then setting b(a_{1}, …, a_{n−1}) = (−1)^{n−1}f_{n−1}(a_{1}, …, a_{n−1}) we get

c(a_{1}, …, a_{n}) = δb(a_{1}, …, a_{n}),

where c is the n-cocycle defined in (i). Hence, c is a Hochschild coboundary.

Thus, the study of admissible A_{∞}-structures on A is closely related to the study of certain components of Hochschild cohomology of A. More precisely, let us denote C_{p,q}^{n}(A) (resp. HH_{p,q}^{n}(A)) the space of reduced Hochschild n-cochains (resp. of nth Hochschild cohomology classes) of internal grading −p and of cohomological grading −q. In other words, C_{p,q}^{n}(A) consists of cochains c : A^{⊗n} → A such that intdeg c(a_{1}, …, a_{n}) = intdeg a_{1} + … + intdeg a_{n} − p, deg c(a_{1}, …, a_{n}) = deg a_{1} + … + deg a_{n} − q. Since, all the operations m_{n} respect the internal grading and have (cohomological) degree 2 − n, we see that the cocycle c defined in Lemma 2.2 lives in C_{0,n−2}^{n}(A).

There is an analogue of Lemma 2.2 for strict A_{∞}-isomorphisms.

Lemma 2.3. Let m and m′ be admissible A_{∞}-structures on A, f, f′ be a pair of strict A_{∞}-isomorphisms from m to m′. Assume that f_{i} = f′_{i} for i < n, where n ⩾ 2.

(i) Set c(a_{1}, …, a_{n}) = (f′_{n} − f_{n})(a_{1}, …, a_{n}). Then c is a Hochschild n-cocycle in C_{0,n−1}^{n}(A). (ii) If ϕ : f → f′ is a homotopy such that ϕ_{i} = 0 for i < n − 1, then for b(a_{1}, …, a_{n−1}) = ±ϕ_{n−1}(a_{1}, …, a_{n−1}) one has c = δb.

3. Calculations

3.1. Hochschild cohomology

In this subsection we calculate the components of the Hochschild cohomology of A = A_{L} that are relevant for the proof of Theorem 1.1.

Let us set R = R_{L} and let R_{+} = ⊕_{n⩾1}R_{n} be the augmentation ideal in R, so that R/R_{+} = k. Recall that the bar-construction provides a free resolution of k as R-module of the form

… → R_{+} ⊗ R_{+} ⊗ R → R_{+} ⊗ R → R → k.  (3.1.1)

For graded R-bimodules M_{1}, …, M_{n} we consider the bar-complex

B^{•}(M_{1}, …, M_{n}) = M_{1} ⊗ T (R_{+}) ⊗ M_{2} ⊗ … T (R_{+}) ⊗ M_{n},

where T (R_{+}) is the tensor algebra of R_{+}. This is just the tensor product over T (R_{+}) of the bar-complexes of M_{1}, …, M_{n} (where M_{1} is considered as a right Rmodule, M_{2}, …, M_{n−1} as R-bimodules, and M_{n} as a left R-module). The grading in this complex is induced by the cohomological grading of the tensor algebra T (R_{+}) defined by deg T^{i}(R_{+}) = −i, so that B^{•}(M_{1}, …, M_{n}) is concentrated in nonnegative degrees and the differential has degree 1. For example, B^{•}(k, R) is the bar-resolution (3.1.1) of k.

Proposition 3.1. Under the assumptions of Theorem 1.1 let us consider the graded R-module M = ⊕_{i∈ℤ}H^{d}(X, L^{i}). Let M_{1}, …, M_{n} be graded R-bimodules such that each of them is isomorphic to M as a (graded) right R-module and as a left Rmodule.

(i) The complex B^{•}(k, M) = T (R_{+}) ⊗ M has one-dimensional cohomology, which is concentrated in degree −d − 1 and internal degree 0. (ii) H^{i}(B^{•}(M_{1}, M_{2})) = 0 for i ≠ −d − 1 and H^{−d−1}(B^{•}(M_{1}, M_{2})) is isomorphic to M as a (graded) right R-module and as a left R-module. (iii) H^{i}(B^{•}(M_{1}, …, M_{n})) = 0 for i > −(n − 1)(d + 1). (iv) H^{i}(B^{•}(k, M_{1}, …, M_{n}, k)) = 0 for i > −n(d + 1). In addition, the space H^{−d−1}(B^{•}(k, M_{1}, k)) is one-dimensional and has internal degree 0.

Proof. (i) Localizing the exact sequence (3.1.1) on X and tensoring with L^{m}, where m ∈ ℤ, we obtain the following exact sequence of vector bundles on X:

… ⊕_{n_{1},n_{2}>0} R_{n_{1}} ⊗ R_{n_{2}} ⊗ L^{m−n_{1}−n_{2}} → ⊕_{n>0}R_{n} ⊗ L^{m−n} → L^{m} → 0. (3.1.2)

Each term in this sequence is a direct sum of a number of copies of line bundles L^{n}: for a finite-dimensional vector space V we denote by V ⊗ L^{n} the direct sum of dim V copies of L^{n}. Now let us consider the spectral sequence with E_{1}-term given by the cohomology of all sheaves in this complex and abutting to zero (this sequence converges since we can compute cohomology using Čech resolutions with respect to a finite open affine covering of X). The E_{1}-term consists of two rows: one obtained by applying the functor H^{0}(X, ·) to (3.1.2), another obtained by applying H^{d}(X, ·). The row of H^{0}’s has form

… ⊕_{n_{1},n_{2}>0} R_{n_{1}} ⊗ R_{n_{2}} ⊗ R_{m−n_{1}−n_{2}} → ⊕_{n>0}R_{n} ⊗ R_{m−n} → R_{m} → 0

which is just the m-th homogeneous component of the bar-resolution. Hence, this complex is exact for m ≠ 0. Since the sequence abuts to zero the row of H^{d}’s should also be exact for m ≠ 0. For m = 0 the row of H^{0}’s reduces to the single term H^{0}(X, O_{X}) = k, hence, the row of H^{d}’s in this case has one-dimensional cohomology at −(d + 1)th term and is exact elsewhere. (ii) Consider the filtration on B^{•}(M_{1}, M_{2}) associated with the ℤ-grading on M_{2}. By part (i) the corresponding spectral sequence has the term E_{1} ≃ H^{−d−1}(M_{1} ⊗ T (R_{+})) ⊗ M_{2} ≃ M_{2}. Hence, it degenerates in this term and

H^{∗}(K^{•}) ≃ H^{−d−1}(K^{•}) ≃ M

as a right R-module. Similarly, the spectral sequence associated with the filtration on K^{•} induced by the ℤ-grading on M_{2} gives an isomorphism of left R-modules H^{−d−1}(K^{•}) ≃ M. (iii) For n = 2 this follows from (ii). Now let n > 2 and assume that the assertion holds for n′ < n. We can consider B^{•}(M_{1}, …, M_{n}) as the total complex associated with a bicomplex, where the bidegree (deg_{0}, deg_{1}) on M_{1}⊗T (R_{+})⊗…⊗T (R_{+})⊗M_{n} is given by

deg_{0}(x_{1} ⊗ t_{1} ⊗ … ⊗ t_{n−1} ⊗ x_{n}) = ∑_{i≡0(2)} deg(t_{i}),

deg_{1}(x_{1} ⊗ t_{1} ⊗ … ⊗ t_{n−1} ⊗ x_{n}) = ∑_{i≡1(2)} deg(t_{i}),

where t_{i} ∈ T (R_{+}), x_{i} ∈ M_{i}, deg denotes the cohomological degree on T (R_{+}). Therefore, there is a spectral sequence abbuting to cohomology of B^{•}(M_{1}, …, M_{n}) with the E_{1}-term

H^{∗}(M_{1} ⊗ T (R_{+}) ⊗ M_{2}) ⊗ T (R_{+}) ⊗ H^{∗}(M_{3} ⊗ T (R_{+}) ⊗ M_{4}) ⊗ …,

where the last factor of the tensor product is either M_{n} or H^{∗}(M_{n−1}⊗T (R_{+})⊗M_{n}). Using part (ii) we see that E_{1} is isomorphic to the complex of the form

B^{•}(M′_{1}, …, M′_{n′})[(n − n′)(d + 1)]

with n′ < n. It remains to apply the induction assumption. (iv) Consider first the case n = 1. The complex B^{•}(k, M_{1}, k) = T (R_{+})⊗M_{1}⊗T (R_{+}) is the total complex of the bicomplex (∂_{1} ⊗ id, id ⊗∂_{2}), where ∂_{1} and ∂_{2} are bardifferentials on T (R_{+}) ⊗ M_{1} and M_{1} ⊗ T (R_{+}). Our assertion follows immediately from (i) by considering the spectral sequence associated with this bicomplex.

Now assume that for some n > 1 the assertion holds for all n′ < n. As before we view B^{•}(k, M_{1}, …, M_{n}, k) as the total complex of a bicomplex by defining the bidegree on T (R_{+}) ⊗ M_{1} ⊗ … ⊗ M_{n} ⊗ T (R_{+}) as follows:

deg_{0}(t_{0} ⊗ x_{1} ⊗ t_{1} … ⊗ x_{n} ⊗ t_{n}) = ∑_{i≡0(2)} deg(t_{i}),

deg_{1}(t_{0} ⊗ x_{1} ⊗ t_{1} … ⊗ x_{n} ⊗ t_{n}) = ∑_{i≡1(2)} deg(t_{i}).

Assume first that n is even. Then there is a spectral sequence associated with the above bicomplex abutting to the cohomology of B^{•}(k, M_{1}, …, M_{n}, k) and with the E_{1}-term

T (R_{+})⊗H^{∗}(M_{1} ⊗T (R_{+})⊗M_{2})⊗T (R_{+})⊗…⊗H^{∗}(M_{n−1} ⊗T (R_{+})⊗M_{n})⊗T (R_{+}).

Using (ii) we see that E_{1} is isomorphic to the complex of the form

B^{•}(k, M′_{1}, …, M′_{n/2}, k)[n(d + 1)/2],

so we can apply the induction assumption. If n is odd then we consider another spectral sequence associated with the above bicomplex, so that

E_{1} =H^{∗}(T (R_{+}) ⊗ M_{1}) ⊗ T (R_{+})⊗

H^{∗}(M_{2} ⊗ T (R_{+}) ⊗ M_{3}) ⊗ T (R_{+}) ⊗ … ⊗ H^{∗}(M_{n−1} ⊗ T (R_{+}) ⊗ M_{n}) ⊗ T (R_{+}).

By (i) and (ii) this complex is isomorphic to B^{•}(k, M′_{1}, …, M′_{(n−1)/2}, k)[(n + 1)(d + 1)/2]. Again we can finish the proof by applying the induction assumption.

We will also need the following simple lemma.

Lemma 3.2. Let C^{•} be a complex in an abelian category equipped with a decreasing filtration C^{•} = F^{0}C^{•} ⊃ F^{1}C^{•} ⊃ F^{2}C^{•} ⊃ … such that C^{n} = proj. lim_{i} C^{n}/F^{i}C^{n} for all n. Let gr_{i} C^{•} = F^{i}C^{•}/F^{i+1}C^{•}, i = 0, 1, … be the associated graded factors. Assume that H^{n} gr_{i} C^{•} = 0 for all i > 0 and for some fixed n. Then the natural map H^{n}C^{•} → H^{n} gr_{0} C^{•} is injective.

Proof. Considering an exact sequence of complexes

0 → F^{1}C^{•} → C^{•} → gr_{0} C^{•} → 0

one can easily reduce the proof to the case H^{n} gr_{i} C^{•} = 0 for all i ⩾ 0. In this case we have to show that H^{n}C^{•} = 0. Let c ∈ C^{n} be a cocycle and let c_{i} be its image in C^{n}/F^{i}C^{n}. It suffices to construct a sequence of elements x_{i} ∈ C^{n−1}/F^{i}C^{n−1}, i = 1, 2, …, such that x_{i+1} ≡ x_{i} mod F^{i}C^{n−1} and c_{i} = d(i)x_{i}, where d(i) is the differential on C^{•}/F^{i}C^{•}. Since n-th cohomology of C^{•}/F^{1}C^{•} = gr_{0} C^{•} is trivial we can find x_{1} such that c_{1} = d(1)x_{1}. Then we proceed by induction: once x_{1}, …, x_{i} are chosen an easy diagram chase using the exact triple of complexes

0 → gr_{i} C^{•} → C^{•}/F^{i+1}C^{•} → C^{•}/F^{i}C^{•} → 0

and the vanishing of H^{n}(gr_{i} C^{•}) show that x_{i+1} exists.

Theorem 3.3. Under the assumptions of Theorem 1.1 one has HH_{0,md}^{i}(A) = 0 for i < m(d + 2) and dim HH_{0,d}^{d+2}(A) ⩽ 1, where A = A_{L}.

Proof. Set C^{i} = C_{0,md}^{i}(A) (see 2.2). Note that Hochschild differential maps C^{i} into C^{i+1} (since m_{2} preserves both gradings on A). Recall that the decomposition of A into graded pieces with respect to the cohomological degree has form A = R ⊕ M, where R has degree 0 and M = ⊕_{i∈ℤ}H^{d}(X, L^{i}) has degree d. The natural augmentation of A is given by the ideal A_{+} = R_{+} ⊕ M. Each of the spaces C^{i} decomposes into a direct sum C^{i} = C^{i}(0) ⊕ C^{i}(d), where C^{i}(0) ⊂ Hom(A_{+}^{⊗i}, R), C^{i}(d) ⊂ Hom(A_{+}^{⊗i}, M). More precisely, the space C^{i}(0) consists of linear maps

[T (R_{+}) ⊗ M ⊗ T (R_{+}) ⊗ … ⊗ M ⊗ T (R_{+})]_{i} → R  (3.1.3)

preserving the internal grading, where there are m factors of M in the tensor product and the index i refers to the total number of factors H^{∗}(L^{∗}) (so that the LHS can be considered as a subspace of A_{+}^{⊗i}). Similarly, the space C^{i}(d) consists of linear maps

[T (R^{+}) ⊗ M ⊗ T (R_{+}) ⊗ … ⊗ M ⊗ T (R_{+})]_{i} → M

preserving the internal grading, where there is m + 1 factors of M in the tensor product. Clearly, C^{•}(d) is a subcomplex in C^{•}, so we have an exact sequence of complexes

0 → C^{•}(d) → C^{•} → C^{•}(0) → 0.

Therefore, it suffices to prove that H^{i}(C^{•}(0)) = H^{i}(C^{•}(d)) = 0 for i < m(d+2), and that in the case m = 1 one has in addition H^{d+2}(C^{•}(d)) = 0 and dim H^{d+2}(C^{•}(0)) ⩽ 1.

To compute the cohomology of these two complexes we can use spectral sequences associated with some natural filtrations to reduce the problem to simpler complexes. First, let us consider the decomposition

C^{•}(0) = ∏_{j⩾0} C^{•}(0)_{j},

where C^{i}(0)_{j} ⊂ C^{i}(0) is the space of maps (3.1.3) with the image contained in H^{0}(L^{j}) ⊂ R. The differential on C^{•}(0) has form

δ(x_{j})_{j⩾0} = (∑_{j′⩽j} δ_{j′,j}x_{j′})_{j⩾0}

for some maps δ_{j′,j} : C^{•}(0)_{j′} → C^{•}(0)_{j}, where j′ ⩽ j. By Lemma 3.2 it suffices to prove that one has H^{i}(C^{•}(0)_{j}, δ_{j,j}) = 0 for i < m(d + 2) and all j, while for m = 1 one has in addition H^{d+2}(C^{•}(0)_{j}, δ_{j,j}) = 0 for j > 0 and dim H^{d+2}(C^{•}(0)_{0}, δ_{0,0}) ⩽ 1. But

(C^{•}(0)_{j}, δ_{j,j}) = Hom(K_{m,j}^{•}, R_{j})[−m],

where K_{m}^{•} = B^{•}(k, M, …, M, k) (m copies of M) and K_{m,j}^{•} is its j-th graded component with respect to the internal grading. Here we use the following convention for the grading on the dual complex: Hom(K^{•}, R)^{i} = Hom(K^{−i}, R). Therefore, Proposition 3.1(iv) implies that cohomology of C^{•}(0)_{j} is concentrated in degrees ⩾ m(d + 1) + m = m(d + 2). Moreover, for m = 1 the (d + 2)-th cohomology space is non-zero only for j = 0, in which case it is one-dimensional.

For the complex C^{•}(d) we have to use a different filtration (since M is not bounded below with respect to the internal grading). Consider the decreasing filtration on C^{•}(d) induced by the following grading on T (R^{+}) ⊗ M ⊗ T (R_{+}) ⊗ … ⊗ M ⊗ T (R_{+}):

deg(t_{1} ⊗ x_{1} ⊗ t_{2} ⊗ … ⊗ x_{m+1} ⊗ t_{m+2}) = deg(t_{1}) + deg(t_{m+2}),

where t_{i} ∈ T (R_{+}), x_{i} ∈ M, the degree of R_{+} is taken to be −1. The associated graded complex is

Hom_{gr}(T (R^{+}) ⊗ B^{•}(M, …, M) ⊗ T (R^{+}), M)[−m − 1],

where there are m + 1 factors of M in the bar-construction. It remains to apply Proposition 3.1(iii).

3.2. Some Massey products

In this subsection we will show the nontriviality of the canonical class of A_{∞}- structures on A_{L} and combine it with our computations of the Hochschild cohomology to prove the main theorem.

Note that the canonical class of A_{∞}-structures can be defined in a more general context. Namely, if C is an abelian category with enough injectives then we can define the canonical class of A_{∞}-structures on the derived category D^{+}(C) of bounded below complexes over C. Indeed, one can use the equivalence of D^{+}(C) with the homotopy category of complexes with injective terms and apply Kadeishvili’s construction to the dg-category of such complexes (see [10],1.2 for more details). In the case when C is the category of coherent sheaves the resulting A_{∞}-structure is strictly A_{∞}-isomorphic to the structure obtained using Čech resolutions (since the relevant dg-categories are equivalent). In this context we have the following construction of nontrivial Massey products.

Lemma 3.4. Let C be an abelian category with enough injectives,

0 → F_{0} →^{α_{1}} F_{1} →^{α_{2}} F_{2} → … →^{α_{n}} F_{n} → 0

be an exact sequence in C, where n ⩾ 2, and let β : F_{n} → F_{0}[n − 1] be a morphism in the derived category D^{b}(C) corresponding to the Yoneda extension class in Ext^{n−1}(F_{n}, F_{0}) represented by the above sequence. Assume that Ext^{j−i−1}(F_{j}, F_{i}) = 0 when 0 ⩽ i < j ⩽ n − 1. Then

m_{n+1}(α_{1}, …, α_{n}, β) = ± id_{F_{0}}

for any A_{∞}-structure (m_{i}) on D^{b}(C) from the canonical class.

Proof. Assume first that n = 2. Then m_{3}(α_{1}, α_{2}, β) is the unique value of the welldefined triple Massey product in D^{b}(C) (see [9], 1.1). Using the standard recipe for its calculation (see [2], IV.2) we immediately get that m_{3}(α_{1}, α_{2}, β) = id.

For general n we can proceed by induction. Assume that the statement is true for n − 1. Set F′_{n−1} = ker(α_{n}). Then we have exact sequences

0 → F_{0} →^{α_{1}} F_{1} → … → F_{n−2} →^{α′_{n−1}} F′_{n−1} → 0,  (3.2.1)
0 → F′_{n−1} →^{i} F_{n−1} →^{α_{n}} F_{n} → 0,  (3.2.2)

where m_{2}(α′_{n−1}, i) = i ∘ α′_{n−1} = α_{n−1}. By the definition, we have β = m_{2}(γ, β′), where β′ ∈ Ext^{n−2}(F′_{n−1}, F_{0}) and γ ∈ Ext^{1}(F_{n}, F′_{n−1}) are the extension classes corresponding to these exact sequences. Applying the A_{∞}-axiom to the elements (α_{1}, …, α_{n}, γ, β′) and using the vanishing of m_{n−i+2}(α_{i+1}, …, a_{n}, γ, β′) ∈ Ext^{i−1}(F_{i}, F_{0}) for 0 < i < n, we get

m_{n+1}(α_{1}, …, α_{n}, β) = ±m_{n}(α_{1}, …, α_{n−1}, m_{3}(α_{n−1}, α_{n}, γ), β′).  (3.2.3)

Furthermore, applying the A_{∞}-axiom to the elements (α′_{n−1}, i, α_{n}, γ) we get

m_{3}(α_{n−1}, α_{n}, γ) = ±m_{2}(α′_{n−1}, m_{3}(i, α_{n}, γ)).  (3.2.4)

Next, we claim that sequences (3.2.1) and (3.2.2) satisfy the assumptions of the lemma. Indeed, for (3.2.1) this is clear, so we just have to check that Hom(F_{n−1}, F′_{n−1}) = 0. The exact sequence (3.2.1) gives a resolution F_{0} → … → F_{n−2} of F′_{n−1} and we can compute Hom(F_{n−1}, F′_{n−1}) using this resolution. Now the required vanishing follows from our assumption that Ext^{n−i−2}(F_{n−1}, F_{i}) = 0 for 0 ⩽ i ⩽ n − 2. Therefore, we have

m_{3}(i, α_{n}, γ) = id .

Together with (3.2.4) this implies that

m_{3}(α_{n−1}, α_{n}, γ) = ±α′_{n−1}.

Substituting this into (3.2.3) we get

m_{n+1}(α_{1}, …, α_{n}, β) = ±m_{n}(α_{1}, …, α_{n−1}, α′_{n−1}, β′).

It remains to apply the induction assumption to the sequence (3.2.1).

Proof of Theorem 1.1. (i) Since the algebra A = A_{L} is concentrated in degrees 0 and d, the first potentially nontrivial higher product of an admissible A_{∞}-structure (m_{i}) on A is m_{d+2}. Therefore, by Lemma 2.2 for every such A_{∞}-structure (m_{i}) on A the map m_{d+2} induces a cohomology class [m_{d+2}] ∈ HH_{0,d}^{d+2}(A). We claim that if (m′_{i}) is another admissible A_{∞}-structure on A then (m_{i}) is strictly A_{∞}-isomorphic to (m′_{i}) if and only if [m_{d+2}] = [m′_{d+2}]. Indeed, this follows from Lemma 2.2 and from the vanishing of higher obstructions due to Theorem 3.3 (these obstructions lie in HH_{0,md}^{md+2}(A) where m > 1, and the vanishing follows since md+2 < m(d+2)). In particular, an admissible A_{∞}-structure (m_{i}) is nontrivial if and only if [m_{d+2}] ≠ 0. Since by Theorem 3.3 the space HH_{0,d}^{d+2}(A) is at most one-dimensional, it remains to prove the nontriviality of an admissible A_{∞}-structure from the canonical class. Replacing L by its sufficiently high power if necessary we can assume that there exists d+1 sections s_{1}, …, s_{d+1} ∈ H^{0}(L) without common zeroes. The corresponding Koszul complex gives an exact sequence

0 → O → O^{⊕(d+1)} ⊗_{O} L → O^{⊕\binom{d+1}{2}} ⊗_{O} L^{2} → … → O^{⊕(d+1)} ⊗_{O} L^{d} → L^{d+1} → 0.

By our assumptions this sequence satisfies the conditions required in Lemma 3.4, hence we get a nontrivial (d + 2)-ple Massey product for our A_{∞}-structure. (ii) Applying Lemma 2.3 we see that obstructions for connecting two strict A_{∞}- isomorphisms by a homotopy lie in ⊕_{m⩾1}HH_{0,md}^{md+1}(A). But this space is zero by Theorem 3.3.

Corollary 3.5. Under assumptions of Theorem 1.1 the space HH_{0,d}^{d+2}(A_{L}) is onedimensional.

Proof. Indeed, from Theorem 3.3 we know that dim HH_{0,d}^{d+2}(A_{L}) ⩽ 1. If this space were zero then the above argument would show that all admissible A_{∞}-structures on A_{L} are trivial. But we know that A_{∞}-structures on A_{L} from the canonical class are nontrivial.

Remark. One can ask whether there exists an A_{∞}-structure on A_{L} from the canonical class such that m_{n} = 0 for n > d + 2 or at least m_{n} = 0 for all sufficiently large n. However, even in the case of smooth curves of genus ⩾ 1 the answer is “no”. The proof can be obtained using the construction of a universal deformation of a coherent sheaf (when it exists) using the canonical A_{∞}-structure, outlined in [10]. For example, it is shown there that the products

m_{n+2} : H^{1}(O_{X})^{⊗n} ⊗ H^{0}(L^{n_{1}}) ⊗ H^{0}(L^{n_{2}}) → H^{0}(L^{n_{1}+n_{2}})

appear as coefficients in the universal deformation of the structure sheaf. The base of this family is Spec R, where R ≃ k[[t_{1}, …, t_{g}]] is the completed symmetric algebra of H^{1}(O_{X})^{∨}. If all sufficiently large products were zero, this family would be induced by the base change from some family over an open neighborhood U of zero in the affine space 𝔸^{g}. But this would imply that the embedding of Spec R into the Jacobian (corresponding to the isomorphism of R with the completion of the local ring of the Jacobian at zero) factors through U, which is false.

3.3. Proof of Theorem 1.2

Theorem 1.1(i) easily implies that every admissible A_{∞}-structure on A = A_{L} is (strictly) A_{∞}-isomorphic to some strictly unital A_{∞}-structure. Therefore, it is enough to prove our statement for strictly unital structures. Recall that the group of strict A_{∞}-isomorphisms HG is the group of coalgebra automorphisms of Bar(A_{L}) inducing the identity map A_{L} → A_{L} and preserving two grading on Bar(A_{L}) induced by the two gradings of A_{L}. Thus, we can identify HG with a subgroup of algebra automorphisms of the completed cobar-construction Cobar(A_{L}) = ∏_{n⩾0} T^{n}(A_{L}^{∗}[−1]) (our convention is that passing to dual vector space changes the grading to the opposite one).

By Theorem 1.1(ii) for every strict A_{∞}-automorphism f of an A_{∞}-structure m there exists a homotopy from f to the trivial A_{∞}-automorphism f^{tr}. Let α = α_{f}^{∗} be the automorphism of Cobar(A_{L}) corresponding to f and h = H^{∗} : Cobar(A_{L}) → Cobar(A_{L})[−1] be the map giving the homotopy from f to f^{tr}. The equations dual to (2.1.1) and (2.1.2) in our case have form

h(xy) = h(x)y ± α(x)h(y),
α = id +d ∘ h + h ∘ d,

where d is the differential on Cobar(A_{L}) associated with m. Recall that A_{L} = H^{0} ⊕ H^{1}, where H^{0} = ⊕_{n⩾0}H^{0}(X, L^{n}), H^{1} = ⊕_{n⩽0}H^{1}(X, L^{n}). Since h has degree −1 we have h((H^{1})^{∗}[−1]) = 0 and h((H^{0})^{∗}[−1]) ⊂ T̂((H^{1})^{∗}[−1]). Furthermore, since h preserves the internal degree, we have h(H^{0}(X, L^{n})^{∗}[−1]) = 0 for all n > 0. Let ϵ ∈ (H^{0})^{∗}[−1] ⊂ Cobar(A_{L}) be an element corresponding to the natural projection H^{0} → H^{0}(X, O_{X}) ≃ k. Then we have A_{L}^{∗}[−1] = kϵ ⊕ V, where V = (H^{1})^{∗}[−1] ⊕ (⊕_{n>0}H^{0}(X, L^{n}))^{∗}[−1], and h(V) = 0. Let ⟨V ⟩ ⊂ Cobar(A_{L}) be the subalgebra topologically generated by V. Then h vanishes on ⟨V ⟩. Also, for every x ∈ V we have

dx = ϵx + xϵ mod ⟨V ⟩

since our A_{∞}-structure is strictly unital. Hence, for x ∈ V we have

α(x) = x + h(dx) = x + h(ϵ)x − α(x)h(ϵ),

which implies that

α(x) = (1 + h(ϵ))x(1 + h(ϵ))^{−1}.

In particular, the restriction of α to the subalgebra T̂(H^{1}(X, O_{X})^{∗}[−1]) is the inner automorphism associated with the invertible element 1+h(ϵ) ∈ T̂(H^{1}(X, O_{X})^{∗}[−1]). On the other hand, we have

dϵ = ϵ^{2} mod ⟨V ⟩.

Hence,

α(ϵ) = ϵ + dh(ϵ) + h(ϵ)ϵ − α(ϵ)h(ϵ),

so that

α(ϵ) = (1 + h(ϵ))ϵ(1 + h(ϵ))^{−1} + dh(ϵ) · (1 + h(ϵ))^{−1}.

Thus, α is uniquely determined by h(ϵ). Also, by Lemma 2.1 h(ϵ) can be an arbitrary element of ∏_{n⩾1} T^{n}(H^{1}(X, O_{X})^{∗}[−1]).

References

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[6] K. Lefèvre-Hasegawa, Sur Les A_{∞}-Catégories, Thèse de Doctorat, Univeristé Paris 7, 2002, available at http://www.math.jussieu.fr/˜ lefevre/publ.html

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2003/n1a17/v5n1a17.(dvi,ps,pdf)

A. Polishchuk apolish@uoregon.edu

Department of Mathematics, University of Oregon, Eugene, OR 97403