A-modules over A-algebras and Hochschild cohomology for modules over algebras

Mathematical Notes, vol. 79, no. 5, 2006, pp. 664–674. Translated from Matematicheskie Zametki, vol. 79, no. 5, 2006, pp. 717–728. c Original Russian Text Copyright
2006 by M. V. Ladoshkin.

A∞-Modules over A∞-Algebras and Hochschild Cohomology for Modules over Algebras M. V. Ladoshkin Received July 1, 2003; in final form, June 23, 2005

Abstract—For each left graded module M over a graded algebra A , a Hochschild cochain complex S ∗ (A, M ) whose homology is responsible for the existence of nontrivial structures of A∞ -modules over A∞ -algebras on the given module is constructed. Key words: Hochschild cochain complex, A∞ -module over an A∞ -algebra, Hochschild cohomology, twisting cochain, Stasheff relations.

In [1], Stasheff considered chain complexes of A∞ -spaces and introduced the notion of A∞ -algebra. This notion has become an important tool for various computations in algebraic topology and homological algebra. For example, in [2], A∞ -algebras were applied to describe homologies of differential algebras. In particular, it was shown in [2] that the homology of any differential algebra over a field has the structure of a graded A∞ -algebra. It was also shown in [2] that, for any differential graded module over a differential algebra over a field, the cohomology of this module has the structure of a graded A∞ -module over an A∞ -algebra. On the other hand, in [3] and, later, in [4], a relationship was established between the set of all structures of A∞ -algebras on a fixed graded algebra and the Hochschild cohomology of this algebra, which was considered in [5, 6]. In particular, it was shown in [4, 3] that the vanishing of certain Hochschild cohomology groups means that the algebra under consideration admits only the structure of the trivial graded A∞ -algebra (up to isomorphism). In this paper, for each left graded module M over a graded algebra A , we construct a cochain Hochschild complex S ∗ (A, M ) whose homology is responsible for the existence of a nontrivial structure of an A∞ -module over an A∞ -algebra on the given module. The main result of the paper, Theorem 3.2, says that if the cohomology groups of this Hochschild complex S ∗ (A, M ) vanish in the dimensions (n, 1 − n) , i.e., H n,1−n (S ∗ (A, M )) = 0 , then the A-module M admits only the structure of the trivial graded A∞ -module over an A∞ -algebra (up to isomorphism). In all formulas in this paper, the signs are determined by the standard sign rule, which is used to evaluate tensor products of maps at tensor products of elements. 1. HOCHSCHILD COMPLEXES FOR MODULES OVER ALGEBRAS Let A be a graded algebra over a field, with multiplication π : A ⊗ A → A satisfying the associativity condition π(1 ⊗ π) = π(π ⊗ 1) . Recall that the cochain Hochschild complex (C ∗ (A, A), δ) for the algebra A is defined as [5] C ∗ (A, A) = 664




C n (A, A),

0001-4346/2006/7956-0664

C n (A, A) = Hom(A⊗n , A) ;

c
2006 Springer Science+Business Media, Inc.

(1)

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the coboundary operator δ : C n (A, A) → C n+1 (A, A) is given by δf = π(1 ⊗ f ) +




(−1)n+i f (1 ⊗ · · · ⊗ π ⊗ 1) + (−1)n+1 π(f ⊗ 1),

(2)

where the summation is over the numbers i of positions occupied by π . The homology of the complex (C ∗ (A, A), δ) is called the Hochschild cohomology and denoted by Hoch∗ (A, A) . We assume that the differential δ is compatible with the filtration of the Hochschild complex by powers of maps, i.e., δ : C n,k (A, A) → C n+1,k (A, A) , where C n,k (A, A) = Homk (A⊗n , A) . In [6], the following multiplications in the Hochschild complex were considered:

f ∪1 g =




f ∪ g = π(f ⊗ g), (−1)(n+m)i+(m−1)(i−1) f (1 ⊗ · · · ⊗ 1 ⊗ g ⊗ 1 · · · ⊗ 1),

where f ∈ C m (A, A) , g ∈ C n (A, A) , and the summation is over the positions occupied by g . In the same paper, a relationship between these multiplications and the standard multiplication in the Hochschild complex was established. Now, suppose that (M , µ) is a left graded module over the algebra (A, π) , where the structure map µ : A ⊗ M → M for the module M satisfies the condition µ(1 ⊗ µ) = µ(π ⊗ 1) . By analogy with the Hochschild complex of an algebra, which describes polyadic operations on the algebra, we construct a complex responsible for such operations on the module M over the algebra A . We call this new complex the Hochschild complex for the module M over the algebra A and denote it by S ∗ (A, M ) . We set S ∗ (A, M ) =




S n,k (A, M ),

where

S n,k (A, M ) = Homk (A⊗n ⊗ M , M ),

n,k

and define the differential δ : S n,k (A, M ) → S n+1,k (A, M ) by δf = (−1)n+1 µ(1 ⊗ f ) +


(−1)i+n+1 f (1 ⊗ · · · ⊗ π ⊗ · · · ⊗ 1) + f (1 ⊗ · · · ⊗ µ).

(3)

Let us show that the map defined by (3) is indeed, a differential: δ(δf ) = (−1)n+2 µ(1 ⊗ δf ) +




(−1)i+n+2 δf (1 ⊗ · · · ⊗ π ⊗ · · · ⊗ 1) + δf (1 ⊗ · · · ⊗ µ) ;

taking (3) into account, we obtain δ(δf ) = f (1 ⊗ · · · ⊗ µ(1 ⊗ µ)) + (−1)n+1 µ(1 ⊗ f (1 ⊗ · · · ⊗ µ))
+ (−1)i+n+1 f (1 ⊗ · · · ⊗ π ⊗ · · · ⊗ µ) + (−1)n+2 µ(1 ⊗ f (1 ⊗ · · · ⊗ µ))
+ (−1)n+2+n+1 µ(1 ⊗ µ(1 ⊗ f )) + (−1)n+2 (−1)n+i+2 µ(1 ⊗ f (1 ⊗ · · · ⊗ π ⊗ . . . 1))
+ (−1)i+n+2 f (1 ⊗ · · · ⊗ π ⊗ · · · ⊗ µ) + (−1)(n+1)+(n+2) f (1 ⊗ · · · ⊗ µ(π ⊗ 1))
+ (−1)i+n+2+n+2 µ(1 ⊗ f (1 ⊗ · · · ⊗ π ⊗ · · · ⊗ π ⊗ · · · ⊗ 1))
(−1)(i+n+2)+(k+n+1) f (1 ⊗ · · · ⊗ π ⊗ . . . 1) + (−1)n+3+n+1 µ(π ⊗ f ) +
+ (−1)i+n+2+i+n+1 f (1 ⊗ · · · ⊗ π(π ⊗ 1) ⊗ · · · ⊗ 1)
+ (−1)i+n+2+(i−1)+n+1 f (1 ⊗ · · · ⊗ π(1 ⊗ π) ⊗ · · · ⊗ 1)
+ (−1)i+n+2+(k−1+n+1) f (1 ⊗ · · · ⊗ π ⊗ · · · ⊗ π ⊗ · · · ⊗ 1), MATHEMATICAL NOTES

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where the summation is over the numbers i and k of positions occupied by π . Eliminating the terms which differ only in signs, we obtain δ(δf ) = f (1 ⊗ · · · ⊗ µ(1 ⊗ µ)) − µ(1 ⊗ µ(1 ⊗ f )) − f (1 ⊗ · · · ⊗ µ(π ⊗ 1))
+ µ(π ⊗ f ) + (−1)i+n+2+i+n+1 f (1 ⊗ · · · ⊗ π(π ⊗ 1) ⊗ · · · ⊗ 1)
+ (−1)i+n+2+(i−1)+n+1 f (1 ⊗ · · · ⊗ π(1 ⊗ π) ⊗ · · · ⊗ 1). Since the action µ of the module is compatible with multiplication π in the algebra and the latter is associative, it follows that δ(δf ) = 0, as required. Let us introduce multiplication in the Hochschild complex S ∗ (A, M ) for the module; we denote it by the same symbol ∪ as multiplication in the Hochschild complex for the algebra. For elements f ∈ S n,k (A, M ) and g ∈ S m,t (A, M ) , we define the element f ∪ g ∈ S n+m,t+k (A, M ) by f ∪ g = (−1)(n+m−2)(m−1) f (1 ⊗ · · · ⊗ 1 ⊗ g).

(4)

It is easy to see that this ∪-product is associative, i.e., (f ∪ g) ∪ h = f ∪ (g ∪ h) . Moreover, it satisfies the Leibniz formula δ(f ∪ g) = −δf ∪ g + (−1)n f ∪ δg.

(5)

Indeed, consider the left-hand side of (5): δ(f ∪ g) = (f ∪ g)(1 ⊗ · · · ⊗ µ) + (−1)n+m+1 µ(1 ⊗ (f ∪ g))
+ (−1)i+n+m+1 (f ∪ g)(1 ⊗ · · · ⊗ 1 ⊗ π ⊗ 1 · · · ⊗ 1) = (−1)(n+m−2)(m−1) f (1 ⊗ · · · ⊗ g(1 ⊗ · · · ⊗ µ)) + (−1)(n+m+1)+(n+m−2)(m−1) µ(1 ⊗ f (1 ⊗ · · · ⊗ g))
+ (−1)i+n+m+1+(n+m−2)(m−1) f (1 ⊗ · · · ⊗ g(1 ⊗ · · · ⊗ π ⊗ · · · ⊗ 1)). The right-hand side takes the form (−1)(δf ∪ g) = (−1)(−1)(n+m−1)(m−1) δf (1 ⊗ · · · ⊗ g) = (−1)((−1)(n+m−1)(m−1) f (1 ⊗ · · · ⊗ µ(1 ⊗ g))
+ (−1)(n+m−1)(m−1)+i+n+1 f (1 ⊗ · · · ⊗ 1 ⊗ π ⊗ 1 ⊗ · · · ⊗ g) + (−1)(n+m−1)(m−1)+(n+1) µ(1 ⊗ f (1 ⊗ · · · ⊗ g))) ; (−1)n (f ∪ δg) = (−1)n ((−1)(n+m−1)m f (1 ⊗ ... ⊗ δg)) = (−1)n ((−1)(n+m−1)m f (1 ⊗ · · · ⊗ g(1 ⊗ · · · ⊗ µ)) + (−1)(n+m−1)m+(m+1) f (1 ⊗ · · · ⊗ µ(1 ⊗ g))
+ (−1)(n+m−1)m+i+n+m+1 f (1 ⊗ · · · ⊗ g(1 ⊗ · · · ⊗ π ⊗ · · · ⊗ 1))). Comparing the obtained expressions for the right- and left-hand sides of the Leibniz formula (5), we see that this formula does indeed hold. MATHEMATICAL NOTES

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Note that the Hochschild complex C ∗ (A, A) for an algebra acts on the Hochschild complex S ∗ (A, M ) for a module over this algebra. Let us describe the structure map ∪1 : S ∗ (A, M ) ⊗ C ∗ (A, A) → S ∗ (A, M ) of this action. For f ∈ S m,k (A, M ) and α ∈ C n,t (A, A) , we define f ∪1 α ∈ S m+n−1,t+k (A, M ) by
(−1)i(n−1)+m+n−3 f (1 ⊗ · · · ⊗ 1 ⊗ α ⊗ · · · ⊗ 1). (6) f ∪1 α = The action ∪1 is compatible with the ∪-product in the Hochschild complex for the module over the algebra in the sense that (f ∪ g) ∪1 α = (−1)n f ∪ (g ∪1 α) + (f ∪1 α) ∪ g.

(7)

 The proof of (7) is based on the decomposition of the sum (f ∪ g)(1 ⊗ · · · ⊗ α ⊗ 1 ⊗ · · · ⊗ 1) on the left-hand side of (7) into two parts, each of which is responsible for one term on the right-hand side of (7). 2. THE COHOMOLOGY OF THE HOCHSCHILD COMPLEX FOR A MODULE OVER AN ALGEBRA Recall the basic definitions related to the notion of twisting cochains in Hochschild complexes for algebras [7]. A twisting cochain is an element of the form a = a3,−1 + a4,−2 + · · · + ai,2−i + · · · , where ai,2−i ∈ C i,2−i (A, A) , satisfying the condition δa = a ∪1 a . Here ∪1 is multiplication in the Hochschild complex for algebras. We denote the set of all twisting cochains by T ω(A, A) . Following [8], we say that two twisting cochains a and a are equivalent if there exists an element p = p2,−1 + p3,−2 + · · · + pi,1−i + · · · , where pi,1−i ∈ C i,1−i (A, A) , for which





a − a = δp − p ∪1 a + (−1)dim a −1 a ∪1 p + (−1)dim a −2 a ∪1 (p ⊗ p) + · · · . We denote the set T ω/ ∼ , where ∼ is the equivalence relation specified above, by D(A, A) . It was proved in [3] that if Hochn,2−n (A, A) = 0 for any n > 2 , then D(A, A) = 0 . Below we perform a similar construction for Hochschild complexes S ∗ (A, M ) for modules over algebras. Let a be a cochain from a Hochschild complex for algebras. Definition 2.1. An element of the form m = m2,−1 + m3,−2 + · · · + mi,1−i + · · · ,

where

mi,1−i ∈ S i,1−i (A, M ),

is called an a-element if −δm = m ∪ m + m ∪1 a,

(8)

where the ∪ and ∪1 products are defined by (4) and (6). We denote the set of all a -elements by T ω(a)(A, M ) .

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Definition 2.2. We say that two a -elements m and m are equivalent and write m ∼ m if there exists an element t = t1,−1 + t2,−2 + · · · + ti,−i + · · · , where ti,−i ∈ S i,−i (A, M ) , for which m − m = δt + t ∪1 a + m ∪ t + t ∪ m.

(9)

This relation between a -elements is an equivalence. Let us show this. 1. Reflexivity. It is easy to see that m ∼ m . The role of the intermediary element t is played by the zero element of the complex S ∗ . 2. Symmetry. Let us prove that if m ∼ m , then m ∼ m . Suppose that t = t1,−1 + t2,−2 + · · · + ti,−i + · · · is an intermediary element in the equivalence m ∼ m . Let us show that an intermediary element t = t 1,−1 + t 2,−2 + · · · + t i,−i + · · · in the equivalence m ∼ m , for which
t k ,−k ∪1 an−k+1,1−n+k m n,1−n − mn,1−n = δt n−1,1−n + +




k k ,1−k

m

∪ t n−k ,k−n +

k




t k ,−k ∪ m n−k ,1+k−n ,

(10)

k

can be defined by the recursive formula t 1,−1 = t1,−1 ,

t n,−n = tn,−n +

n−1


t i,−i ∪ tn−i,i−n ,

i=1

or, in index-free form,

t = −t + t ∪ t.

(11)

Since m − m = −(m − m) , we must show that δt + t ∪1 a + m ∪ t + t ∪ m = δt + t ∪1 a + m ∪ t + t ∪ m (we use the index-free form, in which the expression is shorter). Transforming the left-hand side of this formula, replacing t by its expressions given by (10) and (11), and applying (5) and (7), we obtain

((−1)j+1 δtj ∪ ti−j + tj ∪ δti−j ) − ((−1)k (tk ∪1 ai−j+1 ) ∪ tj−k + tk ∪ (tj−k ∪1 ai−j+1 )) j

−




j

(−1) mj ∪ ti−j +

j

−







j ,k

(−1)j+k mj ∪ (tk ∪ ti−j−k ) +

k ,j

(−1)k (tk ∪ tj−k ) ∪ mi−j +




(−1)j mj ∪ ti−j −




j

k ,j




tj ∪ mi−j

j

tj ∪ mi−j .

j

The substitution of the expressions for δt and δt obtained from (9) and (10) into this equality and elimination of the terms that differ only in signs yield



(tj + tj + (−1)k tk ∪ tj−k ) ∪ mi−j − (tj + tj + (−1)k tk ∪ tj−k ) ∪ mi−j . j

k

j

k

According to (10), this expression vanishes, as required. 3. Transitivity. Let us show that if m ∼ m and m ∼ m , then m ∼ m . If t and t are intermediaries in the equivalences m ∼ m and m ∼ m , respectively, then






t = t + t − (−1)dim t +dim t t ∪ t is an intermediary in the equivalence m ∼ m . This can be shown by direct calculation with the use of (5), (7), (10), and (11). Let D(A, M ) = T ω(a)(A, M )/ ∼ , where ∼ is the equivalence relation introduced above. The following proposition is used in what follows. MATHEMATICAL NOTES

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Theorem 2.1. If m ∈ T ω(a)(A, M ) and pn,−n ∈ S n,−n (A, M ) , then there exists an a -element m satisfying the following conditions: (1) mi,1−i = mi,1−i for i < n + 1 ; (2) mn+1,−n = mn+1,−n + δpn,−n ; (3) m ∼ m . Proof. We propose to show that m ∼ m and the intermediary element p in this equivalence is p = 0 + · · · + pn,−n + 0 + · · · . Consider relation (9) separately in each dimension. In the dimensions lower than n , the right-hand side of (9) vanishes, which is equivalent to (1). In the dimension (n + 1, −n) , we obtain mn+1,−n − mn+1,−n = δpn,−n , i.e., condition (2) holds. In the dimension (n + 2, −n − 1) , we have mn+2,−n−1 = mn+2,−n−1 + pn,−n ∪1 a3,−1 + m2,−1 ∪ pn,−n + pn,−n ∪ m2,−1 . Continuing this procedure, we obtain the expression mn+j ,−n−j+1 = mn+j ,−n−j+1 + pn,−n ∪1 a1+j ,−j+1 + mj ,−j+1 ∪ pn,−n + pn,−n ∪ mj ,−j+1 in dimension (n + j , −n − j + 1) . To complete the proof of the theorem, it remains to note that all terms mj ,−j+1 in the expression for mn+j ,−n−j+1 were defined at earlier steps.
The following theorem establishes a relationship between the cohomology of the Hochschild complex for a module over an algebra and the structure of the set of a -elements in this complex. Theorem 2.2. If H n,−n+1 (S ∗ (A, M )) = 0 for n > 1 , then D(A, M ) = 0 . Proof. We must show that, under the assumptions of the theorem, any a -element is equivalent to zero. Consider formula (8) for each dimension separately. In dimension (3, −1) , the left-hand side of (8) contains only δm2,−1 ; therefore, δm2,−1 = 0 , i.e., m2,−1 is a cocycle. By assumption, H n,−n+1 (S ∗ (A, M )) = 0 ; hence there exists a p1,−1 ∈ S 1,−1 (A, M ) for which m2,−1 = δp1,−1 , or m2,−1 − 0 = δp1,−1 . According to Theorem 2.1, we can construct an a -element m such that m2,−1 = 0 and m ∼ m . In dimension (4, −2) , we have δm3,−2 = m2,−1 ∪1 a3,−1 + m2,−1 ∪ m2,−1 ; since m2,−1 = 0 , it follows that δm3,−2 = 0 , and m3,−2 is a cocycle. By assumption, H 3,−2 (S ∗ (A, M )) = 0 ; hence there exists a p2,−2 such that m2,−1 − 0 = δp2,−2 . This situation is similar to that considered above. Continuing the construction, we obtain a sequence of a -elements equivalent to the given a-element but having zeros in several initial dimensions. At the last step, we obtain the zero a -element equivalent to the given one; the intermediary in the equivalence is p = p1,−1 + p2,−2 + · · · , where each pi,−i is determined by the equality δpi,−i = mi+1,−i obtained at the ith step.
3. A∞ -MODULES OVER A∞ -ALGEBRAS AND THE COHOMOLOGIES OF HOCHSCHILD COMPLEXES FOR MODULES OVER ALGEBRAS Recall some definitions related to graded A∞ -algebras and graded A∞ -modules over graded A∞ -algebras [1]. In all formulas of this section, summation is not only over the indices specified under the summation sign but also over the numbers of positions occupied by the operations in parentheses. An A∞ -algebra is defined as a graded module A endowed with operations {mi } : A⊗i → A satisfying the following conditions: mi ((A⊗i )q ) ⊂ Aq+i−2 , i ≥ 2,
(−1)(j+1)(k+1)+i mi−j+1 (1 ⊗ · · · ⊗ 1 ⊗ mj ⊗ · · · ⊗ 1) = 0. j

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(12)

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A morphism from an A∞ -algebra (A, {mi }) to an A∞ -algebra (A , {mi }) is a family of homomorphisms {fi : A⊗i → A } satisfying the conditions


fi ((A⊗i )q ) ⊂ Aq+i−1 ,

i ≥ 1,

(−1)(i−1)+(i−1)(k+1) fi−j+1 (1 ⊗ . . . 1 ⊗ mj ⊗ · · · ⊗ 1)

j

+




(13) (−1)k2 +k4 +... mt (fk1 ⊗ · · · ⊗ fkt ).

k1 +···+kt =i

An A∞ -module over an A∞ -algebra (A, {mi }) is a graded module M endowed with actions {pi : A⊗i ⊗ M → M } , where i ≥ 1 , satisfying the conditions pi : ((A⊗i ⊗ M )q ) ⊂ Mq−i+1 ,
(−1)k(j+1)+i pi−j+1 (1 ⊗ . . . 1 ⊗ mj ⊗ · · · ⊗ 1)
+ (−1)(i−1)(j−1) pi−j (1 ⊗ · · · ⊗ 1 ⊗ pj ) = 0.

(14)

A morphism of A∞ -modules (M , {pi }) and (M  , {pi }) over A∞ -algebras (A, {mi }) and (A , {mi }) , respectively, is a pair of families of homomorphisms 

{fi : A⊗i → A },

{gi : A⊗i ⊗ M → M  },

where i ≥ 0 and each {fi } is a morphism of A∞ -algebras satisfying the conditions  , gi (A⊗i ⊗ M )q ⊂ Mq−i

k(j−1)+i gi−j+1 (1 ⊗ · · · ⊗ 1 ⊗ mj ⊗ · · · ⊗ 1) + (−1)j(i+1)+1 gi−j (1 ⊗ · · · ⊗ pj ) (−1)
+ (−1)k2 +k4 +... pt (fk1 ⊗ · · · ⊗ fkt−1 ⊗ gkt ).

(15)

k1 +···+kt =i+1

Relations (12) and (14) are known as the Stasheff relations for graded A∞ -algebras and graded A∞ -modules over graded A∞ -algebras, respectively. In [4], a method for calculating the Berikashvili functor [9] was suggested; namely, it was proved that there is a bijection between the set of A∞ -algebra structures on a given bigraded algebra and a subcomplex of the Hochschild homology in dimension −1 . On the other hand, it was shown in [3] that there exists a bijection (up to isomorphism of A∞ -algebras) between the set (A, π)(∞) of A∞ -algebra structures on a given graded algebra (i.e., structures for which m1 = 0 and m2 = π , where π is multiplication in the algebra) and the set D(A, A) of twisting cochains in the Hochschild complex modulo the equivalence relation ∼ . Below we perform constructions similar to those of [3] for A∞ -modules over A∞ -algebras and a -elements of Hochschild complexes for modules over algebras. Consider the set of all graded A∞ -module structures over a fixed A∞ -algebra that exist on a left module M over a given algebra A with zero differential. We denote this set by (A, M , µ)(∞) , where M is the module over the algebra A and µ : A ⊗ M → M is the structure map of M . We say that two elements of this set are equivalent if there exists a morphism of A∞ -modules whose zero component g0 is the identity map. Let us prove the following theorem. Theorem 3.1. There is a bijection between the sets (A, M , µ)(∞) and D(A, M ) .

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Proof. Any element (M , {pi }) ∈ (A, M , µ)(∞) is an A∞ -module over the A∞ -algebra with fixed p1 = µ . Setting p˜ = p2 + p3 + · · · , we see that condition (8) from the definition of a -element in the Hochschild complex S ∗ (A, M ) for the module M over the algebra A coincides with the Stasheff relations for the A∞ -module (14), where a ∈ D(A, A) is a twisting cochain from the Hochschild complex for algebras; this cochain corresponds to the given A∞ -algebra. Conversely, each a -element determines p˜ , i.e., an A∞ -module structure on M . To complete the proof, it remains to show that two A∞ -module structures are equivalent if and only if the corresponding a -elements are equivalent. Let us prove this. Suppose that two elements (M , {pi }) and (M , {pi }) ∈ (A, M , µ)(∞) are equivalent, i.e., there exists a morphism from (M , {pi }) to (M , {pi }) whose zero component g0 is the identity map. Since the initial A∞ -algebra is the same for these elements, it follows that the maps {fi } from (13) and (15) have the form fi = 0 for i > 1.

f1 = id,

Thus, we can write condition (15) in the form  gi (A⊗i ⊗ M )q ∈ Mq−i ,

(−1)j(i+1)+1 gi−j (1 ⊗ · · · ⊗ pj ) (−1)k(j−1)+i gi−j+1 (1 ⊗ · · · ⊗ 1 ⊗ mj ⊗ · · · ⊗ 1) +
+ (−1)i+1 pt (1 ⊗ · · · ⊗ 1 ⊗ gkt ),

where the pi : A⊗i ⊗ M → M and pi : A⊗i ⊗ M  → M  are the actions of the A∞ -modules and the gi : A⊗i ⊗ M → M  are maps of A∞ -modules. Let us group together the terms corresponding to small dimensions:
(−1)k(j−1)+i gi−j+1 (1 ⊗ · · · ⊗ 1 ⊗ mj ⊗ · · · ⊗ 1) j>2

+




(−1)k+i gi−1 (1 ⊗ · · · ⊗ m2 ⊗ · · · ⊗ 1) +




(−1)j(i+1)+1 gi−j (1 ⊗ · · · ⊗ pj )

j>1

k i

+ (−1) gi−1 (1 ⊗ · · · ⊗ p1 ) + (−1)gi (1 ⊗ · · · ⊗ 1 ⊗ p0 ) + g0 (pi ⊗ 1) +

i−1


(−1)i−1 ps (1 ⊗ · · · ⊗ gi−s ) + p1 (1 ⊗ g) + pi (1 ⊗ · · · ⊗ g0 ) + p0 (gi ).

s=2

Note that g0 = id and p0 = p0 = 0 , because M and M  coincide as graded modules. Moreover, p1 = p1 = µ , because the A∞ -module structure is compatible with the graded module structure, and m2 = π , because the A∞ -algebra structure is compatible with the graded algebra structure. Applying these observations to the last expression, we obtain

(−1)k(j−1)+i gi−j+1 (1 ⊗ · · · ⊗ 1 ⊗ mj ⊗ · · · ⊗ 1) + (−1)k+i gi−1 (1 ⊗ · · · ⊗ π ⊗ · · · ⊗ 1) j>2

+




k

(−1)j(i+1)+1 gi−j (1 ⊗ · · · ⊗ pj ) + (−1)i gi−1 (1 ⊗ · · · ⊗ µ)

j>1

+

i−1


(−1)i−1 ps (1 ⊗ · · · ⊗ gi−s ) + µ(1 ⊗ g).

(16)

s=2

Using (3), (5), (6), and the notation introduced above, we write (16) in the short form


gi−j+1 ∪1 mj + δgi−1 gi−k ∪ pk − pi + pi + ps ∪ gi−s = 0. j

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This equality, the arbitrariness of i , and (9) mean that p and p are equivalent as a -elements, and g is the intermediary element in the equivalence. The proof of the converse assertion is similar, with the only difference that, instead of decomposing the sums from (15), we must supplement relations (9) by using the properties of associative algebras and modules over them.
Theorems 3.1 and 2.2 imply the following assertion. Theorem 3.2. If the homology groups H n,1−n (S ∗ (A, M )) of the Hochschild complex for the module M over the algebra A vanish for n > 1 , then any A∞ -module structure on M for the A∞ -algebra A is equivalent to the trivial structure. The following example demonstrates the application of Theorem 3.2 to calculations. Consider the polynomial ring Z2 [x, y] in two variables over the field Z2 as a module over the polynomial algebra Z2 [x] in one variable over the field Z2 . Suppose that the structure action µ : Z2 [x] ⊗ Z2 [x, y] → Z2 [x, y] is defined on the generators by

µ(xk ⊗ xt y q ) = xk+t y q .

(17)

Let us study the existence of a structure of an A∞ -module over an A∞ -algebra with trivial structure Z2 [x] on the module Z2 [x, y] . Suppose that f ∈ S ∗ (Z2 [x], Z2 [x, y]) and δf = 0 . Let us show that there exists an element h ∈ S ∗−1 (Z2 [x], Z2 [x, y]) for which δh = f . We set h(xk1 ⊗ · · · ⊗ xkn−1 ⊗ xt y q ) = f (xk1 ⊗ · · · ⊗ xkn−1 ⊗ xt ⊗ y q ). Let us prove that δh(xk1 ⊗ · · · ⊗ xkn ⊗ xt y q ) = f (xk1 ⊗ · · · ⊗ xkn ⊗ xt y q ). Bearing in mind that π is the usual multiplication of polynomials in the algebra Z2 [x] (i.e., π(xk ⊗ xn ) = xk+n ) and µ is defined by (17), we apply (3) (to shorten expressions, we use the notation xn+k y q = µ(xn ⊗ xk y q ) = xn (xk y q )): δh(xk1 ⊗ · · · ⊗ xkn ⊗ xkn+1 y q ) = xk1 h(xk2 ⊗ · · · ⊗ xkn ⊗ xkn+1 y q ) + h(xk1 ⊗ · · · ⊗ xkn−1 ⊗ xkn +kn+1 y q ) +

n−1


h(xk1 ⊗ · · · ⊗ xki +ki+1 ⊗ · · · ⊗ xn ⊗ xkn+1 y q )

i=1 k2

k1

= x f (x +

n−1


⊗ · · · ⊗ xkn ⊗ xkn+1 ⊗ y q ) + f (xk1 ⊗ · · · ⊗ xkn−1 ⊗ xkn +kn+1 ⊗ y q )

f (xk1 ⊗ · · · ⊗ xki +ki+1 ⊗ · · · ⊗ xn ⊗ xkn+1 ⊗ y q )

i=1

= xk1 f (xk2 ⊗ · · · ⊗ xkn ⊗ xkn+1 ⊗ y q ) +

n


f (xk1 ⊗ · · · ⊗ xki +ki+1 ⊗ · · · ⊗ xkn+1 ⊗ y q )

i=1

= f (xk1 ⊗ · · · ⊗ xkn ⊗ xkn+1 y q ). The last equality follows from the assumption δf = 0 ; in particular, 0 = δf (xk1 ⊗ · · · ⊗ xkn+1 ⊗ y q ) = xk1 f (xk2 ⊗ · · · ⊗ xkn+1 ⊗ y q ) +

n


f (xk1 ⊗ · · · ⊗ xki +ki+1 ⊗ · · · ⊗ xkn+1 ⊗ y q ) + f (xk1 ⊗ · · · ⊗ xkn ⊗ xkn+1 y q ).

i=1

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These calculations show that, for any f ∈ Ker δ n , there exists an h ∈ S ∗ (Z2 [x], Z2 [x, y]) such that δh = f . Thus, H n (S ∗ (Z2 [x], Z2 [x, y])) = 0 for n > 1 . Since H n (S ∗ (Z2 [x], Z2 [x, y])) =



H n,i (S ∗ (Z2 [x], Z2 [x, y])),

i

we have H n,1−n (S ∗ (Z2 [x], Z2 [x, y])) = 0. Applying Theorem 3.1 to this result, we see that Z2 [x, y] admits only the trivial A∞ -module structure (up to A∞ -module isomorphism). In conclusion, we show how the correspondence between equivalent a -elements with different twisting cochains can be described in terms of additional structures on the Hochschild complex for modules. Not only the Hochschild complex for algebras but also its tensor powers C ⊗n act on the Hochschild complex for modules. We denote this action by ∪1 , by analogy with the notation introduced in the second section. Let us describe this new action. Suppose that C ∗ (A, A) is a Hochschild complex for algebras, S ∗ (A, M ) is a Hochschild complex for modules, p1 , p2 , . . . , pk ∈ C 8 (A, A) , and m ∈ S ∗ (A, M ) ; then m ∪1 (p1 ⊗ p2 ⊗ · · · ⊗ pk ) =




m((1 ⊗ · · · ⊗ p1 ⊗ · · · ⊗ pk ) ⊗ · · · ⊗ 1).

(18)

Suppose also that a, a ∈ C ∗ (A, A) are equivalent cochains in the Hochschild complex, i.e., we have a, a ∈ [a] , where [a] ∈ D(A, A) . Definition 3.1. We say that an a -element m ∈ S ∗ (A, M ) and an a -element m ∈ S ∗ (A, M ) are equivalent and write m ∼ m if there exists a t ∈ S i,−i (A, M ) for which m − m = δt − t ∪1 a − t ∪ m + (−1)α (m ∪1 p) ∪ t + (−1)α (m ∪1 (p ⊗ p)) ∪ t + (−1)α (m ∪1 (p ⊗ p ⊗ p)) ∪ t + · · · ,

(19)

where α is the number of the position at which the first p occurs and p is the intermediary element in the equivalence a ∼ a . It can be shown that this relation is an equivalence. We denote the set of all a- and a -elements such that a ∼ a by T ω([a])(A, M ) and, as above, write D∗ (A, M ) instead of T ω([a])(A, M )/ ∼ . Consider the set of all structures of A∞ -modules over isomorphic A∞ -algebras (it is assumed that the modules and algebras coincide as graded objects); we denote this set by (A, M , µ)∗ (∞) . There is an equivalence relation (isomorphism of A∞ -modules over A∞ -algebras) on the set (A, M , µ)∗ (∞) . The following assertion is valid; its proof is similar to that of Theorem 2.2. Proposition 3.1. The sets (A, M , µ)∗ (∞)/ ∼ and D∗ (A, M ) are isomorphic. ACKNOWLEDGMENTS The author wishes to express his thanks to S. V. Lapin for his assistance. MATHEMATICAL NOTES

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REFERENCES 1. J. D. Stasheff, “Homotopy associativity of H-space: 1, 2,” Trans. Amer. Math. Soc., 108 (1963), no. 2, 275–313. 2. T. V. Kadeishvili, “The homology theory of fibered spaces revisited,” Uspekhi Mat. Nauk [Russian Math. Surveys], 35 (1980), no. 3(213), 183–188. 3. T. V. Kadeishvili, “The A∞ -algebra structure and the Hochschild and Harrison cohomologies,” Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 91 (1988), 19–27. 4. V. A. Smirnov, “ A∞ -structures and the functor D ,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 64 (2000), no. 5, 145–162. 5. M. Barr, “Harrison homology, Hochschild homology and triples,” J. Algebra, 8 (1968), 314–323. 6. M. Gerstenhaber, “The homology structure of an associative ring,” Ann. of Math., 4 (1963), 267–288. 7. E. Brown, Jr., “Twisted tensor products,” Ann. of Math., 69 (1959), 223–246. 8. V. A. Smirnov, “The functor D for twisted tensor products,” Mat. Zametki [Math. Notes], 20 (1976), no. 4, 465–472. 9. N. A. Berikashvili, “Differentials of spectral sequences,” Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 51 (1976), 1–105. N. P. Ogarev Mordovian State University

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