BIMODULE COMPLEXES VIA STRONG HOMOTOPY ACTIONS

BERNHARD KELLER

To Professor Klaus W. Roggenkamp on the occasion of his sixtieth birthday

Abstract. We present a new and explicit method for lifting a tilting complex to a bimodule complex. The key ingredient of our method is the notion of a strong homotopy action in the sense of Stasheff.

1. Introduction

Let A and B be (associative, unital) algebras over a commutative ring k. Denote by Mod A the category of (right) A-modules. Suppose that P is a B-module endowed with an algebra morphism

A → Hom_{Mod B}(P, P).

Then P becomes an A-B-bimodule and we have the tensor functor

? ⊗_{A} P : Mod A → Mod B ,

which takes the free A-module A_{A} to P. This is the basic fact which allows us to construct Morita equivalences Mod A →^{∼} Mod B.

Now let DB = D Mod B be the (unbounded) derived category of the category of B-modules. Suppose that we have a complex T ∈ DB and an algebra morphism

A → Hom_{DB}(T, T).

It is not clear whether this map comes from an action of A on the components of T, even after replacing T by an isomorphic object of DB. Therefore, the (derived) tensor product by T not well-defined and the analogy with the case of module categories seems to break down. Nevertheless, for complexes T satisfying the ‘Toda condition’

Hom_{DB}(T, T [−n]) = 0 , for all n > 0 ,

J. Rickard succeeded [8] in constructing a triangle functor F : D^{−}A → DB taking A_{A} to T, where D^{−}A ⊂ DA denotes the subcategory of right bounded complexes. This construction was at the basis of the proof of his ‘Morita theorem for derived categories’. Later he showed [9] that if F restricts to an equivalence between the bounded derived categories (and suitable flatness hypotheses hold), then after replacing T by an isomorphic object, it is possible to lift the A-action to the components of T. In his proof, he used the functor F constructed in [8].

Date: September 20, 1999, last modified on January 14, 2000.

1991 Mathematics Subject Classification.

Key words and phrases. A-infinity algebra, Derived category, Tilting.

In [3] and [4], we gave an a priori construction of a lift (up to isomorphism) of T to a complex of bimodules X (under suitable flatness hypotheses). This made it possible to define F = L(? ⊗_{A} X) and to give a new proof [10, Ch. 8] of Rickard’s Morita theorem.

Thus, up to now, there have been two constructions of a bimodule complex X from a complex T as above. Neither of them is very explicit: the first one [9] uses the functor F; the second one [3] uses resolutions over differential graded algebras.

In this paper, we present a new construction, which is surprisingly explicit. In fact, if we assume that T is a right bounded complex of projective B-modules, then essentially the only data we need are homotopies H(f) such that

f = d ∘ H(f) + H(f) ∘ d

for each morphism of complexes f : T → T [−n], n > 0. The details of the construction are concisely presented in section 9. Its relation to Rickard’s original construction is explained in section 10. We also prove a unicity result which improves on [9] and [3].

The essential new ingredient of our method is the notion of a strong homotopy action (=A_{∞}-action) due to Stasheff [11], [12] and Kadeishvili [2], and recently popularized again by Kontsevich [6], [7]. The present article is self-contained but the interested reader may find more information and references on strong homotopy methods in [5]. We will show that if A is projective over k and T is right bounded with projective components and satisfies the Toda condition, then the ‘action up to homotopy’ of A on T may be enriched to a strong homotopy action. It is remarkable that this can be done without changing the underlying complex of T. In a second step, we show that each strong homotopy action on a complex K yields a strict action on a larger (but quasi-isomorphic) complex K′. In fact, K′ may be viewed as a ‘perturbed Hochschild resolution’ of the complex K.

Acknowledgments. The author wishes to thank Prof. K. W. Roggenkamp for his stimulating interest and encouragment.

The work on this article was begun during a stay of the author at the Sonderforschungsbereich ‘Diskrete Strukturen in der Mathematik’ at the University of Bielefeld. The author thanks the Bielefeld representation theory group, and in particular C. M. Ringel and P. Draexler, for their kind hospitality.

The author thanks the referee for his careful reading of the manuscript and his suggestions for improvements.

Contents

1. Introduction 1 2. The main theorem 3 3. First special case 4 4. Second special case 5 5. Proof of unicity 6 6. From homotopy actions to strong homotopy actions 7 6.1. Lifting homotopy actions 7 6.2. Differential coalgebra 7 6.3. Proof of theorem 6.1 9 6.4. Morphisms of strong homotopy actions 10 7. From strong homotopy actions to strict actions 12 8. Proof of existence 15 9. Summary of the construction 15 10. Comparison with Rickard’s construction 17 References 18

2. The main theorem

Let k be a commutative ring and B an (associative, unital) k-algebra. Let T be a complex of (right) B-modules. Let A be another k-algebra. A strict action of A on T is an homomorphism (preserving the unit)

A → Hom_{CB}(T, T) ,

where CB is the category of complexes of B-modules. Equivalently, a strict action is the datum of a complex of A-B-bimodules whose restriction to B equals T. An homotopy action of A on T is an homomorphism

α : A → Hom_{HB}(T, T) ,

where HB is the homotopy category of right B-modules. The pair (T, α) is then an homotopy module. If T and T′ are endowed with homotopy actions α and α′, a morphism of complexes f : T → T′ is compatible with these actions if f ∘ α(a) is homotopic to α′(a) ∘ f for all a ∈ A. Then f is also called a morphism of homotopy modules.

Clearly, each strict action yields an homotopy action. The converse is false, in general. However, we have the

Theorem 2.1. Let k be a commutative ring and A, B two (associative, unital) kalgebras. Let T be a complex of right B-modules endowed with an homotopy action by A. Suppose that A is projective as a k-module, and that T is right bounded with projective components and satisfies

(2.1)  Hom_{HB}(T, T [−n]) = 0 for all n > 0.

a) There is a right bounded complex of projective B-modules X endowed with a strict action by A and a quasi-isomorphism φ : T → X of complexes of B-modules compatible with the homotopy actions by A.

b) If φ : T → X and φ′ : T → X′ are two quasi-isomorphisms as in a), then there is a unique isomorphism ψ : X → X′ in the derived category of A-B-bimodules such that we have ψ ∘ φ = φ′ in the homotopy category of B-modules.

For the case where T is a tilting complex, the theorem was proved by J. Rickard in [9]. The general case is proved in [3]. We give a new proof which, compared to these previous approaches, is much more explicit. We illustrate this by two special cases. The general method is concisely presented in section 9.

3. First special case

Suppose that the assumptions of theorem 2.1 hold. Let us assume that T has non vanishing components at most in degrees 0 and 1:

T = (… → 0 → T_{1} → T_{0} → 0 → …).

We will construct a quasi-isomorphism φ : T → X as in theorem 2.1 a).

Since A is projective over k, we can find a k-linear map α̃ : A → Hom_{CB} (T, T) lifting the given homotopy action α : A → Hom_{HB} (T, T). Now we define m_{2} : A ⊗ T → T by

m_{2}(a, x) = (α̃(a))(x) , a ∈ A, x ∈ T.

Now we have to take into account the non-associativity of m_{2}: Consider the square

A ⊗ A ⊗ T →^{1_{A}⊗m_{2}} A ⊗ T
m_{A} ⊗ 1_{T} ↓  ↓ m_{2}
A ⊗ T →^{m_{2}} T ,

where m_{A} denotes the multiplication of A. The square becomes commutative in the homotopy category. Hence there is a morphism of graded B-modules

m_{3} : A ⊗ A ⊗ T → T

homogeneous of degree −1 such that we have

m_{2}(ab, x) − m_{2}(a, m_{2}(b, x)) = −m_{3}(a, b, d(x)) − d(m_{3}(a, b, x))

for all a, b ∈ A and x ∈ T. We construct a complex X̃ as follows: The underlying graded module of X̃ is

A ⊗ A ⊗ A ⊗ T [2] ⊕ A ⊗ A ⊗ T [1] ⊕ A ⊗ T.

The differential is given by

d(a, b, c, x) = −ab ⊗ c ⊗ x + a ⊗ bc ⊗ x
+ a ⊗ m_{3}(b, c, x) − a ⊗ b ⊗ m_{2}(c, x) + a ⊗ b ⊗ c ⊗ d(x) ,
d(a, b, x) = −ab ⊗ x + a ⊗ m_{2}(b, x) − a ⊗ b ⊗ d(x) ,
d(a, x) = a ⊗ d(x) ,

where a, b, c ∈ A, x ∈ T. We define the complex X to be the truncation

X = τ_{≤1}X̃ = (X̃_{1}/ im d_{2} → X̃_{0}).

We define the morphism of complexes f : T → X̃ by f(x) = 1 ⊗ x. This morphism is compatible with the homotopy actions by A. Indeed, if we define the graded morphism f_{2} : A ⊗ T → X̃ of degree −1 by

f_{2}(a ⊗ x) = 1 ⊗ a ⊗ x ∈ A ⊗ A ⊗ T [1] , a ∈ A, x ∈ T ,

then we have

d(f_{2}(a, x)) + f_{2}(a, d(x)) = f(m_{2}(a, x)) − a f(x) , a ∈ A , x ∈ T.

By composition, f yields a morphism T → X. Its homotopy class is the required quasi-isomorphism φ. Since T → X̃ is compatible with the homotopy action by A and X̃ → X is a morphism of complexes of bimodules, the morphism φ is compatible with the homotopy action by A.

4. Second special case

Suppose that the assumptions of theorem 2.1 hold. Let us assume that T has non vanishing components at most in degrees 0, 1 and 2:

T = (… → 0 → T_{2} → T_{1} → T_{0} → 0 → …).

We will construct a complex of bimodules X and a quasi-isomorphism φ : X → T as in theorem 2.1 a). For this, we construct a morphisms m_{2}, m_{3} as in section 3. Now consider the graded morphism c : A^{⊗3} ⊗ T → T defined by

c(a, b, c, x) = m_{3}(ab, c, x) − m_{3}(a, bc, x) + m_{3}(a, b, m_{2}(c, x)) − m_{2}(a, m_{3}(b, c, x)).

A computation shows that c defines a morphism of complexes A^{⊗3} ⊗ T → T [−1]. By the (2.1), there exists a graded morphism m_{4} : A^{⊗3} ⊗ T → T homogeneous of degree −2 such that we have

c(a, b, c, x) = m_{4}(a, b, c, d_{T} (x)) − d_{T} (m_{4}(a, b, c, x)) ,

a, b, c ∈ A, x ∈ T. We define X̃ to be the complex whose underlying graded module is

⨁_{i=1}^{4} A^{⊗i} ⊗ T [i − 1]

and whose differential is given by

d(a_{0}, …, a_{3}, x) = −(a_{0}a_{1}, a_{2}, a_{3}, x) + (a_{0}, a_{1}a_{2}, a_{3}, x) − (a_{0}, a_{1}, a_{2}a_{3}, x)
+ (a_{0}, m_{4}(a_{0}, a_{1}, a_{2}, a_{3}, x)) − (a_{0}, a_{1}, m_{3}(a_{2}, a_{3}, x))
+ (a_{0}, a_{1}, a_{2}, m_{2}(a_{3}, x)) − (a_{0}, a_{1}, a_{2}, a_{3}, d(x))
d(a_{0}, a_{1}, a_{2}, x) = −(a_{0}a_{1}, a_{2}, x) + (a_{0}, a_{1}a_{2}, x)

+ (a_{0}, m_{3}(a_{1}, a_{2}, x)) − (a_{0}, a_{1}, m_{2}(a_{2}, x)) + (a_{0}, a_{1}, a_{2}, d(x)) ,

d(a_{0}, a_{1}, x) = −(a_{0}a_{1}, x) + (a_{0}, m_{2}(a_{1}, x)) ,
d(a_{0}, x) = (a_{0}, d(x)).

We define f : T → X̃ by x ↦ 1 ⊗ x and we define X to be the truncation

X = τ_{≤2}X̃.

The homotopy class of the composition T → X̃ → X is the required morphism φ. As in section 3, one checks that φ is compatible with the homotopy actions by A.

5. Proof of unicity

We will prove part b) of the main theorem. This could be done by strong homotopy methods as well. The following argument is shorter but less explicit.

Let us first observe that φ and φ′ are homotopy equivalences of complexes of B-modules. So there is a unique morphism f : X → X′ of HB such that f ∘ φ = φ′ in the homotopy category of B-modules. Of course, f is a morphism of homotopy modules. We have to show that it lifts to a unique morphism X → X′ in the derived category of A-B-bimodules D(A^{op} ⊗ B). Let us compute Hom_{D(A^{op}⊗B)}(X, X′). The complex X is quasi-isomorphic to the (sum) total complex of its Hochschild resolution:

… → A ⊗ A^{⊗p} ⊗ X → … → A ⊗ X → 0 , p ≥ 0.

This total complex is right bounded and its components are projective over A^{op} ⊗B since A is projective over k and the components of X are projective over B. So we can compute R Hom_{A−B}^{•}(X, X′) by applying Hom_{A−B}^{•}(?, X′) to the total complex of the Hochschild resolution. Using the isomorphism

Hom_{A−B}^{•}(A ⊗ A^{⊗p} ⊗ X, X′) = Hom_{B}^{•}(A^{⊗p} ⊗ X, X′)

we find that R Hom_{A−B}^{•}(X, X′) isomorphic to the product total complex of the following double complex D

Hom_{B}^{•} (X, X′) → Hom_{B}^{•} (A ⊗ X, X′) → Hom_{B}^{•} (A ⊗ A ⊗ X, X′) → ….

We have to to compute H^{0} Tot^{Π} D. For this, we first truncate the columns of D: For a complex of k-modules K, let

τ_{≥0}K = (… → 0 → 0 → K^{0}/ im d^{−1} → K^{1} → K^{2} → …).

Let D_{≥0} be the double complex obtained by applying τ_{≥0} to each column of D and let D_{<0} be the kernel of D → D_{≥0}. We claim that D_{<0} is acyclic. Indeed, the homology of the p-th column of D_{<0} in degree −q is isomorphic to

Hom_{HB}(A^{⊗p} ⊗ X, X′[−q]).

This vanishes for −q < 0 by the projectivity of A over k and assumption (2.1). Hence each column of D_{<0} is acyclic. Moreover, D_{<0} is concentrated in the right half plane. We claim that the product total complex Tot^{Π} D_{<0} is acyclic. Indeed, this complex is the inverse limit of the sequence of the complexes A_{p} = Tot^{Π} F_{p}D_{<0}, p ≥ 0, associated with the column filtration F_{p}D_{<0}. The A_{p} are acyclic by induction on p. Each map A_{p+1} → A_{p} is surjective in each component. It follows that it is surjective in the boundaries and hence in the cycles (which equal the boundaries since the A_{p} are acyclic). Therefore the inverse limit of the A_{p} is acyclic, i.e. D_{<0} is acyclic. So the morphism D → D_{≥0} induces a quasi-isomorphism in the product total complexes. Hence it is enough to compute H^{0} Tot^{Π} D_{≥0}. It is straightforward to check that this group is canonically isomorphic to the group of morphisms of homotopy modules X → X′.

6. From homotopy actions to strong homotopy actions

6.1. Lifting homotopy actions. Suppose that k is a commutative ring, A and B are associative unital k-algebras, and L is a Z-graded B-module. A strong homotopy action [2] of A on L is the datum of graded (B-linear) morphisms

m_{n} : A^{⊗n−1} ⊗ L → L

defined for n ≥ 1 and homogeneous of degree 2 − n such that for each n ≥ 1 and all a_{i} ∈ A, x ∈ L, we have

\begin{aligned}0 & = ∑_{i=1}^{n−2}(−1)^{i−1}m_{n−1}(a_{1}, …, a_{i}a_{i+1}, …, a_{n−1}, x)+ \\ & ∑_{l=1}^{n}(−1)^{n−l}m_{n−l+1}(a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x)). \end{aligned} ; (6.1)

Note that if L has non vanishing components only in degrees 0, …, N, then m_{n} vanishes for n > N + 2. It is instructive to consider the cases n = 1, 2, 3 of (6.1): For n = 1, we obtain

0 = m_{1}m_{1}

so that (L, m_{1}) is a complex. For n = 2, we have

0 = −m_{2}(a, m_{1}(x)) + m_{1}(m_{2}(a, x)) , a ∈ A, x ∈ L ,

so that x ↦ m_{2}(a, x) is a morphism of complexes for all a ∈ A. For n = 3, a, b ∈ A, x ∈ L, we have

(6.2) 0 = m_{3}(a, b, m_{1}(x)) + m_{2}(ab, x) − m_{2}(a, m_{2}(b, x)) + m_{1}(m_{3}(a, b, x)) ,

which expresses the fact that m_{2} is an associative operation up to an homotopy given by m_{3}.

Theorem 6.1. Suppose that L is a graded B-module endowed with three graded morphisms m_{i} : A^{⊗i−1} ⊗ L → L, 1 ≤ i ≤ 3, homogeneous of degree 2 − i and satisfying (6.1) for n ≤ 3. Suppose that we have

(6.3)  Hom_{HB} (A^{⊗n} ⊗ L, L[2 − n]) = 0 for all n ≥ 3.

Then the triple m_{1}, m_{2}, m_{3} may be completed to a strong homotopy action m_{n}, n ≥ 1, of A on L.

In the next subsection, we will set up the dictionary between strong homotopy and differential coalgebra. We will then prove the theorem in 6.3 using this dictionary.

6.2. Differential coalgebra. Passing from strong homotopy notions to differential coalgebra notions is a classical device, cf. [11], [12], [1]. In this subsection, we adapt it to our needs.

Let

C = T (A[1]) = k ⊕ A[1] ⊕ (A[1] ⊗ A[1]) ⊕ …

be the graded tensor algebra over the graded k-module A[1]. It becomes a graded coalgebra for the comultiplication defined by

\begin{matrix}Δ(a_{1}, …, a_{n}) = & 1 ⊗ (a_{1}, …, a_{n}) + ∑_{i=1}^{n−1}(a_{1}, …, a_{i}) ⊗ (a_{i+1}, …, a_{n}) \\ + (a_{1}, …, a_{n}) ⊗ 1. \end{matrix}

The graded coalgebra T (A[1]) admits a unique graded endomorphism b of degree +1 which satisfies

b(x_{1}, x_{2}) = x_{1}x_{2} for x_{1}, x_{2} ∈ A ,
b(x_{1}, …, x_{i}) = 0 for i ≠ 2 ,

and which is a coderivation:

Δ ∘ b = (b ⊗ 1_{C} + 1_{C} ⊗ b) ∘ Δ.

Here and elsewhere, we use the graded tensor product: For graded maps f, g and homogeneous elements x, y, we have

(f ⊗ g)(x ⊗ y) = (−1)^{|g||x|}f(x) ⊗ g(y).

where the bars indicate the degree. Explicitly, the formula for b is

b(x_{1}, …, x_{n}) = ∑_{i=1}^{n−1}(−1)^{i−1}(x_{1}, …, x_{i}x_{i+1}, …, x_{n}) , n ≥ 2.

We have b^{2} = 0 so that (C, d) is a differential graded coalgebra.

Now let L be a Z-graded B-module. Let X be the graded B-module C ⊗ L. So we have

X = L ⊕ (A[1] ⊗ L) ⊕ ⋯ ⊕ (A[1]^{⊗i} ⊗ L) ⊕ ⋯, i ∈ N.

The graded module X = C ⊗ L becomes a cofree graded comodule over C for the comultiplication induced from that of C. So we have

\begin{matrix}δ(a_{1}, …, a_{n−1}, x) = 1 ⊗ (a_{1}, …, a_{n−1}, x)+ & a_{1} ⊗ (a_{2}, …, a_{n−1}, x) + … \\ + (a_{1}, …, a_{n−1}) ⊗ x. \end{matrix}

A coderivation of X of degree e is a graded endomorphism b of degree e such that

δ ∘ b = b_{C} ⊗ 1_{X} + 1_{C} ⊗ b.

Let ε : X → L be the projection. Then the map b ↦ ε ∘ b is a bijection from the set of degree e coderivations of X to the set of graded morphisms of degree e from X to L. Let us describe the inverse map: Let b : X → L be a graded morphism of degree e. It is given by its components (homogeneous of degree e)

b_{i} : A[1]^{⊗i−1} ⊗ L → L , i ≥ 1.

The corresponding coderivation is given by

\begin{aligned}b(a_{1}, …, a_{n−1}, x) & = ∑_{i=1}^{n−2}(−1)^{e(i−1)}(a_{1}, …, a_{i}a_{i+1}, …, a_{n−1}, x) \\ & + ∑_{l=1}^{n}(−1)^{e(n−l)}(a_{1}, …, a_{n−l}, b_{l}(a_{n−l+1}, …, a_{n−1}, x)) , \end{aligned} ; (6.4)

where a_{i} ∈ A, x ∈ L.

Let U be a C-comodule. Since X is cofree, the map f ↦ ε ∘ f is a bijection from the degree e comodule morphisms f : U → X to the degree e graded k-linear morphisms U → L.

Let b be a degree 1 coderivation of C ⊗ X. Then we have

(1_{C} ⊗ b)(b_{C} ⊗ 1_{X}) = −(b_{C} ⊗ 1_{X})(1_{C} ⊗ b)

as morphisms C ⊗ X → C ⊗ X. Since b_{C}^{2} = 0, we deduce that

b^{2} ∘ δ = (1_{C} ⊗ b + b_{C} ⊗ 1_{X})^{2} ∘ δ = (1_{C} ⊗ b^{2}) ∘ δ.

It follows that b^{2} : X → X is a morphism of C-comodules. In particular, we have b^{2} = 0 iff ε ∘ b^{2} = 0. Thanks to (6.4), this last equality translates into

\begin{aligned}0 = & ∑_{i=1}^{n−2}(−1)^{(i−1)}b_{n−1}(a_{1}, …, a_{i}a_{i+1}, …, a_{n−1}, x) \\ & + ∑_{l=1}^{n}(−1)^{(n−l)}b_{n−l+1}(a_{1}, …, a_{n−l}, b_{l}(a_{n−l+1}, …, a_{n−1}, x)) , \end{aligned} ; (6.5)

for all n ≥ 1. If we compare (6.1) to (6.5), we see that the map (m_{n}) ↦ (b_{n}) defined by

b_{n}(a_{1}, …, a_{n−1}, x) = m_{n}(a_{1}, …, a_{n−1}, x)

is a bijection between strong homotopy actions of A on L and degree 1 comodule differentials on X. Note that m_{i} is a graded map A^{⊗i−1} ⊗ L → L of degree 2 − i whereas b_{i} is a graded map A[1]^{⊗i−1} ⊗ L → L of degree 1.

Now assume that b is a degree 1 coderivation of X. Let us analyse the equation b^{2} = 0 in terms of the components b_{i} : A[1]^{⊗i−1} ⊗ L → L. If we identify them with their extensions to coderivations, then the equation b^{2} = 0 translates into

(6.6)  0 = b_{1}b_{n} + b_{2}b_{n−1} + … + b_{n}b_{1}

for all n ≥ 1. Note that for each p ≥ 0, the right hand side takes A[1]^{⊗n+p} ⊗ L to A[1]^{⊗p} ⊗ L. The equation b^{2} = 0 holds iff, for all n ≥ 1, the right hand side of (6.6) induces the zero map A[1]^{⊗n−1} ⊗ L → L. Indeed, in this case we have ε ∘ b^{2} = 0.

6.3. Proof of theorem 6.1. We use the notations of the previous two subsections. If we identify graded maps A^{⊗i−1} ⊗ L → L with their extensions to coderivations X → X, then we are given coderivations b_{1}, b_{2}, b_{3} such that we have

0 =b_{1}^{2} ,
0 =b_{1}b_{2} + b_{2}b_{1} ,
0 =b_{1}b_{3} + b_{2}b_{2} + b_{3}b_{1}.

We have to construct b_{i}, i ≥ 4, such that

(6.7)  b_{1}b_{n} + b_{2}b_{n−1} + ⋯ b_{n−1}b_{2} + b_{n}b_{1} = 0

for all n ≥ 4. Suppose that N ≥ 4 and that b_{1}, …, b_{N−1} have been constructed such that (6.7) holds for all n ≤ N − 1. We are looking for b_{N} such that

0 = b_{1}b_{N} + b_{N}b_{1} + (b_{2}b_{N−1} + b_{3}b_{N−2} + ⋯ + b_{N−1}b_{2}).

Put

c = b_{2}b_{N−1} + b_{3}b_{N−2} + ⋯ + b_{N−1}b_{2}.

Let X_{N} ⊂ X be the C-subcomodule

L ⊕ (A[1] ⊗ L) ⊕ … ⊕ (A[1]^{⊗N−1} ⊗ L).

Note that b_{i} takes X_{N} to X_{N−i+1}. In particular, c takes X_{N} to X_{1} = L and vanishes on X_{N−1}. So it induces a graded morphism of degree 2

X_{N}/X_{N−1} = A[1]^{⊗N−1} ⊗ L → L.

We only have to show that this is a morphism of complexes: Indeed, by our assumption on L, it will then have to be nullhomotopic. The extension of an homotopy to a coderivation yields the required morphism b_{N}. To show that the morphism induced by c commutes with the differential b_{1}, we define B = b_{1} + ⋯ + b_{N−1}. Since (6.7) holds for n ≤ N − 1, we have

(6.8)  B^{2} ≡_{N} c

where ≡_{N} denotes the equality of the restrictions to X_{N}. We conclude that B^{2} vanishes on X_{N−1} and takes X_{N} to L ⊂ X_{N}. This implies that B^{2}B ≡_{N} B^{2}b_{1} and BB^{2} ≡_{N} b_{1}B^{2}. Therefore we have B^{2}b_{1} ≡_{N} b_{1}B^{2} and finally cb_{1} ≡_{N} b_{1}c.

6.4. Morphisms of strong homotopy actions. Suppose that k is a commutative ring, A and B are associative unital k-algebras, and L and M are strong homotopy modules, i.e. Z-graded B-modules endowed with strong homotopy actions by A. A morphism of strong homotopy modules f : L → M is a sequence of graded (B-linear) morphisms

f_{i} : A^{⊗i−1} ⊗ L → M

homogeneous of degree 1 − i such that for each n ≥ 1, we have

\begin{aligned}& ∑_{l=1}^{n} m_{l}(a_{1}, …, a_{l−1}, f_{n−l+1}(a_{l}, …, a_{n−1}, x)) \\ = & ∑_{i=1}^{n−2}(−1)^{i−1}f_{n−1}(a_{1}, …, a_{i}a_{i+1}, …, a_{n−1}, x) \\ & + ∑_{l=1}^{n}(−1)^{n−l}f_{n−l+1}(a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x)) , \end{aligned} ; (6.9)

for all a_{i} ∈ A, x ∈ L. For n = 1, this specializes to

m_{1}f_{1} = f_{1}m_{1}

so that f_{1} is a morphism of complexes. For n = 2, we obtain (6.10) m_{1}(f_{2}(a_{1}, x)) + m_{2}(a_{1}, f_{1}(x)) = f_{1}(m_{2}(a_{1}, x)) − f_{2}(a_{1}, m_{1}(x)) , a_{1} ∈ A , x ∈ L ,

which means that for each a_{1} ∈ A, the morphism f_{1} commutes with the left multiplication by a_{1} up to the homotopy x ↦ f_{2}(a_{1}, x).

A morphism f : L → M of strong homotopy modules is nullhomotopic if there exists an homotopy from f to 0, i.e. a family

h_{i} : A^{⊗i−1} ⊗ L → M , i ≥ 1 ,

of graded morphisms homogeneous of degree −i such that for each n ≥ 1, we have

\begin{aligned}f_{n} = & ∑_{l=1}^{n}(−1)^{l−1}m_{l}(a_{1}, …, a_{l−1}, h_{n−l+1}(a_{l}, …, a_{n−1}, x)) \\ & + ∑_{i=1}^{n−2}(−1)^{i−1}h_{n−1}(a_{1}, …, a_{i}a_{i+1}, …, a_{n−1}, x) \\ & + ∑_{l=1}^{n}(−1)^{n−l}h_{n−l+1}(a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x)) , \end{aligned} ; (6.11)

For n = 1, this equation becomes

f_{1} = m_{1} h_{1} + h_{1} m_{1} ,

which means that f_{1} is nullhomotopic. Two morphisms between strong homotopy modules are homotopic if their difference is nullhomotopic.

We extend our dictionary between strong homotopy and differential graded coalgebra: Let X = C ⊗ L and Y = C ⊗ M be the differential graded comodules associated with L and M, in analogy with subsection 6.2. It is easy to check that the map f ↦ ε∘f, where ε : Y → M is the canonical projection, is a bijection from the set of comodule morphisms to the set of morphisms of graded modules X → M and that under this bijection, the morphisms of differential comodules correspond exactly to the morphisms of strong homotopy modules. If f is a nullhomotopic morphism of differential graded comodules, the map h ↦ ε ∘ h also induces a bijection from the set of homotopies from f to 0 to the set of homotopies from ε ∘ f to 0.

7. From strong homotopy actions to strict actions

Let k be a commutative ring and A and B (associative, unital) k-algebras. Let L and M be strong homotopy modules (cf. 6.4). We define a complex of k-modules Hom^{∞}^{•}(L, M) as follows: its pth component is the k-module of sequences of graded (B-linear) morphisms

f_{n} : A^{⊗n−1} ⊗ L → M , n ≥ 1 ,

of degree p+1−n; its differential maps a sequence f_{n} in Hom^{∞}^{p}(L, M) to the sequence g_{n} defined by

\begin{aligned}g_{n} = & ∑_{l=1}^{n}(−1)^{p(l−1)}m_{l}(a_{1}, …, a_{l−1}, f_{n−l+1}(a_{l}, …, a_{n−1}, x)) \\ & − (−1)^{p} ∑_{i=1}^{n−2}(−1)^{i−1}f_{n−1}(a_{1}, …, a_{i}a_{i+1}, …, a_{n−1}, x) \\ & − (−1)^{p} ∑_{l=1}^{n}(−1)^{n−l}f_{n−l+1}(a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x)) , \end{aligned} ; (7.1)

Lemma 7.1. The square of the above differential vanishes. The group of zero cycles of Hom^{∞}^{•}(L, M) identifies with the group of morphisms of strong homotopy modules L → M. Zero boundaries correspond exactly to the nullhomotopic morphisms.

Proof. We use the correspondence of 6.4: The set Hom^{∞}^{p}(L, M) is in bijection with the set of graded comodule morphisms f : C ⊗ L → C ⊗ M of degree p and the differential corresponds to the map

f ↦ b ∘ f − (−1)^{p}f ∘ b.

This shows that we have a well-defined complex. The rest follows upon inspection of (6.9) and (6.11). √

Let Shmod denote the category of strong homotopy A-modules over B and let Bimod denote the category of complexes of A-B-bimodules. We have an obvious functor

R : Bimod → Shmod

which maps a complex of A-B-bimodules to the underlying Z-graded B-module endowed with the homotopy action given by the differential, the multiplication, and m_{n} = 0 for all n ≥ 3. We will construct a left adjoint. We use the notations of 6.2. For X ∈ Shmod, let LX be the complex whose underlying graded A-B-bimodule is A ⊗ C ⊗ X and whose differential is

\begin{aligned}d(a_{0}, a_{1}, …, a_{n−1}, x) = & − (a_{0}a_{1}, a_{2}, …, a_{n−1}, x) \\ & + ∑_{i=1}^{n−2}(−1)^{i−1}(a_{0}, …, a_{i}a_{i+1}, …, a_{n−1}, x) \\ & + ∑_{l=1}^{n}(−1)^{n−l}(a_{0}, a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x)). \end{aligned} (7.2)

Lemma 7.2. The square of the above differential vanishes.

Proof. Define a differential on the graded module A ⊗ C by

d(a_{0}, a_{1}, …, a_{n−1}) = −(a_{0}a_{1}, a_{2}, …, a_{n−1}) + a_{0} ⊗ d_{C}(a_{1}, a_{2}, …, a_{n−1}).

It is not hard to check that its square vanishes. On the other hand, C ⊗ X is endowed with the differential of (6.4). Now the morphism

1_{A} ⊗ Δ ⊗ 1_{X} : A ⊗ C ⊗ X → (A ⊗ C) ⊗ (C ⊗ X)

defines an isomorphism onto a graded submodule and the differential given on A ⊗ C ⊗ X is induced by the one on (A ⊗ C) ⊗ (C ⊗ X).

√

We define

φ : Hom^{∞}^{•}(X, RY) →^{∼} Hom^{•}(LX, Y)

as the composition

Hom^{∞}^{•}(X, RY) = Hom_{k}^{•}(C ⊗ X, Y) →^{∼} Hom_{A}^{•}(A ⊗ C ⊗ X, Y) = Hom^{•}(LX, Y).

More explicitly, a morphism of strong homotopy modules f corresponds to a morphism of graded modules f : C ⊗ X → Y. By definition, the image of f under φ maps a ⊗ c ⊗ x to af(c, x). Of course, φ is an isomorphism of graded k-modules. Its inverse maps g to c ⊗ x ↦ g(1 ⊗ c ⊗ x). Note that the fact that A has a unit is crucial for this.

Lemma 7.3. The isomorphism φ is compatible with the differentials. In particular, the functors L and R are adjoints and induce a pair of adjoint functors in the homotopy categories.

Proof. The claim follows from the equalities

d(φ(f)(a_{0}, …, a_{n−1}, x))

= a_{0}d(f(a_{1}, …, a_{n−1}, x))

φ(f)(d(a_{0}, …, a_{n−1}, x))

= −(a_{0}a_{1})f(a_{2}, …, a_{n−1}, x)

+ ∑_{i=1}^{n−2}(−1)^{i−1}a_{0}f(a_{1}, …, a_{i}a_{i+1}, … a_{n−1}, x)

+ ∑_{l=1}^{n}(−1)^{n−l}a_{0}f(a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x))

φ(d(f))(a_{0}, …, a_{n−1}, x)

= a_{0}d(f(a_{1}, …, a_{n−1}, x)) − (−1)^{p}a_{0}(a_{1}f(a_{2}, … a_{n−1}, x))

− (−1)^{p} ∑_{i=1}^{n−2}(−1)^{i−1}a_{0}f(a_{1}, …, a_{i}a_{i+1}, … a_{n−1}, x)

− (−1)^{p} ∑_{l=1}^{n}(−1)^{n−l}a_{0}f(a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x)).

√

The functors R and L induce a pair of adjoint functors between the homotopy categories of Shmod and Bimod. A quasi-isomorphism of Shmod is a morphism f : X → Y such that f_{1} is a quasi-isomorphism of the underlying complexes. Then clearly the functor R preserves quasi-isomorphisms. Since A is projective over k, the functor L also preserves quasi-isomorphisms. Hence if we define the derived categories D Shmod and D Bimod to be the localizations of the homotopy categories with respect to the quasi-isomorphisms, then L and R induce a pair of adjoint functors between the derived categories:

D Bimod ; L ↑↓ R ; D Shmod

Let Y be a complex of A-B-bimodules. It is easy to check that LRY = A⊗T (A[1])⊗ Y is isomorphic to the Hochschild resolution of Y and that the adjunction morphism

LRY = A ⊗ T (A[1]) ⊗ Y → Y

identifies with the augmentation of the Hochschild resolution. In particular, the adjunction is a quasi-isomorphism. It follows that the functor R : D Bimod → D Shmod is fully faithful. If X is a strong homotopy module, the adjunction morphism X → RLX = A ⊗ T (A[1]) ⊗ X is the morphism of strong homotopy modules whose component in degree i is the morphism

f_{i} : A^{⊗i−1} ⊗ X → A ⊗ A[1]^{⊗i−1} ⊗ X ⊂ A ⊗ T (A[1]) ⊗ X

given by

(a_{1}, …, a_{i−1}) ⊗ x ↦ 1 ⊗ (a_{1}, …, a_{i−1}) ⊗ x.

We say that X is H-unital if the adjunction morphism is a quasi-isomorphism. We deduce the

Proposition 7.4. The functor

R : D Bimod → D Shmod

is an equivalence onto the full subcategory of H-unital strong homotopy modules. Its inverse is induced by the functor L.

For the applications, we need a criterion for H-unitality:

Lemma 7.5. Let X be a strong homotopy module. Then the following are equivalent:

(i) X is H-unital.

(ii) The morphism of complexes of k-modules m_{2}(1,?) : X → X induces the identity in homology.

Proof. Suppose that (i) holds. The square

X → RLX
m_{2}(1,?) ↓  ↓ m_{2}(1,?)
X → RLX

commutes in the homotopy category of complexes of k-modules thanks to (6.10). By assumption, the adjunction morphism X → RLX is a quasi-isomorphism. The right vertical arrow is the identity (since the A-module structure on RLX = A ⊗ C ⊗ X is induced from that of A). So if we apply the homology functor to the diagram, we see that (ii) holds.

Suppose that (ii) holds. Consider the filtrations

F_{p}RLX = A ⊗ X ⊕ (A ⊗ A[1] ⊗ X) ⊕ ⋯ ⊕ (A ⊗ A[1]^{⊗p} ⊗ X) , p ≥ 0 ,

and

F_{p}X = X , p ≥ 0.

The morphism f : X → RLX, x ↦ 1 ⊗ x, is compatible with the filtrations. The E_{1}-term of the spectral sequence associated with F_{p}RLX is the Hochschild resolution of the graded A-module H^{∗}X. By our assumption, this module is unital and thus the Hochschild resolution is quasi-isomorphic to the module H^{∗}X and the map x ↦ 1 ⊗ x induces a quasi-isomorphism. It follows that f induces an isomorphism in the E_{2}-terms of the spectral sequences. Since the filtrations are bounded below and exhaustive, the spectral sequences converge (by the classical convergence theorem [13, 5.5.1]) and f is a quasi-isomorphism. √

8. Proof of existence

We prove part a) of theorem 2.1. We put m_{1} = d : T → T and construct m_{2}, m_{3} as in section 3. Since A is projective over k, the vanishing condition (6.3) follows from (2.1). Hence by theorem 6.1, the triple m_{1}, m_{2}, m_{3} may be completed to a strong homotopy action m_{n}, n ≥ 1, of A on T. Let us denote by T̃ ∈ Shmod the corresponding strong homotopy module, cf. 6.4. In the homotopy action of A on T, the unit of A acts by the identity, so that m_{2}(1,?) : T̃ → T̃ is homotopic to the identity. By lemma 7.5, the strong homotopy module T̃ is H-unital. So by proposition 7.4, it comes from a complex of bimodules. More precisely, the canonical morphism of strong homotopy modules

f : T̃ → RLT̃

is a quasi-isomorphism. This means that f_{1} is a quasi-isomorphism, which, by (6.10), is compatible with the homotopy actions of A on T̃ and LT̃. We put X = LT̃ = A ⊗ T (A[1]) ⊗ T and φ = f_{1}.

Note that if T is a bounded complex, we can truncate X to a bounded complex, as we did in sections 3 and 4.

9. Summary of the construction

For the convenience of the reader, we sum up the explicit procedure that we propose for turning a tilting complex into a complex of bimodules.

Suppose that the hypotheses of theorem 2.1 are satisfied: k is a commutative ring, A, B are two (associative, unital) k-algebras and T is a complex of right Bmodules endowed with an homotopy action by A. We suppose that A is projective as a k-module, and that T is right bounded with projective components and satisfies

(9.1)  Hom_{HB}(T, T [−n]) = 0 for all n > 0.

In a first step, we define homogeneous graded B-module morphisms

m_{n} : A^{⊗n} ⊗ T → T , n ≥ 1 ,

of degree 2 − n. The construction works by induction on n. The cases n = 1, n = 2 and n = 3 have to be treated separately.

The case n = 1. By definition, the morphism m_{1} equals the differential of the complex T.

The case n = 2. Since A is projective over k, we can find a k-linear map α̃ : A → Hom_{CB} (T, T) lifting the given homotopy action α : A → Hom_{HB} (T, T). Now we define m_{2} : A ⊗ T → T by

m_{2}(a, x) = (α̃(a))(x) , a ∈ A, x ∈ T.

The case n = 3. To define m_{3}, we consider the square

A ⊗ A ⊗ T →^{1_{A}⊗m_{2}} A ⊗ T
m_{A} ⊗ 1_{T} ↓  ↓ m_{2}
A ⊗ T →^{m_{2}} T ,

where m_{A} denotes the multiplication of A. The square becomes commutative in the homotopy category. Hence there is a morphism of graded B-modules

m_{3} : A ⊗ A ⊗ T → T

homogeneous of degree −1 such that we have

m_{2}(ab, x) − m_{2}(a, m_{2}(b, x)) = −m_{3}(a, b, d(x)) − d(m_{3}(a, b, x))

for all a, b ∈ A and x ∈ T.

The case n > 3. Suppose that m_{1}, …, m_{n−1} have been constructed. Consider the morphism of graded B-modules

c : A^{⊗n−1} ⊗ T → T

of degree 3 − n defined by

\begin{matrix}c(a_{1}, …, & a_{n−1}, x) = ∑_{i=1}^{n−2}(−1)^{i−1}m_{n−1}(a_{1}, …, a_{i}a_{i+1}, …, a_{n−1}, x)+ \\ ∑_{l=2}^{n−1}(−1)^{n−l}m_{n−l+1}(a_{1}, …, a_{n−l}, m_{l}(a_{n−l+1}, …, a_{n−1}, x)). \end{matrix} ; (9.2)

It was shown in section 6.3 that c is a morphism of complexes of degree 3 − n. Now by our assumptions on T and A, all morphisms of complexes of B-modules of degree < 0 from A^{⊗n−1} ⊗ T to T are nullhomotopic. We can therefore choose a graded morphism m_{n} : A^{⊗n−1} ⊗ T → T of degree 2 − n such that

0 = c + m_{1} m_{n} − (−1)^{n} m_{n} ∘ (1^{⊗n−1} ⊗ m_{1}).

This ends the inductive construction of m_{n}.

In a second step, we define a bimodule complex X quasi-isomorphic to T as a complex of right B-modules and which ‘strictifies’ the homotopy action of A on T. The underlying graded A-B-bimodule of X is defined by

X^{n} = ⨁_{−p+q=n} A ⊗ A^{⊗p} ⊗ T^{q} , n ∈ Z.

The differential of X is defined by

d(a_{0}, a_{1}, …, a_{p−1}, x) = − (a_{0}a_{1}, a_{2}, …, a_{p−1}, x)

+ ∑_{i=1}^{p−2}(−1)^{i−1}(a_{0}, …, a_{i}a_{i+1}, …, a_{p−1}, x)

+ ∑_{l=1}^{p}(−1)^{p−l}(a_{0}, a_{1}, …, a_{p−l}, m_{l}(a_{p−l+1}, …, a_{p−1}, x)).

It is shown in lemma 7.2 that we do have d^{2} = 0 for this differential and in section 8 that X is quasi-isomorphic to T as a complex of right B-modules and ‘strictifies’ the homotopy action on T.

10. Comparison with Rickard’s construction

We keep the notations and assumptions of section 9 and assume moreover that T is a tilting complex [8]. Let us recall Rickard’s construction [9] for replacing the complex T by a bimodule complex X: One starts by proving that A^{op} ⊗ T is a tilting complex over A^{op} ⊗ B and that its endomorphism ring is A^{op} ⊗ A. Then, according to the main result of [8], there is a canonical functor

F : D^{−}(A^{op} ⊗ A) → D^{−}(A^{op} ⊗ B)

sending A^{op} ⊗ A to A^{op} ⊗ T. One then puts

X = F (A)

where A is viewed as a complex of bimodules concentrated in degree 0. To actually compute F (A), one proceeds as follows: First, we have to replace A by a projective resolution over A^{op} ⊗ A. Let us take the Hochschild resolution:

A ⊗ A ← A ⊗ A ⊗ A ← … ← A^{⊗p+2} ⊗ A ← ….

Now, using the action up to homotopy of A on T, we deduce a complex

A ⊗ T ← A ⊗ A ⊗ T ← … ← A^{⊗p+1} ⊗ T ← …, (∗)

whose components and differentials belong to the homotopy category of A^{op} ⊗ Bmodules, and more precisely to its additive subcategory generated by the tilting complex A^{op} ⊗ T. In [8], Rickard shows how such a complex may be lifted to a (filtered) complex of A^{op} ⊗ B-modules. This complex is X = F (A). What the construction presented in section 9 achieves is precisely a lift of (∗) to the complex X.

References

[1] T. V. Kadeishvili, The category of differential coalgebras and the category of A(∞)-algebras, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 77 (1985), 50–70.

[2] , Twisted tensor products for the category of A(∞)-algebras and A(∞)-modules (Russian), Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 83 (1986), 26–45.

[3] Bernhard Keller, A remark on tilting theory and DG algebras, Manuscripta Math. 79 (1993), no. 3-4, 247–252.

[4] , Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102.

[5] , Introduction to A-infinity algebras and modules, Notes of a minicourse given in Ioannina, Greece, March 1999, available at the author’s homepage, 1999.

[6] M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 120–139.

[7] , Triangulated categories and geometry, Course at the Ecole Normale Supérieure, Paris, 1998.

[8] Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. 39 (1989), 436– 456.

[9] , Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48.

[10] A. Zimmermann S. Koenig, Derived equivalences for group rings, Lecture Notes in Mathematics, vol. 1685, Springer–Verlag, 1998.

[11] J. D. Stasheff, Homotopy associativity of H-spaces, I, Trans. Amer. Math. Soc. 108 (1963), 275–292.

[12] , Homotopy associativity of H-spaces, II, Trans. Amer. Math. Soc. 108 (1963), 293– 312.

[13] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, 1994.

UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France

E-mail address: keller@math.jussieu.fr www.math.jussieu.fr/  keller