HOCHSCHILD COHOMOLOGY AND DERIVED PICARD GROUPS

BERNHARD KELLER

Dedicated to Idun Reiten on the occasion of her sixtieth birthday

Abstract. We interpret Hochschild cohomology as the Lie algebra of the derived Picard group and deduce that it is preserved under derived equivalences.

1. Introduction

The Hochschild cohomology groups HH^{i} (A, A) of an algebra A over a field k can be interpreted as higher extension groups of the bimodule A by itself or as morphisms from A to A[i] in the derived category D(A^{op} ⊗ A) of A-A-bimodules. This last interpretation shows that they are preserved under derived equivalences [11], i.e. if X is a complex of A-B-bimodules such that the total derived tensor product by X is an equivalence DA → DB, then X yields a natural isomorphism from HH^{i} (A, A) to HH^{i} (B, B). This isomorphism is compatible with the cup product, since the cup product corresponds to the composition of morphisms in the derived category of bimodules. However, it is not clear whether the isomorphism given by X also respects the Gerstenhaber bracket on Hochschild cohomology [4]. We will show that this is indeed the case by providing an intrinsic interpretation of the Gerstenhaber bracket in terms of derived categories. The basic idea is to view Hochschild cohomology as an analogue of the Lie algebra associated with an algebraic group (more precisely, a group-valued functor). This group will be the derived Picard group [17] [13] [16] of A (more precisely, the functor which sends a commutative differential graded k-algebra R to the R-relative derived Picard group of A ⊗_{k} R). Our interpretation generalizes the fact that the first Hochschild cohomology group of a finite-dimensional algebra A is the Lie algebra of the group of outer automorphisms of A.

The author thanks the referee for his careful reading of the manuscript and for many helpful remarks.

2. Reminder on derived equivalences

2.1. Derived categories and the Hochschild cohomology algebra. Let k be a field and A a k-algebra, i.e. an associative unital k-algebra. Let Mod A denote the category of right A-modules. Let DA denote the (unbounded) derived category of Mod A. Thus, the objects of DA are all complexes

… → M^{p} →^{d} M^{p+1} → …, p ∈ Z , d^{2} = 0 ,

of right A-modules and its morphisms are obtained from morphisms of complexes by formally inverting all quasi-isomorphisms, i.e. morphisms of complexes inducing isomorphisms in homology. Let us recall the most basic examples of morphisms in the derived category: We identify an A-module L with the complex whose 0component is L and whose components in all other degrees vanish. Then, if L and M are A-modules, the group of morphisms in DA from L to M identifies with the group of A-linear maps from L to M and, more generally, we have a natural isomorphism

Date: January 2002, last modified on September 25, 2003.

1991 Mathematics Subject Classification. 16E40, 18E30, 16D90, 18G10.

Key words and phrases. Hochschild cohomology, Derived category, Derived Picard group.

Hom_{DA}(L, M[i]) →^{∼} Ext_{A}^{i}(L, M)

for each i ∈ Z, where, for a complex K, we denote by K[i] the complex with components K[i]^{p} = K^{i+p} and differential (−1)^{i}d_{K}. By convention, Ext-groups vanish in negative degrees.

In particular, we can identify the Hochschild cohomology groups HH^{∗}(A, A) with groups of morphisms in the derived category of A-A-bimodules: Indeed, since A is flat over k, we have a canonical isomorphism

HH^{i} (A, A) →^{∼} Ext_{A^{op}⊗A}^{i}(A, A)

and thus a canonical isomorphism

HH^{i} (A, A) →^{∼} Hom_{D(A^{op}⊗A)}(A, A[i]) , i ∈ Z.

Under this isomorphism, the cup product on Hochschild cohomology corresponds to the graded composition in the derived category. More precisely, the cup product of the cohomology classes corresponding to f : A → A[j] and g : A → A[i] corresponds to the composition f[i] ∘ g.

2.2. Derived equivalence. Let A and B be two k-algebras. We recall one version of Rickard’s Morita theorem for derived categories [11] [12].

Theorem. The following are equivalent

(i) There is a triangle equivalence F : DA →^{∼} DB.

(ii) There are bimodule complexes X ∈ D(A^{op} ⊗ B) and Y ∈ D(B^{op} ⊗ A) and isomorphisms

X ⊗_{B}^{L} Y →^{∼} A in D(A^{op} ⊗ A) and Y ⊗_{A}^{L} X →^{∼} B in D(B^{op} ⊗ B).

Here the symbol ⊗^{L} denotes the total derived tensor functor [14]. The implication from (ii) to (i) is easy: Indeed, the functor

F = ? ⊗_{A}^{L} X : DA → DB

is an equivalence whose inverse is given by ?⊗_{B}^{L} Y, cf. [12]. The implication from (ii) to (i) is considerably more delicate. One can also show [12] that if X ∈ D(A^{op} ⊗ B) is a bimodule complex such that the associated functor ? ⊗_{A}^{L} X : DA → DB is an equivalence, then (ii) holds for X and

Y = R Hom_{B}(X, B).

Thus the essential datum is that of X. We call such X an invertible bimodule complex and Y its inverse. Two algebras A and B are called derived equivalent if the conditions of the theorem hold. If we consider other variants of the derived categories (e.g. the bounded derived categories), we obtain the same equivalence relation on the class of k-algebras, cf. [11]. Of course, derived equivalence generalizes Morita equivalence. The following example, a particular case of Koszul duality [2], [5], shows that this generalization is non trivial.

2.3. An example. Let V be a vector space of dimension n + 1, denote by S^{i} the i-th symmetric power of its dual space and by Λ^{i} the i-th exterior power of V. Let A be the algebra of upper triangular matrices

A = ( \begin{matrix}S^{0} & S^{1} & … & S^{n} \\ 0 & S^{0} & … & S^{n−1} \\ ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & … & S^{0} \end{matrix} )

and B the algebra of lower triangular matrices

B = ( \begin{matrix}Λ^{0} & 0 & … & 0 \\ Λ^{1} & Λ^{0} & … & 0 \\ ⋮ & ⋮ & … & ⋮ \\ Λ^{n} & Λ^{n−1} & … & Λ^{0} \end{matrix} ) .

Then A is derived equivalent to B, but for n > 2, A is not Morita equivalent to B. If E_{i} denotes the simple A-module corresponding to the projection on the (i+1)-th diagonal component, i = 0, …, n, then the complex

… → 0 → E_{n} →^{0} E_{n−1} →^{0} … →^{0} E_{0} → 0 …

is the restriction to A of a bimodule complex X ∈ D(A^{op} ⊗ B), unique up to isomorphism [7], whose associated tensor functor is an equivalence DA → DB. Historically, this example comes from geometry: Beilinson showed in [1] that the derived categories of both A and B are triangle equivalent to the derived category of the category of quasicoherent sheaves on the projective space P(V).

2.4. Invariance of the algebra structure. The following theorem is due to Dieter Happel [6] in the special case of derived equivalences coming from tilting modules and to Jeremy Rickard [12] in the general case. Let A and B be derived equivalent algebras and X ∈ D(A^{op} ⊗ B) and invertible bimodule complex

Theorem. There is a canonical algebra isomorphism

φ_{X} : HH^{∗}(A, A) → HH^{∗}(B, B).

After the preparations we have made, it is easy to construct φ_{X}: Indeed, let Y be the inverse of X with isomorphisms u : Y ⊗_{A}^{L} X →^{∼} B and v : X ⊗_{B}^{L} Y →^{∼} A. Let φ_{X,u} send f : A → A[i] to

u[i] ∘ (Y ⊗_{A}^{L} f ⊗_{A}^{L} X) ∘ u^{−1} : B → B[i].

If u′ is another isomorphism from Y ⊗_{A}^{L} X to B, then u′ = zu for an invertible central element z of B. So we have

φ_{X,u′}(f) = z[i] φ_{X,u}(f) z^{−1}

and this equals φ_{X,u}(f) since the center of B is central in HH^{∗}(B, B). We define φ_{X} = φ_{X,u}.

3. Reminder on the Gerstenhaber bracket

Let A be an algebra. The Hochschild complex of A is the complex

C^{·}(A, A) = (A → Hom_{k}(A, A) → … → Hom_{k}(A^{⊗p}, A) → …)

with A in degree 0 whose differential maps a ∈ A to [a,?], and, more generally, a p-cochain c ∈ Hom_{k}(A^{⊗p}, A) to the (p + 1)-cochain dc defined by

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

The homology in degree i of C^{·}(A, A) is HH^{i} (A, A). For a p-cochain c_{1}, a q-cochain c_{2} and an integer 0 ≤ i ≤ p − 1, define a (p + q − 1)-cochain c_{1} •_{i} c_{2} by

(c_{1} •_{i} c_{2})(a_{1}, …, a_{p+q−1}) = c_{1}(a_{1}, …, a_{i}, c_{2}(a_{i+1}, …, a_{i+q}), a_{i+q+1}, …, a_{p+q−1}).

Then the Gerstenhaber product is defined by

(c_{1} • c_{2}) = ∑_{i=0}^{p−1}(−1)^{i(q−1)}c_{1} •_{i} c_{2}.

The Gerstenhaber product is not associative. However, the associator

A(c_{1}, c_{2}, c_{3}) = (c_{1} • c_{2}) • c_{3} − c_{1} • (c_{2} • c_{3})

is super symmetric in c_{2} and c_{3} endowed with suitable degrees (super=Z/2Zgraded). Namely, we have

A(c_{1}, c_{2}, c_{3}) = (−1)^{(q−1)(r−1)} A(c_{1}, c_{3}, c_{2}) ,

where c_{2} is a q-cochain and c_{3} an r-cochain. Therefore the super commutator of the Gerstenhaber bracket behaves like the commutator of an associative product, i.e. it satisfies the super Jacobi identity. More precisely, we have the

Lemma. [4] Endowed with the Gerstenhaber bracket defined by

[c_{1}, c_{2}] = c_{1} • c_{2} − (−1)^{(p−1)(q−1)}c_{2} • c_{1} , c_{1} ∈ C^{p}(A, A) , c_{2} ∈ C^{q}(A, A) ,

and the differential −d, the graded space C^{·+1}(A, A) becomes a differential graded Lie algebra. In particular, the homology HH^{∗+1}(A, A) becomes a Z-graded super Lie algebra.

For example, let V be a vector space and A the algebra of polynomial functions on V. Then HH^{1} (A, A) identifies with the space Der_{k}(A, A) of k-linear derivations of A and HH^{i+1} (A, A) is canonically isomorphic to the exterior power Λ_{A}^{i+1} Der_{k}(A, A). Under this isomorphism, the Gerstenhaber bracket corresponds to the Nijenhuis-Schouten-bracket, which is the natural extension of the commutator of derivations.

4. The Lie algebra of the derived Picard group

We will interpret Hochschild cohomology with the Gerstenhaber bracket as the Lie algebra of a ‘generalized algebraic group’, namely a group valued functor defined on a category of commutative algebras. For this, let us recall the construction of the Lie algebra of an algebraic group: Let G be an algebraic group over k considered as a group-valued functor

G : {commutative k-algebras} → {groups} , R ↦ G(R).

Then the Lie algebra of G is the space of tangent vectors at the origin, i.e.

Lie(G) = ker(G(k[ε]/(ε^{2})) → G(k)).

The bracket is induced by the commutator in G. To make this last statement more intuitive, consider the example where G = GL_{n}. We have

Lie(GL_{n}) = {1 + εX | X ∈ M_{n}(k) }

and the Lie bracket is determined by the identity

(1 + ε_{1}X_{1})(1 + ε_{2}X_{2})(1 + ε_{1}X_{1})^{−1}(1 + ε_{2}X_{2})^{−1} = 1 + ε_{1}ε_{2} [X_{1}, X_{2}]

in GL_{n}(k[ε_{1}, ε_{2}]/(ε_{1}^{2}, ε_{2}^{2})).

We will now define a group valued functor DPic_{A} whose Lie algebra will be the Hochschild cohomology of A. Since this Lie algebra is graded, the category on which DPic_{A} is defined should include the category of graded commutative algebras. It turns out that a reasonable category is cdg k, the category of commutative differential graded k-algebras. To define

DPic_{A} : cdg k → {groups} ,

we need the relative derived category (cf. [9, Sect. 7]): Let R be a commutative differential graded algebra (for example the algebra k[ε]/(ε^{2}), where ε has any integer degree and d = 0). Let E be a (typically non commutative) differential graded Ralgebra. The R-relative derived category D_{R}E has as objects all differential graded E-modules (these are precisely the complexes of E-modules if E is concentrated in degree 0). The morphisms of D_{R}E are obtained from morphisms of differential graded E-modules by formally inverting all R-relative quasi-isomorphisms, i.e. all morphisms s : L → M of differential graded E-modules whose restriction to R is an homotopy equivalence. For example, the relative derived category D_{k}E equals the usual derived category DE of the differential graded algebra E.

Rouquier-Zimmermann [17] [13] and Yekutieli [16] have independently defined the derived Picard group of a ring. We generalize this as follows: Let R be a commutative differential graded algebra and A an algebra. A bimodule complex U ∈ D_{R}(R ⊗ A^{op} ⊗ A) is R-semifree if its underlying graded R-module is free; it is invertible if it is R-semifree^{1} and there exists an R-semifree bimodule complex V ∈ D_{R}(R ⊗ A^{op} ⊗ A) such that there are isomorphisms

U ⊗_{A⊗R}^{L,rel} V →^{∼} R ⊗ A and V ⊗_{A⊗R}^{L,rel} U →^{∼} R ⊗ A

in D_{R}(R ⊗ A^{op} ⊗ A). The R-relative derived Picard group of A is the set of isomorphism classes of invertible bimodule complexes U in D_{R}(R ⊗ A^{op} ⊗ A). This set is endowed with the group law induced by the derived tensor product. This group is denoted by DPic_{A}(R). It is functorial with respect to R so that we do obtain a functor

cdg k → {groups}.

If A is derived equivalent to B and X ∈ D(A^{op} ⊗ B) is an invertible bimodule complex with inverse Y, then we have an isomorphism

DPic_{A}(R) →^{∼} DPic_{B}(R) , U ↦ Y ⊗_{A}^{L} U ⊗_{A}^{L} X.

In this sense, DPic_{A} is also functorial with respect to invertible bimodule complexes X ∈ D(A^{op} ⊗ B).

We now define the Lie algebra of DPic_{A}. Fix a degree i ∈ Z and let R be the commutative differential graded algebra k[ε]/(ε^{2}), where ε is of degree −i and d = 0. By definition, LieDPic_{A}^{i} is the set of isomorphism classes U of D_{R}(R⊗A^{op} ⊗A) such that U is free as a graded R-module and U ⊗_{R} k is isomorphic to A in D(A^{op} ⊗ A). The graded space LieDPic_{A}^{∗} is endowed with a super Lie bracket defined as for algebraic groups (cf. 5.6). For a super Lie algebra L, we denote by L^{op} the super Lie algebra with the opposite bracket. Its appearance in the theorem below is due to the fact that we consider right modules.

Theorem. There is a canonical isomorphism of graded super Lie algebras

HH^{∗+1}(A, A)^{op} →^{∼} LieDPic_{A}^{∗}

functorial with respect to invertible bimodule complexes X ∈ D(A^{op} ⊗B). In particular, the Gerstenhaber bracket on HH^{∗+1}(A, A) is preserved under derived equivalence.

The rest of the article is devoted to the proof of the theorem. In fact, we will prove it more generally for a differential graded k-algebra A.

1presumably, all semifreeness conditions are redundant.

5. Proof of the main theorem

5.1. Outline of the proof. Let A be a dg algebra. The idea is to construct an intermediate ‘differential graded formal group G’ whose Lie algebra is the Hochschild complex with the Gerstenhaber bracket and which acts on the relative derived category via bimodules. This group is the group G of automorphisms of the cobar construction C^{+} of A, where the cobar construction is viewed as a differential graded counital (but not coaugmented) coalgebra (cf. 5.7). It naturally acts on the category of differential graded comodules over C^{+}. Via the bar-cobar-adjunction at the module level, this action translates into an action of G on the derived category via bimodules. On the other hand, the Lie algebra of G is the Lie algebra of coderivations of C^{+} and, by Stasheff’s interpretation [15], this Lie algebra is the Hochschild complex endowed with the Gerstenhaber bracket. This programme yields a Lie algebra morphism

\xymatrix{Z^{0}(C^{·+1}(A, A) ⊗ 𝔪) \ar[r] & LieDPic(A, R)}

for each augmented commutative dg algebra R = k ⊕ 𝔪 with 𝔪^{2} = 0. To check that it induces an isomorphism, we need to identify the set of deformation classes LieDPic(A, R) with the group

Hom_{D(A⊗A^{op})}(A, A ⊗ 𝔪[1]).

We will also need to know how the Lie algebra structure is reflected under this identification. This is what we study first, in the paragraphs 5.2 – 5.6 below.

5.2. Infinitesimal deformations of modules. Let k be a field and S a commutative dg k-algebra. Suppose that R is an augmented commutative dg S-algebra and denote by 𝔫 the kernel of the augmentation R → S. Thus we have the decomposition R = S ⊕ 𝔫.

Let A be a (typically noncommutative) dg S-algebra, free as a graded S-module. Then A ⊗_{S} R is a dg R-algebra. We consider the reduction functor

D_{R}(A ⊗_{S} R) → D_{S}A , L ↦ L ⊗_{R} S.

We will study the fibers of this functor: Let M be a dg A-module which is free as a graded S-module. Let F be the category whose objects are the deformations of M, i.e. the pairs (L, u) formed by a dg A ⊗ R-module L, free as a graded R-module, and an isomorphism of D_{S}A

u : L ⊗_{R} S →^{∼} M.

Morphisms from (L, u) to (L′, u′) are given by morphisms v : L → L′ of D_{R}(A⊗_{S} R) such that u′ ∘ (v ⊗_{R} S) = u. We denote by

Defo(M, R → S)

the set of isomorphism classes of F. We denote by

Defo′(M, R → S)

the set of isomorphism classes of weak deformations of M, i.e. dg A ⊗ R-modules L free as graded R-modules such that L ⊗_{R} S is isomorphic to M. Note that we have an obvious forgetful map

Defo(M, R → S) → Defo′(M, R → S).

The group of automorphisms of M in D_{S}A acts on Defo(M, R → S) via (L, u).f = (L, f^{−1} ∘ u) and the forgetful map clearly induces a bijection

Defo(M, R → S)/ Aut_{D_{R}A}(M) →^{∼} Defo′(M, R → S).

From now on and to the end of this section, we suppose that 𝔫^{2} = 0. We will parametrize the connected components of F. Let (L, u) be an object of F. Since L is free as a graded R-module, the morphism of complexes L⊗_{R} 𝔫 → L𝔫 is invertible and the sequence

0 → L ⊗_{R} 𝔫 → L → L ⊗_{R} S → 0

is an exact sequence of dg A-modules which splits as a sequence of dg S-modules. Thus it gives rise to a canonical triangle of D_{S}A (but not of D_{R}A !)

(5.1)  L ⊗_{R} 𝔫 → L → L ⊗_{R} S →^{ε′} (L ⊗_{R} 𝔫)[1]

Since 𝔫^{2} = 0, we have a canonical isomorphism of dg modules

L ⊗_{R} 𝔫 →^{∼} (L ⊗_{R} S) ⊗_{S} 𝔫.

Therefore, we can define a canonical morphism ε(L, u) of D_{S}A by the commutative square

\xymatrix{L ⊗_{R} S \ar[r]^{ε′} \ar[d]_{u} & L ⊗_{R} 𝔫[1] \ar[r]^{∼} & (L ⊗_{R} S) ⊗_{S} 𝔫[1] \ar[d]_{u⊗𝔫[1]} \\ M \ar[rr]^{ε(L,u)} &  & M ⊗_{S} 𝔫[1]}

Clearly the morphism ε(L, u) only depends on the isomorphism class of (L, u) in the category F.

Proposition. The map Φ : (L, u) ↦ ε(L, u) induces a bijection

Defo(M, R → S) →^{∼} Hom_{D_{S}A}(M, M ⊗_{S} 𝔫[1]).

Clearly, the bijection of the proposition is equivariant with respect to the action of Aut_{D_{S}A}(M). Therefore we have the

Corollary. The map Φ : (L, u) ↦ ε(L, u) induces a bijection

Defo′(M, R → S) →^{∼} Hom_{D_{S}A}(M, M ⊗_{S} 𝔫[1])/ Aut_{D_{S}A}(M).

Proof. We construct a map Ψ which will turn out to be the inverse bijection. We may and will assume that M is S-relatively cofibrant in the category of dg Amodules, i.e. it satisfies the S-relative variant of property (P) of [8, 3.1]. This means [9, 7.5] that M admits an increasing filtration by dg A-submodules M_{n}, n ≥ 0, such that each inclusion M_{n} ⊂ M_{n+1} splits as a morphism of graded Amodules and the subquotient M_{n+1}/M_{n} is isomorphic to a direct summand of a module K ⊗_{S} A, where K is a dg S-module. Let a morphism

f : M → M ⊗_{S} 𝔫[1]

of D_{S}A be given. Since M is S-relatively cofibrant as a dg A-module, the morphism f is realized by a map f̃ of dg A-modules. We put L = M ⊗_{S} R = (M ⊗_{S} 𝔫) ⊕ M as a graded A ⊗_{S} R-module and we define its differential by

d_{L} = [ \begin{matrix}d_{M⊗_{S}𝔫} & f \\ 0 & d_{𝔫} \end{matrix} ] .

Clearly L is free as a graded R-module (since M is free as a graded S-module) and we have an obvious isomorphism u : L⊗_{R}S → M. So we have constructed an object (L, φ) of F by choosing a representative f̃of the homotopy class f : M → M ⊗_{S}𝔫[1]. Let us check that the connected component of (L, φ) is independent of the choice of the representative. Let f̃′ be another choice and let h : M → M ⊗_{S} 𝔫 be a morphism of graded A-modules such that f′ = f + d ∘ h + h ∘ d. Let (L′, u′) be the object of F constructed from f′. Then L′ equals L as a graded A ⊗_{S} R-module and both equal M ⊗_{S} 𝔫 ⊕ M. The matrix

[ \begin{matrix}1_{M⊗_{S}𝔫} & h \\ 0 & 1_{M} \end{matrix} ]

defines an isomorphism v of dg A ⊗_{S} R-modules from L to L′ and we clearly have u′ = u ∘ (v ⊗_{R} S). By definition, the map Ψ sends f to the connected component of (L, u). The easy check that Φ ∘ Ψ is the identity is left to the reader. To finish the proof, it is enough to check that Ψ is surjective. For this, let an object (L′, u′) of F be given. We may and will assume that L′ is R-relatively cofibrant. Put M′ = L′ ⊗_{R} S. Since L′ is free as a graded R-module, there is an isomorphism of graded R-modules M′ ⊗_{S} R →^{∼} L′ which lifts the identity of M′. The differential of L′ then yields a differential of M′ ⊗_{S} R = (M′ ⊗_{S} 𝔫) ⊕ M′ given by a matrix

[ \begin{matrix}d_{M′⊗_{S}𝔫} & f′ \\ 0 & d_{M′} \end{matrix} ] .

Now define f : M → M ⊗_{S} 𝔫[1] by the commutative square of D_{S}A

\xymatrix{M′ \ar[d]_{u′} \ar[r]^{f′} & M′ ⊗_{S} 𝔫[1] \ar[d]^{u′⊗_{S}𝔫[1]} \\ M \ar[r]^{f} & M ⊗_{S} 𝔫[1]}

Since M′ is S-relatively cofibrant, there is a morphism of dg modules u_{1} : M′ → M lifting u′. Moreover, f ∘ u_{1} is homotopy equivalent to (u_{1} ⊗_{S} 𝔫[1]) ∘ f′. Choose an homotopy h between the two. Then the matrix

[ \begin{matrix}u_{1} ⊗ 1_{𝔫} & h \\ 0 & u_{1} \end{matrix} ]

defines a map

ũ : L′ = (M′ ⊗_{S} 𝔫) ⊕ M′ → (M ⊗_{S} 𝔫) ⊕ M = L.

This is in fact a morphism of dg A⊗_{S} R-modules and clearly it gives a morphism of F. Since the triangle (5.1) does not exist in the category D_{R}A, it is not immediate that ũ is invertible. However, starting from an inverse of u′, we can analogously construct a morphism ṽ : L → L′ of D_{R}A. Then the reduction of ũṽ modulo 𝔫 is homotopic to the identity. Let H be an homotopy. The homotopy H is a morphism of graded A-modules, and we can lift it to a morphism of graded A ⊗_{S} R-modules. We see that ũṽ is homotopic to a morphism w whose reduction modulo 𝔫 is the identity. Since L′ and L are free over R and 𝔫 is nilpotent, it follows that w is invertible. So ũṽ is homotopic to an invertible morphism. Thus ũṽ is invertible in the homotopy category. Similarly, we see that ṽũ is invertible in the homotopy category.

We conclude that (L′, u′) is in the isomorphism class of F which is the image of f under Ψ. Hence Ψ is surjective. √

5.3. An exact sequence. Let T be a commutative dg algebra, S an augmented dg T -algebra and R = S ⊕ 𝔫 an augmented dg S-algebra. Thus R becomes an augmented dg T -algebra R = T ⊕ 𝔪 and we have a commutative diagram with exact rows

[figure: 0; 𝔫; R; S; 0; 0; 𝔪; R; T; 0.]

Let A be a dg T -algebra free as a graded T -module and let M be a dg A-module free as a graded T -module. Then the tensor product ? ⊗_{R} S yields a natural map

Defo(M, R → T) → Defo(M, S → T).

On the other hand, if we have a representative (L′, u′) of an element of Defo(M ⊗_{T} S, R → S), we obtain an element of Defo(M, R → T) by taking L = L′ and u the composition

L ⊗_{S} T = L′ ⊗_{S} T ⟶^{u′⊗_{S}T} (M ⊗_{T} S) ⊗_{S} T = M.

The following lemma is immediate from these definitions.

Lemma. a) The sequence

Defo(M ⊗_{T} S, R → S) → Defo(M, R → T) → Defo(M, S → T) → ∗

is exact in the sense that the second map is surjective and its fibre over the base point is the image of the first map.

b) The sequence

∗ → Defo′(M ⊗_{T} S, R → S) → Defo′(M, R → T) → Defo′(M, S → T) → ∗

is exact in the sense that the second map is surjective, and its fiber over the base point equals the image of the first map, which is injective.

5.4. A base change isomorphism. Let T be a commutative dg algebra, S an augmented T -algebra and R = S ⊕ 𝔫 an augmented S-algebra. Thus R becomes an augmented T -algebra R = T ⊕ 𝔪. Let R′ = T ⊕ 𝔫 so that we have a commutative diagram with exact rows

[figure: 0; 𝔫; R; S; 0; 0; 𝔫; R′; T; 0.]

Let A be a dg T -algebra free as a graded T -module and let M be a dg A-module free as a graded T -module. The tensor products ? ⊗_{R′} R and ? ⊗_{T} S yield a natural map

Defo(M, R′ → T) → Defo(M ⊗_{T} S, R → S).

Lemma. If 𝔫^{2} = 0, this map is an isomorphism.

Proof. We have a commutative square

Defo(M, R′ → T)  →  Defo(M ⊗_{T} S, R → S)
↓  ↓
Hom_{D_{T} A}(M, M ⊗_{T} 𝔫[1]) → Hom_{D_{S}A}(M ⊗_{T} S, M ⊗_{T} S ⊗_{S} 𝔫[1])

whose vertical maps are given by proposition (5.2) and whose horizontal maps are given by the tensor functor. So the vertical maps are bijective. And the lower horizontal arrow identifies with the adjunction isomorphism

Hom_{D_{T} A}(M, M ⊗_{T} 𝔫[1]) → Hom_{D_{S}A}(M ⊗_{T} S, M ⊗_{T} 𝔫[1])

where we view M ⊗_{T} 𝔫[1] on the left hand side as the restriction to A of the A⊗_{T} Smodule M ⊗_{T} 𝔫[1] on the right hand side. √

Corollary. With the above notations, suppose that the map

Defo(M ⊗_{T} S, R → S) → Defo′(M ⊗_{T} S, R → S)

is bijective. Then we have an exact sequence

∗ → Defo(M, R′ → T) → Defo′(M, R → T) → Defo′(M, S → T) → ∗

in the sense that the second map is surjective and its fibre over the base point is the image of the first map, which is injective.

This follows from the lemma and from part b) of lemma 5.3.

5.5. Infinitesimal deformations of the bimodule A. Let A be a dg k-algebra, T a commutative dg k-algebra and S = T ⊕ 𝔫 an augmented commutative dg T -algebra with 𝔫^{2} = 0.

We consider the module M = A ⊗_{k} T over the algebra B = A^{op} ⊗ A ⊗ T.

Lemma. Suppose that H^{∗}T is of finite total dimension. Then the canonical map

Defo(M, S → T) → Defo′(M, S → T)

is bijective.

Proof. By (5.2), we have to show that the action of Aut_{D_{T} B}(A ⊗ T) on

Hom_{DB}(M, M ⊗_{T} 𝔫[1])

is trivial. We claim that we have an isomorphism

R Hom_{B}(A ⊗ T, A ⊗ T) ⊗_{T} 𝔫[1] →^{∼} R Hom_{B}(A ⊗ T, A ⊗ T ⊗_{T} 𝔫[1]).

Indeed, we have an adjunction isomorphism

R Hom_{B}(A ⊗ T, A ⊗ T) ⊗_{T} 𝔫[1] →^{∼} R Hom_{A^{e}}(A, A ⊗ T) ⊗_{T} 𝔫[1].

Now since H^{∗}T is of finite total dimension, we have an isomorphism

R Hom_{A^{e}}(A, A ⊗ T) →^{∼} R Hom_{A^{e}}(A, A) ⊗ T.

Combining the two preceding isomorphisms, we obtain an isomorphism

R Hom_{B}(A ⊗ T, A ⊗ T) ⊗_{T} 𝔫[1] →^{∼} R Hom_{A^{e}}(A, A) ⊗ 𝔫[1].

Now we have isomorphisms

R Hom_{A^{e}}(A, A) ⊗ 𝔫[1] →^{∼} R Hom_{A^{e}}(A, A ⊗ 𝔫[1]) →^{∼} R Hom_{B}(A ⊗ T, A ⊗ T ⊗_{T} 𝔫[1]) ,

where we have first used that H^{∗}𝔫 is of finite total dimension and then the adjunction, as above. The claim follows. Now Aut_{D_{T} B}(A ⊗ T) is the group of invertible elements of the zeroth homology of

R Hom_{B}(A ⊗ T, A ⊗ T).

The claim follows since this dg algebra is commutative up to homotopy. √

5.6. Definition of the Lie bracket. Let A be a dg k-algebra. Let 𝔪 be a dg k-module whose homology is of finite total dimension. Let R = k ⊕ 𝔪 denote the augmented commutative dg algebra with 𝔪^{2} = 0. We consider A as an A-A-bimodule. We define

G(𝔪) = Defo′(A, k ⊕ 𝔪 → k).

In other words, G(𝔪) is formed by the isomorphism classes of objects in D_{R}(A^{op} ⊗ A ⊗ R) whose reduction modulo 𝔪 is isomorphic to A in D(A^{op} ⊗ A). Note that, according to lemma 5.5, we have a canonical bijection

Defo(A, k ⊕ 𝔪 → k) →^{∼} Defo′(A, k ⊕ 𝔪 → k).

We will view D(A^{op} ⊗ A) (resp. D_{R}(A^{op} ⊗ A ⊗ R)) as a monoidal category for the derived tensor product over A (resp. for the relative derived tensor product over A ⊗ R). The monoidal structure of D_{R}(A^{op} ⊗ A ⊗ R) induces a monoid structure on G(𝔪) and the bijection

G(𝔪) ←^{∼} Defo(A, k ⊕ 𝔪 → k) → Hom_{D(A^{op}⊗A)}(A, A ⊗ 𝔪[1])

is a monoid morphism. In particular, G(𝔪) is an abelian group, functorial in 𝔪.

For two dg k-modules 𝔪_{1} and 𝔪_{2} whose homology is of finite total dimension, we define a Lie bracket

G(𝔪_{1}) × G(𝔪_{2}) → G(𝔪_{1} ⊗ 𝔪_{2})

as follows : Let L_{1} and L_{2} represent elements of G(𝔪_{1}) resp. G(𝔪_{2}). Put R_{i} = k ⊕ 𝔪_{i}. Let U_{i} be the image of L_{i} in D_{R}(R ⊗ A^{op} ⊗ A) where R = R_{1} ⊗ R_{2} (note that the kernel of R → k is not of square zero !). The U_{i} are invertible objects of a monoidal category. Let V denote the commutator of U_{1} with U_{2}. Then V yields an element of Defo′(A, R → k). We have a canonical map

G(𝔪_{1} ⊗ 𝔪_{2}) → Defo′(A, R → k)

and we claim that it is injective and that V lies in its image. Indeed, the image of V in Defo′(A, R_{1} → k) = G(𝔪_{1}) vanishes since G(𝔪_{1}) is commutative. Thus V lies in

G(𝔪_{2} ⊕ 𝔪_{1} ⊗ 𝔪_{2}) = Defo′(A, k ⊕ (𝔪_{2} ⊕ 𝔪_{1} ⊗ 𝔪_{2}))

by (5.3). The image of V in G(𝔪_{2}) = Defo′(A, k⊕𝔪_{2} → k) also vanishes since G(𝔪_{2}) is commutative. So again by 5.3, V lies in Defo′(A, k⊕(𝔪_{1}⊗𝔪_{2}) → k) = G(𝔪_{1}⊗𝔪_{2}).

5.7. From coalgebra automorphisms to bimodules. Let R be a commutative dg k-algebra and A a (typically noncommutative) dg R-algebra. Denote by SA the graded R-module with (SA)^{p} = A^{p+1}. We recall the bar construction of A relative to R. It is the dg R-coalgebra C^{+} defined as follows: Its underlying graded space is

R ⊕ SA ⊕ (SA ⊗_{R} SA) ⊕ … ⊕ (SA)^{⊗_{R}p} ⊕ ….

The comultiplication is defined by

Δ(a_{1}, …, a_{p}) = 1_{R} ⊗ (a_{1}, …, a_{p}) + ∑_{i=1}^{p−1}(a_{1}, …, a_{i}) ⊗ (a_{i+1}, …, a_{p})
+(a_{1}, …, a_{p}) ⊗ 1_{R} .

Moreover C^{+} is endowed with the counit η : C^{+} → R given by the natural projection and the coaugmentation ε : R → C^{+} given by the natural inclusion. The composition of the projection C^{+} → SA with the canonical morphism s : SA → A of degree +1 yields a homogeneous morphism τ : C^{+} → A of degree +1. A coderivation of C^{+} is a homogeneous R-linear map D : C^{+} → C^{+} such that Δ∘D = D⊗1+1⊗D. Note that this implies that η∘D = 0. Let Coder(C^{+}, C^{+}) denote the graded R-module whose p-th component is formed by the coderivations of degree p. Then the composition with τ is a bijection onto the space of homogeneous R-linear morphisms from C^{+} to A

(5.2)  Coder(C^{+}, C^{+}) →^{∼} Hom_{R}(C^{+}, A) , D ↦ τ ∘ D.

In particular, C^{+} admits a unique coderivation d_{C^{+}} of degree +1 such that τ ∘ d_{C^{+}} restricted to (SA)^{⊗_{R}p} vanishes for p ≠ 1, 2, and equals

μ ∘ (s ⊗ s) : (SA) ⊗_{R} (SA) → A

for p = 2 and −d_{A} ∘ s for p = 1. Here μ denotes the multiplication of A. The facts that μ : A ⊗ A → A is a morphism of complexes and that μ is associative are equivalent to the fact that d_{C^{+}}^{2} = 0.

Proposition ([15]). Endowed with the supercommutator and the differential D ↦ [d_{C^{+}}, D] the graded space Coder(C^{+}, C^{+}) becomes a differential graded Lie algebra which is isomorphic to the Gerstenhaber Lie algebra by the map (5.2).

For two homogeneous R-linear morphisms f, g : C^{+} → A, we define f ⋆ g = μ ∘ (f ⊗ g) ∘ Δ. Then we have

(5.3)  τ ⋆ τ = d ∘ τ + τ ∘ d

For a dg right A-module M and a dg left C^{+}-comodule N, we denote by M ⊗_{τ} N the dg R-module M ⊗_{R} N with the differential defined by

d(x ⊗ y) = d(x) ⊗ y + (−1)^{p}x ⊗ y + ∑ x τ(y_{(1)}) ⊗ y_{(2)} ,

where x is homogeneous of degree p and δ(y) = ∑ y_{(1)} ⊗ y_{(2)} (Sweedler’s notation). Similarly, for a left A-module M and a right C^{+}-comodule N, we define N ⊗_{τ} M to be N ⊗_{R} M with the differential defined by

d(x ⊗ y) = d(x) ⊗ y + (−1)^{p}x ⊗ y − ∑ x_{(1)} ⊗ τ(x_{(2)}) y ,

(note the sign in front of ∑). The fact that the squares of these differentials vanish follows from equation (5.3).

The dg R-module A⊗_{τ} C^{+} inherits a right C^{+}-comodule structure from C^{+} and a left A-module structure from A. It yields the dg A-A-bimodule

A ⊗_{τ} C^{+} ⊗_{τ} A.

It is not hard to check that up to the signs of the differentials, this is the (sum) total dg module associated with the bar resolution of the A-A-bimodule A. In particular, we have a canonical quasi-isomorphism (which is even an homotopy equivalence of left dg A-modules or right dg A-modules)

A ← A ⊗_{τ} C^{+} ⊗_{τ} A.

Now let φ : C^{+} → C^{+} be an automorphism of the dg counital R-coalgebra C^{+}. Define C_{φ}^{+} to be the dg C^{+}-C^{+}-bicomodule whose left comultiplication is that of C^{+} whereas the right comultiplication is (1 ⊗ φ) ∘ Δ. We define the bimodule

X(φ) = A ⊗_{τ} C_{φ}^{+} ⊗_{τ} A.

Note that the underlying graded module of X(φ) is A⊗_{R}C^{+}⊗_{R}A but the differential is twisted by φ. Now let ψ : C^{+} → C^{+} be another automorphism. Then we have a natural morphism of dg C^{+}-C^{+}-bicomodules

C_{ψφ}^{+} → C_{φ}^{+} ⊗_{τ} A ⊗_{τ} C_{ψ}^{+} , c ↦ ∑ c_{(1)} ⊗ 1_{A} ⊗ φ(c_{(2)}).

It induces a morphism of dg A-A-bimodules

(5.4)  X(ψφ) → X(φ) ⊗_{A} X(ψ).

Proposition. a) As a left dg A-module, X(ψ) is relatively cofibrant. In particular, we have a canonical isomorphism

X(φ) ⊗_{A}^{L,rel} X(ψ) →^{∼} X(φ) ⊗_{A} X(ψ)

in D_{R}(A^{op} ⊗ A).

b) The morphism (5.4) is a homotopy equivalence of dg R-modules.

In the next section, we will deduce this from results of [10]. Note that we obtain a morphism of groups from the group

Aut_{R−coalg}(C^{+})^{op}

to the group of autoequivalences of the relative derived category D_{R}A.

Now suppose that R is an augmented dg k-algebra and R = k⊕𝔪 the corresponding decomposition. Suppose that φ⊗_{R} k : C^{+} ⊗_{R} k → C^{+} ⊗_{R} k is the identity. Then clearly X(φ) ⊗_{R} k is isomorphic to X(1) ⊗_{R} k as a dg A-A-bimodule and we have a canonical isomorphism X(1) ⊗_{R} k → A in D(A^{op} ⊗ A). So we obtain a canonical isomorphism u_{φ} : X(φ) → A in D(A^{op} ⊗ A) and an object (X(φ), u_{φ}) of the fiber category F associated with the reduction functor D_{R}(R ⊗ A^{op} ⊗ A) → D(A^{op} ⊗ A) (cf. section 5.2).

5.8. Modules and comodules. Let R be a commutative dg k-algebra and A a dg R-algebra of the form A = A′ ⊗_{k} R for some dg k-algebra A′. We define A^{+} to be the augmented algebra R ⊕ A and C^{+} to be the coaugmented coalgebra defined in the previous section. We still denote by τ : C^{+} → A^{+} the composition of the morphism τ of the previous section with the inclusion A → A^{+}. Denote by Comod C^{+} the category of dg counital right C^{+}-comodules.

\xymatrix{Mod A^{+} \ar[d]^{B} \\ Comod C^{+} \ar[u]^{Ω}}

given by

BM = M ⊗_{τ} C^{+} , ΩN = N ⊗_{τ} A^{+}.

One can check [10] that they form an adjoint pair. Let W denote the class of morphisms s in Mod A^{+} whose restriction to Mod R is an homotopy equivalence and let W′ be the class of morphisms s of Comod C^{+} such that Ωs belongs to W.

Theorem. a) The dg A-module ΩN is relatively cofibrant for each dg C^{+}-

comodule N.

b) We have BW ⊂ W′ and ΩW′ ⊂ W and the functors B and Ω induce quasi-inverse equivalences between the localized categories

(Mod A^{+})[W^{−1}] →^{∼} (Comod C^{+})[W′^{−1}].

Theorem. The restriction functor Mod A → Mod A^{+} induces an equivalence from D_{R}(A) onto the full subcategory of (Mod A^{+})[W^{−1}] whose objects are the dg modules M such that M ⊗_{τ} C^{+} is R-relatively acyclic (i.e. its underlying dg R-module is contractible).

These theorems are proved in [10] in the case where R = k (the first one corresponds to Theorem 2.2.2.2 and the second one to Proposition 4.1.2.10 in [10]). We omit the proof in the general case since it is similar. Note however the following: If M is a dg A-module, then according to the first theorem, we have a canonical R-relative quasi-isomorphism

M ⊗_{τ} C^{+} ⊗_{τ} A^{+} → M.

The existence of such a quasi-isomorphism is not surprising. Indeed, the decomposition A^{+} = A ⊕ R yields an R-split short exact sequence

0 → M ⊗_{τ} C^{+} ⊗_{τ} A → M ⊗_{τ} C^{+} ⊗_{τ} A^{+} → M ⊗_{τ} C^{+} ⊗ R → 0.

The last term identifies with the augmented bar resolution of M. It is therefore relatively R-acyclic. The first term identifies with the bar resolution of M. It is therefore relatively R-quasi-isomorphic to M.

Now suppose that φ : C^{+} → C^{+} is an automorphism of dg R-coalgebras. Then it induces a selfequivalence F_{φ} of Comod C^{+} given by twisting by φ, i.e. if N is a dg C^{+}-comodule, then F_{φ}N is the dg comodule with the same underlying graded module and the same differential but with the new comultiplication δ_{φ} = (1 ⊗ φ) ∘ δ. Clearly F_{φ} preserves the subcategory of R-relatively acyclic comodules. Thus the composition ΩF_{φ}B preserves the image of D_{R}(A) in (Mod A^{+})[W^{−1}]. More precisely, if M is a dg A^{+}-module then

ΩF_{φ}BM = M ⊗_{τ} C_{φ}^{+} ⊗_{τ} A^{+}

and if M comes from a (unital) dg A-module, then the last term is R-relatively quasi-isomorphic to its submodule M ⊗_{τ} C_{φ}^{+} ⊗_{τ} A since

M ⊗_{τ} C_{φ}^{+} ⊗_{τ} R = M ⊗_{τ} C^{+}

is R-relatively acyclic. So for each M ∈ Mod A, we have a canonical R-relative quasi-isomorphism

M ⊗_{τ} C_{φ}^{+} ⊗_{τ} A → ΩF_{φ}BM.

If ψ is another automorphism, by composition, we obtain a canonical R-relative quasi-isomorphism

M ⊗_{τ} C_{φ}^{+} ⊗_{τ} A ⊗_{τ} C_{ψ}^{+} ⊗_{τ} A → ΩF_{ψ}BΩF_{φ}BM.

On the other hand, the adjunction morphism 1 → BΩ yields a morphism

ΩF_{ψ}BΩF_{φ}BM ← ΩF_{ψ}F_{φ}BM = ΩF_{ψφ}BM ,

which is also an R-relative quasi-isomorphism. To prove proposition 5.7, it remains to be checked that the morphism (5.4) constructed in (5.7) makes the following square commutative

\xymatrix{A ⊗_{τ} C_{φ}^{+} ⊗_{τ} A ⊗_{τ} C_{ψ}^{+} ⊗_{τ} A \ar[r] & ΩF_{ψ}BΩF_{φ}BA \\ A ⊗_{τ} C_{ψφ}^{+} ⊗_{τ} A \ar[u] \ar[r] & ΩF_{ψφ}BA \ar[u]}

This is left to the reader.

5.9. Proof of the main result. Let A be a dg k-algebra and R a commutative augmented dg k-algebra. We write R = k ⊕ 𝔪, where 𝔪 is the kernel of the augmentation.

Let C^{+} be the bar construction of A relative to k (cf. 5.7). Then the bar construction of A ⊗ R relative to R identifies with C^{+} ⊗ R. We put

LieAut(C^{+}, R) = ker(Aut_{R}(C^{+} ⊗ R) → Aut_{k}(C^{+}))

where Aut_{R} denotes the group of automorphisms of dg counital R-coalgebras.

We define LieDPic(A, R) to be the group of isomorphism classes (cf. section 4) of invertible dg bimodules X ∈ D_{R}(A^{op} ⊗ A ⊗ R) free as graded R-modules such that X ⊗_{R} k is isomorphic to A in D(A^{op} ⊗ A).

By section 5.7, we obtain a morphism of groups

Φ : LieAut(C^{+}, R)^{op} → LieDPic(A, R)

which is clearly functorial in R. As in the case of algebraic groups, one obtains canonical Lie brackets on the restrictions of these functors to the subcategory of augmented dg k-algebras R = k ⊕ 𝔪 with 𝔪^{2} = 0 (cf. section 5.6) and Φ is compatible with the bracket.

Lemma. If R = k ⊕ 𝔪 with 𝔪^{2} = 0, there is natural isomorphism

Z^{0}(Coder_{k}(C^{+}, C^{+}) ⊗_{k} 𝔪) → LieAut(C^{+}, R).

This is a variant of a classical result on infinitesimal deformations. The easy proof is left to the reader. As we recalled from [15] in section 5.7, we have a natural isomorphism of dg Lie algebras

C^{·+1}(A, A) →^{∼} Coder_{k}(C^{+}, C^{+}).

So for 𝔪^{2} = 0, we obtain morphisms

Z^{0}(C^{·+1}(A, A) ⊗ 𝔪) → LieAut(C^{+}, R) → LieDPic(A, R)^{op}.

compatible with the bracket. Now by 5.2, we have an isomorphism

Φ : LieDPic(A, R) →^{∼} Hom_{D(A^{op}⊗A)}(A, A ⊗ 𝔪[1]).

It is easy to see that the composition

Z^{0}(C^{·+1}(A, A) ⊗ 𝔪) → LieDPic(A, R) →^{∼} H^{0}(C^{·+1}(A, A) ⊗ 𝔪).

is the canonical surjection. So we have a commutative square

\xymatrix{Z^{0}(C^{·+1}(A, A) ⊗ 𝔪) \ar[d] \ar[r] & LieDPic(A, R) \ar[d]^{∼} \\ H^{0}(C^{·+1}(A, A) ⊗ 𝔪) \ar[r]^{∼} & Hom_{D(A^{op}⊗A)}(A, A ⊗ 𝔪[1])}

We see that if we transport the Gerstenhaber bracket to the lower right hand corner, then the map

Φ : LieDPic(A, R)^{op} → Hom_{D(A^{op}⊗A)}(A, A ⊗ 𝔪[1])

becomes an isomorphism which respects the bracket and is functorial with respect to R and with respect to invertible bimodule complexes X ∈ D(A^{op} ⊗ B).

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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