Homology, Homotopy and Applications, vol.4(2), 2002, pp.63–82

CROSSED EXTENSIONS OF ALGEBRAS AND HOCHSCHILD COHOMOLOGY

HANS-JOACHIM BAUES and ELIAS GABRIEL MINIAN

(communicated by Clas Löfwall)

Abstract We introduce the notion of crossed n-fold extensions of an algebra B by a bimodule M and prove that such extensions represent classes in the Hochschild cohomology of B with coefficients in M. Moreover we consider this way characteristic classes of chain (resp. cochain) algebras in Hochschild cohomology.

To Jan–Erik Roos on his sixty–fifth birthday

1. Introduction

Crossed modules over groups were introduced by J.H.C.Whitehead [12]. Mac Lane–Whitehead [11] observed that a crossed module over a group G with kernel a G-module M represents an element in the cohomology H^{3}(G, M). This result was generalized by Huebschmann [7] by showing that crossed n-fold extensions over G by M represent elements in H^{n+1}(G, M).

In this paper we prove similar results for the Hochschild cohomology HH^{n+1}(B, M) of an algebra B with coefficients in a B-bimodule M. We show that crossed modules over algebras as introduced in [2] can be used to define crossed n-fold extensions of B by M which represent elements in HH^{n+1}(B, M) for n ⩾ 2.

Our results are also available for graded algebras. In particular we show that each chain (resp. cochain) algebra C yields canonically a crossed module over the homology (resp. cohomology) algebra B = HC and this crossed module represents a characteristic class ⟨C⟩ in the Hochschild cohomology of HC. The characteristic class ⟨C⟩ determines all triple Massey products which are secondary operations on HC determined by C. We can consider ⟨C⟩ as an analogue of the first k-invariant of a connected space X (in the Postnikov decomposition) which is an element in the cohomology of the fundamental group G = π_{1}X. Berrick–Davydov [6] recently studied the class ⟨C⟩ without using crossed modules over algebras. We compute also the characteristic class ⟨A ⊗ B⟩ of the tensor product of chain algebras A and B.

Received November 3, 2000, revised August 21, 2001; published on July 12, 2002. 2000 Mathematics Subject Classification: 18G99. Key words and phrases: Hochschild cohomology, Massey products, crossed module. \begin{matrix}c \\ ◯ \end{matrix} 2002, Hans-Joachim Baues and Elias Gabriel Minian. Permission to copy for private use granted.

2. Hochschild cohomology

Let k be a field. Classical Hochschild cohomology is defined for algebras and also for graded algebras over k. We consider here the graded and the non-graded case at the same time. In this paper an algebra B will mean an associative (graded) algebra with unit k → B. A B-bimodule is a (graded) k-vector space V which is a left and a right B-module such that for a, b ∈ B and x ∈ V we have (ax)b = a(xb). For example B can be considered as a B-bimodule via the multiplication in B. Given two (graded) k-vector spaces V and W we denote the tensor product V ⊗_{k} W simply by V ⊗ W.

Recall that the Hochschild cohomology of B with coefficients in a B-bimodule M is the family of extension groups

HH^{∗}(B, M) = Ext_{B−B}^{∗}(B, M)  (2.1)

between the B-bimodules B and M.

One can associate to B the bar complex B_{∗}(B), where B_{n}(B) = B^{⊗(n+2)} with differential d : B_{n}(B) → B_{n−1}(B) given by

d(x_{0} ⊗ … ⊗ x_{n+1}) = ∑_{i=0}^{n}(−1)^{i}(x_{0} ⊗ … ⊗ x_{i−1} ⊗ x_{i}x_{i+1} ⊗ x_{i+2} ⊗ … ⊗ x_{n+1})

The bar complex is acyclic for any B. This follows from the existence of a homotopy h between the identity of B_{∗}(B) and the zero map. The homotopy h : B_{n}(B) → B_{n+1}(B) is defined by h(x) = 1 ⊗ x.

The differential of the bar complex is B-bilinear. Thus we get the standard resolution of the B-bimodule B. Using this resolution one can identify the cohomology groups HH^{n}(B, M) with the cohomology of the complex

F^{n}(B, M) = Hom_{B−B}(B^{⊗(n+2)}, M) = Hom_{k}(B^{⊗n}, M)  (2.2)

with differential δ : F^{n}(B, M) → F^{n+1}(B, M) given by

(δf)(x_{1} ⊗ … ⊗ x_{n+1}) =x_{1}f(x_{2} ⊗ … ⊗ x_{n+1})

+ (∑_{i=1}^{n}(−1)^{i}f(x_{1} ⊗ … ⊗ x_{i}x_{i+1} ⊗ … ⊗ x_{n+1}))

+ (−1)^{n+1}f(x_{1} ⊗ … ⊗ x_{n})x_{n+1}.

3. Crossed modules over algebras and HH^{3}

We recall from [2] the following definition of crossed modules over algebras.

Definition 3.1. A Crossed module over a k-algebra is a triple (V, A, ∂) where A is a (graded) k-algebra, V is a (graded) A-bimodule and ∂ : V → A is a map of A-bimodules such that (∂v)w = v(∂w) for v, w ∈ V. A map (α, β) : (V, A, ∂) → (V′, A′, ∂′) between crossed modules consists of a map α : V → V′ of k-vector spaces and a map β : A → A′ of k-algebras such that ∂′α = β∂ and α(avb) = β(a)α(v)β(b) for a, b ∈ A and v ∈ V.

Given a crossed module ∂ : V → A, we consider B = coker(∂) and M = ker(∂) in the category of (graded) k-vector spaces. The algebra structure of A induces an algebra structure on B and the A-bimodule structure on V induces a B-bimodule structure on M given by π(a)m = am and mπ(a) = ma where π : A → coker(∂) is the projection. This multiplication is well defined since (a+∂(v))m = am+∂(v)m = am + v∂(m) = am. Hence a crossed module yields an exact sequence

\xymatrix{0 \ar[r] & M \ar[r]^{i} & V \ar[r]^{∂} & A \ar[r]^{π} & B \ar[r] & 0}

in which all the maps are maps of A-bimodules. Here the A-bimodule structure on M and B is induced by the map π. We call ∂ : V → A a crossed module over the k-algebra B with kernel M. Let Cross(B, M) be the category of such crossed modules. Morphisms are maps between crossed modules which induce the identity on M and B.

For a category C, let π_{0}C be the class of connected components in C. An object in π_{0}C is also termed a connected class of objects in C. In fact, π_{0} Cross(B, M) is a set, as implied by the following result, which extends the well-known facts that HH^{1}(B, M) is given by derivations and HH^{2}(B, M) classifies the singular algebra extensions of M by B (cf. [10]). The proof of this result can also be found in [9].

Theorem 3.2. There exists a bijection

ψ : π_{0} Cross(B, M) → HH^{3}(B, M).

Proof. We define ψ : π_{0} Cross(B, M) → HH^{3}(B, M) as follows. Given

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{i} & V \ar[r]^{∂} & A \ar[r]^{π} & B \ar[r] & 0})

choose k-linear sections s : B → A, πs = 1 and q : Im(∂) → V, ∂q = 1. For x, y ∈ B, we have π(s(x)s(y) − s(xy)) = 0 and then s(x)s(y) − s(xy) ∈ Im(∂). Take g(x, y) = q(s(x)s(y) − s(xy)) ∈ V and define

θ_{E}(x, y, z) = s(x)g(y, z) − g(xy, z) + g(x, yz) − g(x, y)s(z) (§)

Since ∂ is a map of A-bimodules it follows that ∂(θ_{E}(x, y, z)) = 0 and therefore θ_{E}(x, y, z) ∈ M = ker(∂). Thus we have defined a k-linear map θ_{E} : B^{⊗3} → M which is a cocycle with respect to the coboundary map δ in (2.2). In fact, one easily checks that δ(θ_{E}) = 0. We define ψ : π_{0} Cross(B, M) → HH^{3}(B, M) by taking ψ(E) to be the class of θ_{E} in HH^{3}(B, M).

One has to check that ψ is a well defined function from π_{0} Cross(B, M) to HH^{3}(B, M), i.e. the class of θ_{E} in HH^{3}(B, M) does not depend on the sections s and q. Moreover, if E → E′ is a map in Cross(B, M), then θ_{E} = θ_{E′} in HH^{3}(B, M).

We show first that the class of θ_{E} does not depend on the section s. Suppose \overline{s} : B → A is another section of π and let \overline{θ}_{E} be the map defined using \overline{s} instead of s. Since s and \overline{s} are both sections of π there exists a linear map h : B → V with s − \overline{s} = ∂h. We have

(θ_{E} − \overline{θ}_{E})(x, y, z) =h(x)(s(y)s(z) − s(yz)) − (s(x)s(y) − s(xy))h(z)
+ \overline{s}(x)(g − \overline{g})(y, z) − (g − \overline{g})(xy, z)
+ (g − \overline{g})(x, yz) − (g − \overline{g})(x, y)\overline{s}(z) (∗)

where g(x, y) = q(s(x)s(y) − s(xy)) and \overline{g}(x, y) = q(\overline{s}(x)\overline{s}(y) − \overline{s}(xy)). We define a map b : B^{⊗2} → V as follows.

b(x, y) = s(x)h(y) − h(xy) + h(x)s(y) − h(x)∂h(y)

Since ∂b = ∂(g−\overline{g}) then (g−\overline{g}−b) is a map from B^{⊗2} to M. Moreover we can replace (g − \overline{g}) by b without changing the equality (∗) in HH^{3}(B, M) since the difference is the coboundary δ(g − \overline{g} − b). After replacing (g − \overline{g}) by b, since ∂ : V → A is a crossed module we obtain the following equality in HH^{3}(B, M).

(∗) ≡∂h(x)s(y)h(z) − h(x)s(y)∂h(z) − ∂h(x)h(yz) + h(x)∂h(yz)
+ ∂h(x)h(y)\overline{s}(z) − h(x)∂h(y)\overline{s}(z) = 0

That proves that the class of θ_{E} does not depend on the section s.

Consider a map E → E′ as follows.

[figure: 0; M; i; V; ∂; α; A; π; β; B; 0; 0; M; i′; V′; ∂′; A′; π′; B; 0]

Let s : B → A and q : Im(∂) → V be sections of π and ∂ and let s′ : B → A′ and q′ : Im(∂′) → V′ be sections of π′ and ∂′. Then

(θ_{E} − θ_{E′})(x, y, z) =α(s(x)q(s(y)s(z) − s(yz)) − q(s(xy)s(z) − s(xyz))
+ q(s(x)s(yz) − s(xyz)) − q(s(x)s(y) − s(xy))s(z))
− s′(x)q′(s′(y)s′(z) − s′(yz)) + q′(s′(xy)s′(z) − s′(xyz))

− q′(s′(x)s′(yz) − s′(xyz)) + q′(s′(x)s′(y) − s′(xy))s′(z) (∗)

Since π′βs = 1 then βs is another section for π′ and therefore we can now replace s′ by βs and we obtain the following equality in HH^{3}(B, M).

(∗) ≡βs(x)((αq − q′β)(s(y)s(z) − s(yz))) − (αq − q′β)(s(xy)s(z).s(xyz))

+ (αq − q′β)(s(x)s(yz) − s(xyz)) − (αq − q′β)(s(x)s(y) − s(xy))βs(z)

Thus (θ_{E} − θ_{E′})(x, y, z) = δϕ(x, y, z) for some ϕ : B^{⊗2} → M. This proves that θ_{E} = θ_{E′} in HH^{3}(B, M) and that the class of θ_{E} in HH^{3}(B, M) does not depend on the sections s and q. Therefore ψ is well defined.

The bijectivity of ψ follows from the following lemma.

Lemma 3.3. Given c ∈ HH^{3}(B, M) there exists a crossed module E_{c} ∈ Cross(B, M) such that θ_{E_{c}} = c. Moreover if E ∈ Cross(B, M) and θ_{E} = c in HH^{3}(B, M) there exists a map of crossed modules E_{c} → E.

Before we proceed with the proof, we show a construction which will be useful to prove the lemma.

Construction 3.4. Free crossed module. Given a (graded) k-algebra A, a (graded) k-vector space V and a k-linear map d : V → A (of degree 0) we obtain the free crossed module with basis (V, d) as follows. Define

\xymatrix{A ⊗ V ⊗ A ⊗ V ⊗ A \ar[r]^{d_{2}} & A ⊗ V ⊗ A \ar[r]^{d_{1}} & A}

by

d_{2}(a ⊗ x ⊗ b ⊗ y ⊗ c) = (a(dx)b ⊗ y ⊗ c) − (a ⊗ x ⊗ b(dy)c)
d_{1}(a ⊗ x ⊗ b) = a(dx)b

for a, b, c ∈ A and x, y ∈ V. Since d_{1}d_{2} = 0 then d_{1} induces

∂ : W = A ⊗ V ⊗ A/Im(d_{2}) → A.

It is easy to see that (W, A, ∂) is a crossed module which has the universal property of the free crossed module with basis (V, d).

Proof of Lemma 3.3. Let

T (B) = ⨁_{n⩾0} B^{⊗n}

be the tensor algebra generated by B as a k-vector space and let π : T (B) → B be the map of algebras given by π(a_{1} ⊗ … ⊗ a_{n}) = a_{1} … a_{n}. Let V = B ⊗ B and let d : V → T (B) be the linear map defined by

d(x ⊗ y) = x ⊗ y − xy

and consider (W, T (B), ∂) the free crossed module with basis (V, d). The cokernel of this crossed module is the algebra B. Let N be the kernel of ∂.

Now consider the bar resolution (B^{⊗(n+2)}, d) and the following commutative diagram of vector spaces

\xymatrix{0 \ar[r] & N \ar[r] \ar[d]_{h} & W \ar[r]^{∂} \ar[d]^{h_{1}} & T (B) \ar[d]^{h_{0}} \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & B^{⊗5}/Im(d_{4}) \ar[r]^{d_{3}} & B^{⊗4} \ar[r]^{d_{2}} & B^{⊗3} \ar[r]^{d_{1}} & B^{⊗2} \ar[r]^{μ} & B \ar[r] & 0}

Here the map h_{0} : T (B) → B^{⊗3} is not a bimodule map but a derivation defined by h_{0}(b) = 1 ⊗ b ⊗ 1 for b ∈ B and h_{0}(xy) = π(x)h_{0}(y) + h_{0}(x)π(y) for x, y ∈ T (B). The map h_{1} : W → B^{⊗4} is the bimodule map defined by h_{1}(x ⊗ a ⊗ b ⊗ y) = π(x) ⊗ a ⊗ b ⊗ π(y) for x, y ∈ T (B) and a, b ∈ B. It is easy to see that d_{2}h_{1} = h_{0}∂. By restricting h_{1} to N we obtain the map of B-bimodules h : N → B^{⊗5}/Im(d_{4}).

An element c ∈ HH^{3}(B, M) can be seen as a map c : B^{⊗5}/Im(d_{4}) → M of Bbimodules. Composing with h we obtain the map c̃ = ch : N → M of B-bimodules.

Consider the pushout of T (B)-bimodules

\xymatrix{N \ar[d]^{P ush}_{c̃} \ar[r] & W \ar[d]^{r} \ar[ddr]^{∂} \\ M \ar[r] \ar[drr]_{0} & \overline{W} \ar@{.>}[dr]^{\overline{∂}} \\ & & T (B)} .

We show that E_{c} = ( \xymatrix{0 \ar[r] & M \ar[r] & \overline{W} \ar[r]^{\overline{∂}} & T (B) \ar[r] & B \ar[r] & 0}) is the desired crossed module.

The free crossed module F = ( \xymatrix{0 \ar[r] & N \ar[r] & W \ar[r]^{∂} & T (B) \ar[r] & B \ar[r] & 0}) induces a cocycle θ_{F} : B^{⊗5}/Im(d_{4}) → N as a map of B-bimodules by the formula (§) via the linear sections s : B → T (B) given by s(b) = b and q : Im(∂) = ker(π) → W defined by q(x ⊗ y − xy) = \overline{1 ⊗ x ⊗ y ⊗ 1} ∈ W. Now θ_{E_{c}} can be computed from θ_{F} since the map \overline{q} = rq : Im(\overline{∂}) = ker(π) → \overline{W} is a section of \overline{∂}, that is

θ_{E_{c}} = chθ_{F} : B^{⊗5}/Im(d_{4}) → M.

Since hθ_{F} = 1 : B^{⊗5}/Im(d_{4}) → B^{⊗5}/Im(d_{4}) it follows that θ_{E_{c}} = c.

Suppose now we have a crossed module

E = ( \xymatrix{0 \ar[r] & M \ar[r] & V \ar[r]^{α} & A \ar[r]^{p} & B \ar[r] & 0})

such that ψ(E) = c ∈ HH^{3}(B, M). That implies that for certain choice of linear sections s : B → A and q : Im(α) → V we have θ_{E} = c where θ_{E} is constructed by the formula (§). This induces a map of crossed modules

[figure: 0; N; ch; W; ∂; \overline{g}; T (B); π; \overline{s}; B; 0; 0; M; V; A; B; 0]

where the map \overline{s} : T (B) → A is the map of algebras induced by s and the map \overline{g} : W → V is induced by the linear map g : B^{⊗2} → V, g(x, y) = q(s(x)s(y)−s(xy)). By the universal property of the pushout

\xymatrix{N \ar[d]^{P ush}_{c̃} \ar[r] & W \ar[d]^{r} \\ M \ar[r] & \overline{W}},

there is a map E_{c} → E in Cross(B, M).

Any differential graded k-algebra induces a crossed module as we can see in the following construction.

Construction 3.5. The characteristic class of a cochain algebra. Let C be a differential graded k-algebra with differential of degree +1, that is C = ⨁_{i⩾0} C^{i} with C^{i}C^{j} ⊆ C^{i+j} and a differential d : C → C of degree +1 satisfying d(xy) = (dx)y + (−1)^{|x|}xd(y) and d^{2} = 0. Consider the graded k-vector spaces V = coker(d)[1] and A = ker(d). Here we define for a graded vector space W the shifted graded vector space W [1] by

(W [1])^{n+1} = W^{n}

The elements in (W [1])^{n+1} are denoted by s(w), where w ∈ W^{n}. Hence for the cokernel of the differential W = coker(d) we obtain the shifted object V = coker(d)[1]. We denote by s(\overline{x}) ∈ coker(d)[1] the element corresponding to x ∈ C via the projection C ↠ coker(d). Then d induces a map of graded k-vector spaces

∂ : V = coker(d)[1] → A = ker(d)

carrying s(\overline{x}) to dx. The multiplication in C induces a structure of k-algebra on A. Moreover it induces a structure of A-bimodule on V by setting

a ∗ s(\overline{x}) = (−1)^{|a|}s(\overline{ax})
s(\overline{x}) ∗ b = s(\overline{xb})

In fact, for y = dz and a ∈ A we have (−1)^{|a|}ay = d(az) and therefore the multiplication is well defined. We now check that ∂ : V → A is a crossed module. Given a ∈ A and s(\overline{x}) ∈ V we have

∂(a ∗ s(\overline{x})) = (−1)^{|a|}∂(s(\overline{ax})) = (−1)^{|a|}d(ax) = ad(x) = a∂(s(\overline{x}))

In the same way one can check that ∂(s(\overline{x})∗a) = ∂(s(\overline{x}))a. Given now s(\overline{x}), s(\overline{y}) ∈ V we have

∂(s(\overline{x})) ∗ s(\overline{y}) = (dx) ∗ s(\overline{y}) = (−1)^{|x|+1}s(\overline{(dx)y}) = s(\overline{x(dy)}) = s(\overline{x}) ∗ (dy) =
s(\overline{x}) ∗ ∂(s(\overline{y})).

Thus the DG-algebra C induces a crossed module (V, A, ∂), the cokernel of which is the algebra H^{∗}(C) and the kernel is the H^{∗}(C)-bimodule whose underlying k-vector space is H^{∗}(C)[1] and where the left multiplication is twisted, i.e. x ∗ y = (−1)^{|x|}xy and the right multiplication is the ordinary one. We denote this H^{∗}(C)-bimodule by \overline{H^{∗}(C)}[1]. The crossed module ∂ : V → A represents by 3.2 an element ⟨C⟩ ∈ HH^{3}(H^{∗}(C), \overline{H^{∗}(C)}[1]) which is termed the characteristic class of the cochain algebra C (compare with [6]).

Construction 3.6. The characteristic class of a chain algebra. For a chain algebra C = {C_{i}, i ⩾ 0} concentrated in non-negative degrees with differential d : C → C of degree −1 satisfying d(xy) = (dx)y + (−1)^{|x|}xd(y) and d^{2} = 0 we proceed in the same way as in 3.5. We consider the graded vector spaces V = coker(d)_{⩾1}[−1] and A = ker(d). Here coker(d)_{⩾1}[−1] denotes the shifted graded vector space from coker(d)_{⩾1} similarly as above. The differential d induces as before a crossed module

∂ : V = coker(d)_{⩾1}[−1] → A = ker(d)

the cokernel of which is the algebra H_{∗}(C) and the kernel is the H_{∗}(C)-bimodule whose underlying vector space is H_{⩾1}(C)[−1] and where the left multiplication is twisted and the right multiplication is the ordinary one. We denote this bimodule by \overline{H_{⩾1}(C)}[−1]. The crossed module represents by 3.2 an element ⟨C⟩ ∈ HH^{3}(H_{∗}(C), \overline{H_{⩾1}(C)}[−1]) which is termed the characteristic class of the chain algebra C.

Defination and Remark 3.7. Massey triple products of crossed modules. Let

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{i} & V \ar[r]^{∂} & A \ar[r]^{π} & B \ar[r] & 0})

be a crossed module over B with kernel M. Given a, b, c ∈ B with ab = bc = 0, we define the Massey triple product ⟨a, b, c⟩ ∈ M/(aM + Mc) as follows. Let s : B → A be a k-linear section of π, i.e. πs = 1 and let q : Im(∂) → V be a k-linear section of ∂. Since ab = 0 then s(a)s(b) ∈ ker(π) and we can take q(s(a)s(b)) ∈ V. In the same way, since bc = 0, we consider q(s(b)s(c)) ∈ V. Now consider the element {a, b, c} = s(a)q(s(b)s(c)) − q(s(a)s(b))s(c) ∈ V. Since ∂({a, b, c}) = 0 this element is in fact in M and we define

⟨a, b, c⟩ = \overline{\{a, b, c\}} ∈ M/(aM + Mc),

where \overline{\{a, b, c\}} denotes the class of \{a, b, c\} in the quotient. One can check that ⟨a, b, c⟩ does not depend on the choice of s and q. Moreover it depends only on the class of E in π_{0} Cross(B, M) and the elements a, b and c. In fact ⟨a, b, c⟩ can be computed from HH^{3}(B, M) by taking

⟨a, b, c⟩ = \overline{θ_{E}(a, b, c)}

where θ_{E} is any cocycle representing the class of ψ(E) ∈ HH^{3}(B, M). Note that for any DG-algebra C and any Massey triple x, y, z ∈ H^{∗}(C) the Massey product defined here in terms of ∂ in 3.5 coincides with the classical one.

Remark 3.8. Connection with Baues–Wirsching cohomology of categories. Given a monoid C one can consider C as a category C with one object ∗. Let M : C × C^{op} → Vect_{k} be a functor, where Vect_{k} denotes the category of k-vector spaces. Then the C-bimodule M induces a natural system on C also denoted by M (see [5]) and we have the Baues–Wirsching cohomology groups of C with coefficients in M denoted by H^{n}(C, M). On the other hand one can consider the k-algebra k[C] and the k[C]-bimodule iM induced by the C-bimodule M. It is easy to see that HH^{n}(k[C], iM) = H^{n}(C, M). For n = 3 this isomorphism induces a bijection

π_{0}Track(C, M) = π_{0} Cross(k[C], iM).

Here Track(C, M) denotes the category of track extensions over C with kernel M (cf. [3],[4]).

In the last section of this paper we define the ⊙-product of crossed modules in order to compute the characteristic class of a tensor product of differential algebras.

4. crossed n-fold extensions and main result

We introduce in this section the groups Opext^{n}(B, M) of crossed n-fold extensions of a k-algebra B by a B-bimodule M, n ⩾ 2. These extensions are analogous to crossed extensions of groups (cf. [7]). Our result 4.3 shows that the connected classes of such extensions represent cohomology classes in HH^{n+1}(B, M).

Definition 4.1. Let B be a k-algebra and M a B-bimodule. For n ⩾ 2, a crossed n-fold extension of B by M is an exact sequence

\xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r]^{∂_{n−1}} & … \ar[r]^{∂_{2}} & M_{1} \ar[r]^{∂_{1}} & A \ar[r]^{π} & B \ar[r] & 0}

of k-vector spaces with the following properties.

1. (M_{1}, A, ∂_{1}) is a crossed module with cokernel B,

2. M_{i} is a B-bimodule for 1 < i ⩽ n − 1 and ∂_{i} and f are maps of B-bimodules.

Note that the map ∂_{1} is a map of A-bimodules since (M_{1}, A, ∂_{1}) is a crossed module and it makes sense to require ∂_{2} to be a map of B-bimodules since the kernel of ∂_{1} is naturally a B-bimodule.

Definition 4.2. Given a crossed n-fold extension of B by M

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r]^{∂_{n−1}} & … \ar[r]^{∂_{2}} & M_{1} \ar[r]^{∂_{1}} & A \ar[r]^{π} & B \ar[r] & 0})

and a crossed n-fold extension of B by M′

E′ = ( \xymatrix{0 \ar[r] & M′ \ar[r]^{f′} & M′_{n−1} \ar[r]^{∂′_{n−1}} & … \ar[r]^{∂′_{2}} & M′_{1} \ar[r]^{∂′_{1}} & A′ \ar[r]^{π′} & B \ar[r] & 0})

a map from E to E′ is a sequence (α, δ_{n−1}, …, δ_{1}, β) such that α : M → M′ and δ_{i} : M_{i} → M′_{i} are morphisms of B-bimodules for i ⩾ 2, (δ_{1}, β) : (M_{1}, A, ∂_{1}) → (M′_{1}, A′, ∂′_{1}) is a map of crossed modules which induces the identity on B and the whole diagram commutes.

Let E^{n}(B, M) be the following category. The objects are the crossed n-fold extensions of B by M and the morphisms are the maps between such extensions that induce the identity on M. We denote Opext^{n}(B, M) = π_{0}E^{n}(B, M). Of course Opext^{2}(B, M) coincides with π_{0} Cross(B, M).

We will exhibit a natural structure of Abelian group on Opext^{n}(B, M) and prove the main result of this section.

Theorem 4.3. There exists an isomorphism of Abelian groups

Opext^{n}(B, M) = HH^{n+1}(B, M), n ⩾ 2.

Definition 4.4. For n ⩾ 3 we define the element 0 ∈ Opext^{n}(B, M) as the class of the extension

[figure: 0; M; M; 0; …; 0; B; B; 0].

Remark 4.5. If B is a projective algebra or M is injective as a B-bimodule, then Opext^{n}(B, M) = 0. In general, if

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r]^{∂_{n−1}} & … \ar[r]^{∂_{2}} & M_{1} \ar[r]^{∂_{1}} & A \ar[r]^{π} & B \ar[r] & 0})

and there is a map g : M_{n−1} → M such that gf = 1_{M}, then E = 0 in Opext^{n}(B, M), n ⩾ 3.

Proposition 4.6. Given E ∈ Opext^{n}(B, M) and a map α : M → M′ of Bbimodules, there exists an extension αE ∈ Opext^{n}(B, M′) and a morphism of the form (α, δ_{n−1}, …, β) from E to αE. Moreover, αE is unique in Opext^{n}(B, M′) with this property.

Proof. Let E = ( \xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r]^{∂_{n−1}} & … \ar[r]^{∂_{2}} & M_{1} \ar[r]^{∂_{1}} & A \ar[r]^{π} & B \ar[r] & 0}). Consider the following pushout of B-bimodules

[figure: M; α; P ush; M_{n−1}; i; M′; 0; \overline{M_{n−1}}; M_{n−2}]

Take αE = ( \xymatrix{0 \ar[r] & M′ \ar[r] & \overline{M_{n−1}} \ar[r] & . . . . . . \ar[r] & M_{1} \ar[r]^{∂_{1}} & A \ar[r] & B \ar[r] & 0}) in

Opext^{n}(B, M′) and the morphism (α, i, 1, …, 1) : E → αE.

Given E′ ∈ Opext^{n}(B, M′) and a morphism of the form (α, δ_{n−1}, …, β) : E → E′, by properties of the pushout we find a map (1, j, δ_{n−2}, …, β) : αE → E′ and therefore αE = E′ ∈ Opext^{n}(B, M′).

Defination and Remark 4.7. By 4.6, a morphism of B-bimodules α : M → M′ induces a well defined function

α_{∗} : Opext^{n}(B, M) → Opext^{n}(B, M′)

by α_{∗}(E) = αE.

Lemma 4.8. If E = (\xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r] & …}) ∈ Opext^{n}(B, M), then fE = 0 ∈ Opext^{n}(B, M_{n−1}).

Proof. Consider the morphism of extensions

[figure: 0; M; f; f; M_{n−1}; g; (1,g); M_{n−2}; Identity; …; 0; M_{n−1}; (1,0); M_{n−1} ⊕ M_{n−2}; p_{2}; M_{n−2}; …]

By definition the row in the bottom corresponds to fE, therefore by 4.5 fE = 0.

Definition 4.9. Given two crossed n-fold extensions of B

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r]^{∂_{n−1}} & … \ar[r]^{∂_{2}} & M_{1} \ar[r]^{∂_{1}} & A \ar[r]^{π} & B \ar[r] & 0})

and

E′ = ( \xymatrix{0 \ar[r] & M′ \ar[r]^{f′} & M′_{n−1} \ar[r]^{∂′_{n−1}} & … \ar[r]^{∂′_{2}} & M′_{1} \ar[r]^{∂′_{1}} & A′ \ar[r]^{π′} & B \ar[r] & 0})

the sum of E and E′ over B is denoted by E ⊕_{B} E′ and corresponds to the following crossed n-fold extension

[figure: 0; M ⊕ M′; M_{n−1} ⊕ M′_{n−1}; …]

[figure: M_{1} ⊕ M′_{1}; (∂_{1},∂′_{1}); A ×_{B} A′; q; B; 0].

Here the algebra A×_{B} A′ is defined as follows. The elements of it are the pairs (a, a′) with a ∈ A and a′ ∈ A′ such that πa = π′a′, addition and multiplication is defined coordinatewise. The map q : A ×_{B} A′ → B is the map q(a, a′) = π(a) = π′(a′). The action of A ×_{B} A′ on M_{1} ⊕ M′_{1} is also defined coordinatewise. It is easy to check that this defines a crossed module (M_{1} ⊕ M′_{1}, A ×_{B} A′, (∂_{1}, ∂′_{1})).

Definition 4.10. Given E, E′ ∈ Opext^{n}(B, M) with n ⩾ 3, we define the Baer Sum E + E′ ∈ Opext^{n}(B, M) as follows.

E + E′ = ∇_{M}(E ⊕_{B} E′)

where ∇_{M} : M ⊕ M → M is the codiagonal.

Theorem 4.11. For n ⩾ 3 the set Opext^{n}(B, M) equipped with the Baer sum is an abelian group with the zero element defined as in 4.4. The inverse of an extension

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r]^{g} & . . . . . . \ar[r] & M_{1} \ar[r]^{∂_{1}} & A \ar[r] & B \ar[r] & 0})

is the extension

(−1_{M})E = ( \xymatrix{0 \ar[r] & M \ar[r]^{−f} & M_{n−1} \ar[r]^{g} & . . . . . . \ar[r] & M_{1} \ar[r]^{∂_{1}} & A \ar[r] & B \ar[r] & 0})

Moreover, the maps α_{∗} : Opext^{n}(B, M) → Opext^{n}(B, M′) are morphisms of groups.

Proof. Follows the classical one (cf. [10]). One has to check that

1. (α + β)E = αE + βE

2. α(E + E′) = αE + αE′

The Baer sum in Opext^{2}(B, M) is defined in a slightly different way. Recall that the elements in Opext^{2}(B, M) are classes of crossed modules with cokernel B and kernel M. The class of 0 ∈ Opext^{2}(B, M) is the class of the extension

[figure: 0; M; M; 0; B; B; 0]

Now given

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{i} & V \ar[r]^{∂} & A \ar[r]^{π} & B \ar[r] & 0})

and

E′ = ( \xymatrix{0 \ar[r] & M \ar[r]^{i′} & V′ \ar[r]^{∂′} & A′ \ar[r]^{π′} & B \ar[r] & 0}),

the Baer sum E + E′ is the class of the extension

E + E′ = ( \xymatrix{0 \ar[r] & M \ar[r]^{j} & V + V′ \ar[r]^{˜∂} & A ×_{B} A′ \ar[r]^{q} & B \ar[r] & 0})

where q : A ×_{B} A′ → B is defined as in 4.9 and V + V′ is the pushout of k-vector spaces

\xymatrix{M ⊕ M \ar[d]^{P ush}_{∇} \ar[r]^{i⊕i′} & V ⊕ V′ \ar[d]^{r} \ar[ddr]^{(∂,∂′)} \\ M \ar[r]^{j} \ar[drr]_{0} & V + V′ \ar@{.>}[dr]^{˜∂} \\ & & A ×_{B} A′} .

The structure of (A×_{B} A′)-bimodule on V +V′ is induced by the structure on V ⊕V′ (coordinatewise) via the quotient map r : V ⊕ V′ → V + V′ by (a, a′)r(v, v′) = r(av, a′v′) and r(v, v′)(a, a′) = r(va, v′a′). Note that the multiplication is well defined since (a, a′) ∈ A ×_{B} A′ and therefore π(a) = π′(a′). It is easy to check that ˜∂: V + V′ → A ×_{B} A′ is a crossed module.

Remark 4.12. With this structure of abelian group in Opext^{2}(B, M) the bijection

ψ : Opext^{2}(B, M) → HH^{3}(B, M)

of 3.2 is an isomorphism of groups.

Definition 4.13. Given a short exact sequence of B-bimodules

\xymatrix{0 \ar[r] & M \ar[r]^{α} & M′ \ar[r]^{β} & M′′ \ar[r] & 0}

we define a connecting homomorphism (n ⩾ 2)

δ : Opext^{n}(B, M′′) → Opext^{n+1}(B, M)

as follows. Given an extension E = ( \xymatrix{0 \ar[r] & M′′ \ar[r]^{f} & M_{n−1} \ar[r] & …} ), take δ(E)

to be the class of the extension ( \xymatrix{0 \ar[r] & M \ar[r]^{α} & M′ \ar[r]^{fβ} & M_{n−1} \ar[r] & …} ).

Note that δ is a well defined homomorphism for all n ⩾ 2.

Theorem 4.14. A short exact sequence

\xymatrix{0 \ar[r] & M \ar[r]^{α} & M′ \ar[r]^{β} & M′′ \ar[r] & 0}

of B-bimodules induces a long exact sequence of abelian groups (n ⩾ 2)

[figure: Opext^{n}(B, M); α_{∗}; Opext^{n}(B, M′); β_{∗}; Opext^{n}(B, M′′); δ]
\xymatrix{Opext^{n+1}(B, M) \ar[r] & …}

Proof. To prove exactness at Opext^{n}(B, M′) with n ⩾ 3 note first that β_{∗}α_{∗} = (βα)_{∗} = 0. Now let

E = ( \xymatrix{0 \ar[r] & M′ \ar[r]^{f} & M_{n−1} \ar[r]^{g} & M_{n−2} \ar[r] & … \ar[r] & M_{1} \ar[r]^{∂} & A \ar[r] & B \ar[r] & 0}) ∈ Opext^{n}(B, M′)

and βE = 0. We suppose first that there is a map βE → 0, i.e.

βE = ( \xymatrix{0 \ar[r] & M′′ \ar[r]^{h} & \overline{M_{n−1}} \ar[r]^{g′} & M_{n−2} \ar[r] & … \ar[r] & M_{1} \ar[r]^{∂} & A \ar[r] & B \ar[r] & 0})

and there is a map r : \overline{M_{n−1}} → M′′ such that rh = 1. The following diagram shows that E = α\overline{E}.

[figure: 0; M; fα; α; ker rt; g; i; M_{n−2}; Id; …; 0; M′; f; β; M_{n−1}; t; g; M_{n−2}; Id; …; 0; M′′; h; \overline{M_{n−1}}; g′; M_{n−2}; …]

Suppose now that there is a map 0 → βE. In this case it is easy to see that E = 0. The general case follows combining these both cases. Suppose for example there exists an

extension Ẽ = ( \xymatrix{0 \ar[r] & M′′ \ar[r]^{l} & M̃_{n−1} \ar[r] & M̃_{n−2} \ar[r] & …} ) ∈ Opext^{n}(B, M′′)

and maps Ẽ → βE and Ẽ → 0. In this case we construct the extension \overline{E} with α\overline{E} = E as follows. There exists a retraction r : M̃_{n−1} → M′′ such that rl = 1. Consider the pushout of B-bimodules

\xymatrix{M̃_{n−1} \ar[d]^{P ush}_{r} \ar[r] & \overline{M}_{n−1} \ar[d]^{\overline{r}} \\ M′′ \ar[r] & \overline{M′′}}

and take \overline{E} = ( \xymatrix{0 \ar[r] & M \ar[r]^{fα} & Ker \overline{r}t \ar[r]^{g} & M_{n−2} \ar[r] & …} ).

For n = 2 exactness at Opext^{2}(B, M′) follows from 3.2.

To prove exactness at Opext^{n+1}(B, M) for n ⩾ 2 note first that δ(E) has the form

δ(E) = ( \xymatrix{0 \ar[r] & M \ar[r]^{α} & M′ \ar[r]^{fβ} & M_{n−1} \ar[r] & … \ar[r] & M_{1} \ar[r]^{∂} & A \ar[r] & B \ar[r] & 0})

and therefore αδ(E) = 0 by 4.8. Now let

E = ( \xymatrix{0 \ar[r] & M \ar[r]^{f} & M_{n−1} \ar[r]^{g} & M_{n−2} \ar[r] & … \ar[r] & M_{1} \ar[r]^{∂} & A \ar[r] & B \ar[r] & 0}) ∈ Opext^{n}(B, M)

with αE = 0. Applying the same argument as above, we can suppose that there is a map αE → 0, i.e.

αE = ( \xymatrix{0 \ar[r] & M′ \ar[r]^{l} & \overline{M_{n−1}} \ar[r]^{g′} & M_{n−2} \ar[r] & … \ar[r] & M_{1} \ar[r]^{∂} & A \ar[r] & B \ar[r] & 0})

and there is a map t : \overline{M_{n−1}} → M′ such that tl = 1.

Consider the following diagram

[figure: 0; M; f; M_{n−1}; g; tr; P ush; M_{n−2}; M_{n−3}; Id; …; 0; M; α; M′; j; \overline{M_{n−2}}; M_{n−3}; …]

The map j can be factored j = hβ for some h : M′′ → \overline{M_{n−2}} and therefore E = δ(E′) with

E′ = ( \xymatrix{0 \ar[r] & M′′ \ar[r]^{h} & \overline{M_{n−2}} \ar[r] & M_{n−3} \ar[r] & …} ).

To prove exactness at Opext^{n}(B, M′′) for n ⩾ 2 consider the following diagrams. The first row of the first diagram corresponds to E ∈ Opext^{n}(B, M′) and the second row corresponds to βE ∈ Opext^{n}(B, M′′).

[figure: 0; M′; β; f; M_{n−1}; M_{n−2}; Id; …; 0; M′′; h; \overline{M_{n−1}}; M_{n−2}; …]

[figure: 0; M; (1_{M} ,0); M ⊕ M′; 0+f; α+1_{M′}; M_{n−1}; M_{n−2}; Id; …; 0; M; α; M′; hβ; \overline{M_{n−1}}; M_{n−2}; …]

Proof of 4.3. The result is true for n = 2 by 3.2. For n ⩾ 3 we use theorem 4.14. Since the category of B-bimodules has enough injectives, we can find a short exact sequence

\xymatrix{0 \ar[r] & M \ar[r]^{α} & M′ \ar[r]^{β} & M′′ \ar[r] & 0}

with M′ injective. By 4.14 and 4.5 we have

Opext^{n+1}(B, M) = Opext^{n}(B, M′′).

On the other hand we have HH^{n+2}(B, M) = HH^{n+1}(B, M′′) by the long exact sequence of cohomology. Hence the result follows by induction from 3.2.

Remark 4.15. Theorem 4.3 is the analogue of a corresponding result for the cohomology of groups. In fact, using crossed modules in the category of groups as introduced by J.H.C.Whitehead [12] one can consider crossed extensions of groups which represent elements in the cohomology of groups (cf. Huebschmann [7]).

5. The characteristic class of a tensor product of differential algebras

In this section we define the ⊙-product of crossed modules. The definition of ∂_{1} ⊙ ∂_{2} is used below for the computation of the characteristic class of a tensor product of differential algebras (see 3.5 above).

Definition 5.1. Let ∂_{1} : V_{1} → A_{1} and ∂_{2} : V_{2} → A_{2} be crossed modules. Consider the diagram of (graded) vector spaces

\xymatrix{V_{1} ⊗ V_{2} \ar[r]^{d_{2}} & (V_{1} ⊗ A_{2}) ⊕ (A_{1} ⊗ V_{2}) \ar[r]^{d_{1}} & (A_{1} ⊗ A_{2})}  (*)

where d_{1} and d_{2} are defined as follows.

d_{2}(v_{1} ⊗ v_{2}) = ∂_{1}v_{1} ⊗ v_{2} − v_{1} ⊗ ∂_{2}v_{2}
d_{1}(v_{1} ⊗ a_{2}) = ∂_{1}v_{1} ⊗ a_{2}
d_{1}(a_{1} ⊗ v_{2}) = a_{1} ⊗ ∂_{2}v_{2}

Since d_{1}d_{2} = 0 we obtain a map ∂ induced by d_{1}:

∂ : W = \frac{(V_{1} ⊗ A_{2}) ⊕ (A_{1} ⊗ V_{2})}{Im(d_{2})} → A_{1} ⊗ A_{2}

Note that the diagram (*) is in fact a diagram of (A_{1} ⊗ A_{2})-bimodules. Here the (A_{1} ⊗ A_{2})-bimodule structure on V_{1} ⊗ V_{2} is given by

(a_{1} ⊗ a_{2})(v_{1} ⊗ v_{2}) = (−1)^{|a_{2}||v_{1}|}(a_{1}v_{1} ⊗ a_{2}v_{2})
(v_{1} ⊗ v_{2})(a_{1} ⊗ a_{2}) = (−1)^{|v_{2}||a_{1}|}(v_{1}a_{1} ⊗ v_{2}a_{2})

Thus the map ∂ : W → A_{1}⊗A_{2} is a map of (A_{1}⊗A_{2})-bimodules. We show now that ∂ is a crossed module. Given w, w′ ∈ W we have to check that ∂(w)w′ = w∂(w′). For v_{1}, v′_{1} ∈ V_{1}, a_{2}, a′_{2} ∈ A_{2} we have

∂(\overline{v_{1} ⊗ a_{2}})(\overline{v′_{1} ⊗ a′_{2}}) = (∂_{1}v_{1} ⊗ a_{2})(\overline{v′_{1} ⊗ a′_{2}})

= (−1)^{|a_{2}||v′_{1}|}(\overline{(∂_{1}v_{1})v′_{1} ⊗ a_{2}a′_{2}}) = (\overline{v_{1} ⊗ a_{2}})∂(\overline{v′_{1} ⊗ a′_{2}})

We have similar equation for ∂(\overline{a_{1} ⊗ v_{2}})(\overline{a′_{1} ⊗ v′_{2}}). Now for (\overline{v_{1} ⊗ a_{2}}) and (\overline{a_{1} ⊗ v_{2}}) we have

∂(\overline{v_{1} ⊗ a_{2}})(\overline{a_{1} ⊗ v_{2}}) = (∂_{1}v_{1} ⊗ a_{2})(\overline{a_{1} ⊗ v_{2}}) = (−1)^{|a_{2}||a_{1}|}(\overline{∂_{1}(v_{1}a_{1}) ⊗ a_{2}v_{2}})
= (−1)^{|a_{2}||a_{1}|}(\overline{v_{1}a_{1} ⊗ ∂_{2}(a_{2}v_{2})}) = (\overline{v_{1} ⊗ a_{2}})∂(\overline{a_{1} ⊗ v_{2}})

Thus ∂ : W → A_{1} ⊗ A_{2} is a crossed module termed the ⊙-product of ∂_{1} and ∂_{2} and is denoted by ∂_{1} ⊙ ∂_{2}. Notation. Given a crossed module ∂ : V → A we denote the cokernel of ∂ by π_{0}(∂) and the kernel by π_{1}(∂).

Proposition 5.2. The ⊙-product of two crossed modules ∂_{1} and ∂_{2} satisfies

π_{0}(∂_{1} ⊙ ∂_{2}) = π_{0}(∂_{1}) ⊗ π_{0}(∂_{2})  (1)
π_{1}(∂_{1} ⊙ ∂_{2}) = (π_{0}(∂_{1}) ⊗ π_{1}(∂_{2})) ⊕ (π_{1}(∂_{1}) ⊗ π_{0}(∂_{2}))  (2)

Proof. To prove (1) consider the map

ψ : π_{0}(∂_{1} ⊙ ∂_{2}) → π_{0}(∂_{1}) ⊗ π_{0}(∂_{2})

defined by ψ(\overline{a_{1} ⊗ a_{2}}) = \overline{a_{1}} ⊗ \overline{a_{2}}. It is easy to check that this map is well defined and is and isomorphism.

To prove (2) consider the map

ϕ : (π_{0}(∂_{1}) ⊗ π_{1}(∂_{2})) ⊕ (π_{1}(∂_{1}) ⊗ π_{0}(∂_{2})) → π_{1}(∂_{1} ⊙ ∂_{2})

given by ϕ(\overline{a_{1}} ⊗ v_{2}) = \overline{a_{1} ⊗ v_{2}} and ϕ(v_{1} ⊗ \overline{a_{2}}) = \overline{v_{1} ⊗ a_{2}}.

We show that ϕ is well defined. For a_{1} = ∂_{1}v_{1} and v_{2} ∈ ker(∂_{2}) we have

\overline{∂_{1}v_{1} ⊗ v_{2}} = \overline{v_{1} ⊗ ∂_{2}v_{2}} = 0

The same procedure for a_{2} = ∂_{2}v_{2} and v_{1} ∈ ker(∂_{1}). Moreover ∂(\overline{a_{1} ⊗ v_{2}}) = 0 if v_{2} ∈ ker(∂_{2}) and ∂(\overline{v_{1} ⊗ a_{2}}) = 0 for v_{1} ∈ ker(∂_{1}).

To prove that ϕ is and isomorphism consider a k-linear section of ∂_{1}, q_{1} : Im(∂_{1}) → V_{1}. Suppose \overline{v_{1} ⊗ a_{2}} ∈ W and ∂_{1}v_{1} ⊗ a_{2} = 0. Then (q_{1}∂_{1}v_{1} ⊗ a_{2}) = 0 and therefore v_{1} ⊗ a_{2} = (v_{1} − q_{1}∂_{1}v_{1}) ⊗ a_{2} and v_{1} − q_{1}∂_{1}v_{1} ∈ ker(∂_{1}). The same procedure for \overline{a_{1} ⊗ v_{2}}. This implies that ϕ is an isomorphism.

Proposition 5.3. Let ∂_{1} and ∂_{2} be crossed modules with cokernel B_{i} and kernel M_{i}, i = 1, 2. Then the class

⟨∂_{1} ⊙ ∂_{2}⟩ ∈ π_{0} Cross(B_{1} ⊗ B_{2}, (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2})) =
HH^{3}(B_{1} ⊗ B_{2}, (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2}))

depends only on the classes ⟨∂_{1}⟩ ∈ HH^{3}(B_{1}, M_{1}) and ⟨∂_{2}⟩ ∈ HH^{3}(B_{2}, M_{2}). Moreover one obtains a group homomorphism

Γ : HH^{3}(B_{1}, M_{1}) ⊕ HH^{3}(B_{2}, M_{2}) → HH^{3}(B_{1} ⊗ B_{2}, (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2}))

defined by Γ(⟨∂_{1}⟩, ⟨∂_{2}⟩) = ⟨∂_{1} ⊙ ∂_{2}⟩.

Proof. To check that ⟨∂_{1} ⊙ ∂_{2}⟩ depends only on the class of ∂_{1} and ∂_{2} consider a map α : ∂_{1} → ∂′_{1} in Cross(B_{1}, M_{1})

[figure: M_{1}; V_{1}; ∂_{1}; α_{1}; A_{1}; α_{0}; B_{1}; M_{1}; V′_{1}; ∂′_{1}; A′_{1}; B_{1}]

Then α induces a map

α ⊙ 1 : ∂_{1} ⊙ ∂_{2} → ∂′_{1} ⊙ ∂_{2}

given by (α ⊙ 1)_{0} : A_{1} ⊗ A_{2} → A′_{1} ⊗ A_{2}, (α ⊙ 1)_{0}(a_{1} ⊗ a_{2}) = α_{0}(a_{1}) ⊗ a_{2} and (α ⊙ 1)_{1} : (V_{1} ⊗ A_{2}) ⊕ (A_{1} ⊗ V_{2}) → (V′_{1} ⊗ A_{2}) ⊕ (A′_{1} ⊗ V_{2}) defined by

(α ⊙ 1)_{1}(v_{1} ⊗ a_{2}) = α_{1}(v_{1}) ⊗ a_{2}
(α ⊙ 1)_{1}(a_{1} ⊗ v_{2}) = α_{0}(a_{1}) ⊗ v_{2}

It is easy to check that (α ⊙ 1) is a well defined map in Cross(B_{1} ⊗ B_{2}, (B_{1} ⊗ M_{2})⊕(M_{1}⊗B_{2})) from ∂_{1}⊙∂_{2} to ∂′_{1}⊙∂_{2}. The same argument applies to β : ∂_{2} → ∂′_{2}. That proves the first part of the proposition.

To prove that Γ is a well defined homomorphism one has to check that

⟨(∂_{1} + ∂′_{1}) ⊙ 0⟩ = ⟨∂_{1} ⊙ 0⟩ + ⟨∂′_{1} ⊙ 0⟩ (∗)

and the same for ⟨0 ⊙ (∂_{2} + ∂′_{2})⟩. The sum (∂_{1} + ∂′_{1}) ∈ HH^{3}(B_{1}, M_{1}) and the element 0 ∈ HH^{3}(B_{2}, M_{2}) are defined explicitly in section 4 below (Baer Sum in Opext^{2}(B, M)).

It is easy to check that (∗) holds. In fact the class ⟨(∂_{1} + ∂′_{1}) ⊙ 0⟩ ∈ HH^{3}(B_{1} ⊗ B_{2}, (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2})) corresponds to the class of the crossed module

∂ : ((V_{1} + V′_{1}) ⊗ B_{2}) ⊕ (B_{1} ⊗ M_{2}) → (A_{1} ×_{B_{1}} A′_{1}) ⊗ B_{2}

with ∂((v_{1} + v′_{1}) ⊗ b_{2}) = (∂_{1}v_{1}, ∂′_{1}v′_{1}) ⊗ b_{2} and ∂(b_{1} ⊗ m_{2}) = 0. Here (V_{1} + V′_{1}) and (A_{1} ×_{B_{1}} A′_{1}) are defined as in section 4 below.

We can describe the ⊙-product in terms of classical cohomology products

HH^{n}(B_{1}, M_{1}) ⊗ HH^{m}(B_{2}, M_{2}) → HH^{n+m}(B_{1} ⊗ B_{2}, M_{1} ⊗ M_{2})

(cf. [10],Chapter X). Given f ∈ HH^{3}(B_{1}, M_{1}) we denote by f ⊗ 1_{B_{2}} ∈ HH^{3}(B_{1} ⊗ B_{2}, M_{1} ⊗ B_{2}) the tensor product of f with 1_{B_{2}} ∈ HH^{0}(B_{2}, B_{2}) given by the map

HH^{3}(B_{1}, M_{1}) ⊗ HH^{0}(B_{2}, B_{2}) → HH^{3}(B_{1} ⊗ B_{2}, M_{1} ⊗ B_{2})

In similar way we define for an element g ∈ HH^{3}(B_{2}, M_{2}) the element 1_{B_{1}} ⊗ g ∈ HH^{3}(B_{1} ⊗ B_{2}, B_{1} ⊗ M_{2}).

Proposition 5.4. Let ∂_{1} and ∂_{2} be crossed modules with cokernel B_{i} and kernel M_{i}, i = 1, 2. There is an equivalence of crossed modules

∂_{1} ⊙ ∂_{2} = i_{1}(1_{B_{1}} ⊗ ∂_{2}) + i_{2}(∂_{1} ⊗ 1_{B_{2}})

where

i_{1} : HH^{3}(B_{1} ⊗ B_{2}, B_{1} ⊗ M_{2}) → HH^{3}(B_{1} ⊗ B_{2}, (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2}))

and

i_{2} : HH^{3}(B_{1} ⊗ B_{2}, M_{1} ⊗ B_{2}) → HH^{3}(B_{1} ⊗ B_{2}, (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2}))

are induced by the inclusions i_{1} : B_{1} ⊗ M_{2} → (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2}) and i_{2} : M_{1} ⊗ B_{2} → (B_{1} ⊗ M_{2}) ⊕ (M_{1} ⊗ B_{2}).

Proof. The crossed module i_{1}(1_{B_{1}} ⊗ ∂_{2}) corresponds by definition to the crossed module

p_{1} : (B_{1} ⊗ V_{2}) ⊕ (M_{1} ⊗ B_{2}) → B_{1} ⊗ A_{2}

with p_{1}(b_{1} ⊗ v_{2}) = b_{1} ⊗ ∂_{2}v_{2} and p_{1}(m_{1} ⊗ b_{2}) = 0. The crossed module i_{2}(∂_{1} ⊗ 1_{B_{2}}) corresponds to

p_{2} : (B_{1} ⊗ M_{2}) ⊕ (V_{1} ⊗ B_{2}) → A_{1} ⊗ B_{2}

with p_{2}(v_{1} ⊗ b_{2}) = ∂_{1}v_{1} ⊗ b_{2} and p_{2}(b_{1} ⊗ m_{2}) = 0.

By definition of Baer Sum it is easy to check that i_{1}(1_{B_{1}} ⊗ ∂_{2}) + i_{2}(∂_{1} ⊗ 1_{B_{2}}) is isomorphic to the crossed module ∂_{1} ⊙ ∂_{2}.

Now let A and B be DG-algebras with differentials d_{A} and d_{B} of degree -1. Consider the tensor product A ⊗ B which is a DG-algebra with differential defined as follows.

d_{A⊗B}(x_{i} ⊗ y_{j}) = d_{A}x_{i} ⊗ y_{j} + (−1)^{i}x_{i} ⊗ d_{B}y_{j}

for x_{i} ∈ A_{i} and y_{j} ∈ B_{j}.

Theorem 5.5. The characteristic class ⟨A⊗B⟩ ∈ HH^{3}(H_{∗}(A⊗B), \overline{H_{∗}(A ⊗ B)}[−1]) can be computed as

⟨A ⊗ B⟩ = (ϕ_{1})_{∗}(1 ⊗ ⟨B⟩) + (ϕ_{2})_{∗}(⟨A⟩ ⊗ 1)

where

ϕ_{1} : H_{∗}(A) ⊗ \overline{H_{∗}(B)}[−1] → \overline{H_{∗}(A ⊗ B)}[−1]
ϕ_{2} : \overline{H_{∗}(A)}[−1] ⊗ H_{∗}(B) → \overline{H_{∗}(A ⊗ B)}[−1]

are defined by ϕ_{1}(a ⊗ s^{−1}(b)) = (−1)^{|a|}s^{−1}(a ⊗ b) and ϕ_{2}(s^{−1}(a) ⊗ b) = s^{−1}(a ⊗ b).

Proof. Let ∂_{A}, ∂_{B} and ∂_{A⊗B} be the crossed modules induced by A, B and A ⊗ B. There exists a morphism of crossed modules Υ : ∂_{A}⊙∂_{B} → ∂_{A⊗B} defined as follows.

\xymatrix{\frac{(coker(d_{A})[−1] ⊗ ker(d_{B})) ⊕ (ker(d_{A}) ⊗ coker(d_{B})[−1])}{Im(d_{2})} \ar[d]_{Υ_{1}} \ar[r]^{∂} & ker(d_{A}) ⊗ ker(d_{B}) \ar[d]^{Υ_{0}} \\ coker(d_{A⊗B})[−1] \ar[r]^{∂_{A⊗B}} & ker(d_{A⊗B})}

The top row in the diagram corresponds to the ⊙-product ∂_{A}⊙∂_{B} and the bottom row corresponds to the crossed module ∂_{A⊗B}. The map Υ_{0} : ker(d_{A}) ⊗ ker(d_{B}) → ker(d_{A⊗B}) is defined by

Υ_{0}(x_{i} ⊗ y_{j}) = x_{i} ⊗ y_{j} x_{i} ∈ (ker(d_{A}))_{i}, y_{j} ∈ (ker(d_{B}))_{j}

The map Υ_{1} is defined as follows. For s^{−1}(\overline{x_{i}}) ∈ (coker(d_{A})[−1])_{i} and y_{j} ∈ (ker(d_{B}))_{j} we define Υ_{1}(s^{−1}(\overline{x_{i}}) ⊗ y_{j}) to be the element

s^{−1}(\overline{x_{i} ⊗ y_{j}}) ∈ (coker(d_{A⊗B})[−1])_{i+j}.

For x_{i} ∈ (ker(d_{A}))_{i} and s^{−1}(\overline{y_{j}}) ∈ (coker(d_{B}][−1])_{j} we define

Υ_{1}(x_{i} ⊗ s^{−1}(\overline{y_{j}})) = (−1)^{i}s^{−1}(\overline{x_{i} ⊗ y_{j}}) ∈ (coker(d_{A⊗B})[−1])_{i+j}.

We check that the map Υ_{1} is well defined. Suppose \overline{x_{i}} = \overline{0} ∈ (coker(d_{A}))_{i} i.e. x_{i} = d_{A}a_{i+1} for some a_{i+1} ∈ A_{i+1}. Then

Υ_{1}(s^{−1}(\overline{x_{i}}) ⊗ y_{j}) = s^{−1}(\overline{d_{A}a_{i+1} ⊗ y_{j}}) = s^{−1}(\overline{d_{A⊗B}(a_{i+1} ⊗ y_{j})}) = 0

The same argument for \overline{y_{j}} = \overline{0} ∈ (coker(d_{B}))_{j}. For z = (d_{A}x_{i} ⊗ y_{j} − x_{i} ⊗ d_{B}y_{j}) ∈ Im(d_{2}) we have Υ_{1}(z) = d_{A⊗B}((−1)^{i}x_{i} ⊗ y_{j}). Thus Υ_{1} is well defined.

It is easy to check that the diagram above is a morphism of crossed modules. Moreover Υ : ∂_{A} ⊙ ∂_{B} → ∂_{A⊗B} induces an isomorphism

Υ_{∗} : π_{0}(∂_{A} ⊙ ∂_{B}) = π_{0}(∂_{A}) ⊗ π_{0}(∂_{B}) = H_{∗}(A) ⊗ H_{∗}(B) → π_{0}(∂_{A⊗B}) = H_{∗}(A ⊗ B)

and an epimorphism

Υ_{∗} : π_{1}(∂_{A} ⊙ ∂_{B}) = (\overline{H_{∗}(A)}[−1] ⊗ H_{∗}(B)) ⊕ (H_{∗}(A) ⊗ \overline{H_{∗}(B)}[−1]) → π_{1}(∂_{A⊗B}) =

\overline{H_{∗}(A ⊗ B)}[−1]

The crossed module ∂_{A} ⊙ ∂_{B} induces an element ⟨A⟩ ⊙ ⟨B⟩ = ⟨∂_{A} ⊙ ∂_{B}⟩ ∈ HH^{3}(H_{∗}(A ⊗ B), π_{1}(∂_{A} ⊙ ∂_{B})) which is mapped by Υ to the characteristic class ⟨A ⊗ B⟩ ∈ HH^{3}(H_{∗}(A ⊗ B), \overline{H_{∗}(A ⊗ B)}[−1]) of the chain algebra A ⊗ B, i.e.

⟨A ⊗ B⟩ = Υ_{∗}(⟨A⟩ ⊙ ⟨B⟩)

where the homomorphism

Υ_{∗} : HH^{3}(H_{∗}(A ⊗ B), π_{1}(∂_{A} ⊙ ∂_{B})) → HH^{3}(H_{∗}(A ⊗ B), \overline{H_{∗}(A ⊗ B)}[−1])

is the homomorphism induced by the map Υ_{∗} : π_{1}(∂_{A} ⊙ ∂_{B}) → π_{1}(∂_{A⊗B}). By 5.4 we have ⟨A⟩ ⊙ ⟨B⟩ = i_{1}(1 ⊗ ⟨B⟩) + i_{2}(⟨A⟩ ⊗ 1) and therefore

⟨A ⊗ B⟩ = (ϕ_{1})_{∗}(1 ⊗ ⟨B⟩) + (ϕ_{2})_{∗}(⟨A⟩ ⊗ 1)

with (ϕ_{1})_{∗} = (Υi_{1})_{∗} and (ϕ_{2})_{∗} = (Υi_{2})_{∗}.

For cochain algebras one can prove the following analogous result.

Theorem 5.6. Let A and B be DG-algebras with differentials of degree 1. Then the characteristic class ⟨A ⊗ B⟩ ∈ HH^{3}(H^{∗}(A ⊗ B), \overline{H^{∗}(A ⊗ B)}[1]) of the tensor algebra A ⊗ B can be computed from ⟨A⟩ and ⟨B⟩ as

⟨A ⊗ B⟩ = (ϕ_{1})_{∗}(1 ⊗ ⟨B⟩) + (ϕ_{2})_{∗}(⟨A⟩ ⊗ 1)

where

ϕ_{1} : H^{∗}(A) ⊗ \overline{H^{∗}(B)}[1] → \overline{H^{∗}(A ⊗ B)}[1] ϕ_{2} : \overline{H^{∗}(A)}[1] ⊗ H^{∗}(B) → \overline{H^{∗}(A ⊗ B)}[1]

are the maps defined by ϕ_{1}(a ⊗ s(b)) = (−1)^{|a|}s(a ⊗ b) and ϕ_{2}(s(a) ⊗ b) = s(a ⊗ b).

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

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2002/n2a3/v4n2a3.(dvi,ps,pdf)

Hans-Joachim Baues baues@mpim-bonn.mpg.de

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Elias Gabriel Minian gminian@mpim-bonn.mpg.de

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany