Tensor Hochschild homology and cohomology

Claude Cibils

Abstract

We consider the non commutative setting given by a ring A, an Abimodule M and T the corresponding tensor algebra. We prove that the Hochschild homology of quotients T/I by positive ideals coincides with the homology of A whenever the quiver of M has no oriented cycles.

If the quiver is an arrow (i.e. T is a triangular two by two matrix algebra) the Hochschild cohomology belongs to a long exact sequence. For other quivers, a spectral sequence converging to the Hochschild cohomology will be described in a forthcoming paper.

1991 Mathematics Subject Classification: 16E40, 15A72.

1 Introduction

The purpose of this paper is two-fold: in one direction we prove that the Hochschild homology of quotients of a tensor algebra T over a ring A coincides with the Hochschild homology of A, provided a certain quiver has no oriented cycles. This result has its origin in [4], see also [5] (Proposition 2.3) and [8].

In the other direction, we study the Hochschild cohomology of T when the quiver is only an arrow, which corresponds to the case of a triangular two by two algebra of matrices: the Hochschild cohomology of T can be compared with the Hochschild cohomology of A combined with the extensions groups of the corner two-sided ideal of T, through a long exact sequence. The connecting homomorphism is provided by a cup-product associated to the arrow of the quiver. Actually we use a canonical free resolution of a bimodule M, which provides a complex of cochains computing the Ext functor between M and a bimodule X.

The homology results enlarges considerations of J.-L. Loday in [9] and the cohomology computations generalizes the one-point extensions considered by D. Happel, see [8]. B. Bendiffalah and D. Guin ([1, 2]) have obtained partial results in the cohomological direction for a triangular algebra.

When the quiver is not just an arrow, there is a spectral sequence converging to the Hochschild cohomology of the tensor algebra, but the techniques are more intricate. The spectral sequence is obtained through a filtration given by the trajectories of the quiver that we define in 3.5. In a forthcoming paper we will describe it, as well as the differentials at the first level which are obtained through a cup product with an element provided by the sum of all the arrows of the quiver.

The first section of the present paper provides an introduction to the subject, we highlight relations to known facts and we set apart trivial cases. In the last sections we give the complete statements and proofs of the results.

Acknowledgements : I thank J.-M. Oudom for suggesting the right way of proving the exactness of the resolution of Proposition 4.1.

2 Tensor algebras

In the following we recall the definition of a triangular two by two ring and some well known results. Triangular rings will appear to be special instances of tensor rings; their module categories are morphism categories.

Let A_{1} and A_{2} be rings, M be an A_{2} − A_{1} bimodule. The triangular ring T

is the ring ( \begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix} ), namely the set of triples (a_{1}, a_{2}, m) in A_{1} × A_{2} × M

which multiplies as matrices do, using the bimodule structure of M and the products of A_{1} and A_{2}.

The sub-ring A_{1}×A_{2} has two central orthogonal idempotents: e_{1} = (1, 0, 0) and e_{2} = (0, 1, 0). The proof of the next trivial Lemma is based on the existence of these elements.

Lemma 2.1 The category of left T -modules is equivalent to the category which objects are triples (U_{1}, U_{2}, φ) where U_{1} is an A_{1}-module, U_{2} is an A_{2}-module and φ : M ⊗_{A_{1}}U_{1} ⟶ U_{2} is a morphism of A_{2}-modules. Morphisms of this category are couples consisting of an A_{1} and an A_{2}-morphism (f_{1} : U_{1} ⟶ U′_{1}, f_{2} : U_{2} ⟶ U′_{2}) such that f_{2}φ = φ′(1 ⊗ f_{1}).

Proof. A T -module X provides the triple (e_{1}X, e_{2}X, φ) where φ is given by the

action of ( \begin{matrix}0 & 0 \\ M & 0 \end{matrix} ) on X. Conversely a triple (U_{1}, U_{2}, φ) gives an A_{1} × A_{2}-

module structure on U_{1} ⊕ U_{2}. The action of M is determined by φ. All the verifications are straightforward.

Example 2.2 Let g : A_{2} ⟶ A_{1} be a ring morphism. The ring A_{1} is an A_{2} module, where A_{2} acts on A_{1} via g. The module category of the corresponding triangular algebra is equivalent to the category which objects are morphisms of A_{2}-modules φ : U_{1} ⟶ U_{2}, where U_{1} is an A_{1}-module considered as an A_{2}-module via g, and U_{2} is an A_{2}-module.

Example 2.3 Let k be a field, A_{1} be a K-algebra and M be a right module. Since M is actually a k-A_{1}-bimodule the triangular construction can be performed, the resulting algebra is called a “one-point extension” (see for instance [8]).

Definition 2.4 Let A be a ring and M an A-bimodule. The tensor ring T_{A}(M) = A ⊕ M ⊕ (M ⊗_{A}M) ⊕ (M ⊗_{A}M ⊗_{A}M) ⊕ … has a product given by the natural A-bimodule map

M^{⊗_{A}i} ⊗_{A}M^{⊗_{A}j} ⟶ M^{⊗_{A}i+j}.

The next result recalls that the triangular two by two ring T = ( \begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix} ) is

a tensor ring. Consider A = A_{1} × A_{2} and let \overline{M} be the A-bimodule given by M with actions of A_{2} on the left and of A_{1} on the right extended by zero.

Lemma 2.5 The rings T and T_{A}(M) are isomorphic.

Proof. The key property is that \overline{M} ⊗_{A}\overline{M} = 0. Indeed, m ⊗ m′ = me_{1} ⊗ m′ =

m ⊗ e_{1}m′ = m ⊗ 0 = 0. The map ( \begin{matrix}a_{1} & 0 \\ m & a_{2} \end{matrix} )↦ (a_{1}, a_{2}) + m is a bijective,

and is clearly a ring homomorphism.

Generalizations are immediate, at least in the following direction: let A_{1}, A_{2}, A_{3} be rings and M_{1}, M_{2} be A_{2} − A_{1} and A_{3} − A_{2} bimodules respectively. The ring

( \begin{matrix}A_{1} & 0 & 0 \\ M_{1} & A_{2} & 0 \\ M_{2} ⊗_{A_{2}}M_{1} & M_{2} & A_{3} \end{matrix} )

is isomorphic to the tensor algebra T_{A_{1}×A_{2}×A_{3}}(\overline{M_{1} ⊕ M_{2}}), where \overline{M_{1} ⊕ M_{2}} is M_{1} ⊕ M_{2} with actions extended by zero. Note that

(\overline{M_{1} ⊕ M_{2}})^{⊗_{A}3} = 0.

Objects of the module category are sequences (U_{1}, U_{2}, U_{3}, φ_{1}, φ_{2}), where U_{i} is an A_{i}-module and φ_{i} : M_{i} ⊗_{A_{i}}U_{i} → U_{i+1} is an A_{i+1}-morphism.

Recall that a quiver Q is a finite oriented graph, more precisely Q is given by two finite sets Q_{0} (vertices) and Q_{1} (arrows), and two maps s and t from Q_{1} to Q_{0} assigning the source and the target vertices to each arrow. We define next the quiver of a bimodule; it will provide an appropriate setting for results concerning Hochschild cohomology and homology.

Definition 2.6 Let A be a ring. A central idempotent system Q_{0} of A is a finite set of non zero orthogonal central idempotents of A such that ∑_{s∈Q_{0}} s = 1. The quiver of an A-bimodule M with respect to Q_{0} is an oriented graph with set of vertices Q_{0} and one arrow from the vertex s to the vertex t if and only if tMs ≠ 0. Note that we do not require that the idempotents are primitive: {1} is the trivial central idempotent system.

Remark 2.7 The idea of considering the quiver of a bimodule belongs to P. Gabriel, he used the quiver of radΛ/rad^{2}Λ for a finite-dimensional basic and split k-algebra Λ where radΛ denotes the Jacobson radical of Λ, see for instance [6, 7]. This combinatorial object is at the origin of the developments of the representation theory of non semi-simple algebras during the past decades.

Example 2.8 Let T be a triangular two by two ring ( \begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix} ). The quiver

of \overline{M} with respect to the central idempotent system Q_{0} = \{e_{1}, e_{2}\} of A_{1} × A_{2} is an arrow.

Example 2.9 If a bimodule is non zero, the quiver corresponding to the trivial central idempotent system Q_{0} = {1} is a loop.

Proposition 2.10 Let A be a ring and Q_{0} = {e_{1}, e_{2}} be a central idempotent system of A. Let M be an A-bimodule such that its quiver is an arrow from e_{1}

to e_{2}. Then the tensor ring T_{A}(M) is isomorphic to ( \begin{matrix}Ae_{1} & 0 \\ M & Ae_{2} \end{matrix} ).

Proof. Since {e_{1}, e_{2}} is an idempotent system, there is a ring decomposition A = Ae_{1} × Ae_{2}. Moreover e_{1}Me_{2} = e_{1}Me_{1} = e_{2}Me_{2} = 0, consequently M = e_{2}Me_{1}. We infer that M ⊗_{A}M = 0; this enables to obtain the isomorphism.

Remark 2.11 The next consideration shows that besides the trivial case of a loop, the quiver of a bimodule depends on the choice of the central idempotent system.The 3×3 triangular matrix algebra considered above provides the quiver . → . → . for the natural choice of the central idempotent system {e_{1}, e_{2}, e_{3}}, while {e_{1}, e_{2} + e_{3}} provides the quiver with two vertices, one arrow between them and a loop at the target vertex.

Tensor algebras over separable algebras are almost trivial on a homological point of view, as the following known result shows.

Proposition 2.12 Let k be a field and let A be a separable k-algebra. Let M be an A-bimodule and T = T_{A}(M) be the corresponding tensor algebra.

The projective dimension of T as a T -bimodule is at most one.

Proof. First we will show that for a non necessarily separable k-algebra A there is an exact sequence of T -bimodules

0 ⟶ T ⊗_{A}M ⊗_{A}T ⟶^{g} T ⊗_{A}T ⟶^{f} T ⟶ 0

where f(t_{1}⊗t_{2}) = t_{1}t_{2} and g(t_{1}⊗m⊗t_{2}) = t_{1}m⊗t_{2} − t_{1}⊗mt_{2}.

Consider V = M^{⊗_{A}n} = T^{n} the homogeneous component of T, and note that (T ⊗_{A}T)^{n} = V ⊕ … ⊕ V = nV since

(T ⊗_{A}T)^{n} = (T^{n} ⊗_{A}A) ⊕ (T^{n−1} ⊗_{A}T^{1}) ⊕ … ⊕ (A ⊗_{A}T^{n}).

Similarly, (T ⊗_{A}M ⊗_{A}T)^{n} = (n − 1)V, since

(T ⊗_{A}M⊗_{A}T)^{n} = (T^{n−1}⊗_{A}M⊗_{A}A)⊕(T^{n−2}⊗_{A}M⊗_{A}T^{1})⊕…⊕(A⊗_{A}M⊗_{A}T^{n−1}).

Through these canonical isomorphisms, it is easy to prove that the above sequence becomes a direct sum of sequences

0 ⟶ (n − 1)V ⟶^{g_{n}} nV ⟶^{f_{n}} V ⟶ 0

where

g_{n}(v_{1}, …, v_{n−1}) = (v_{1}, v_{2} − v_{1}, …, v_{n−1} − v_{n−2}, −v_{n−1})

and

f_{n}(v_{1}, …, v_{n}) = v_{1} + … + v_{n}.

Finally let A be a separable k-algebra. By definition A is projective as an Abimodule. Consequently any A-bimodule is projective, hence T ⊗_{A}M ⊗_{A}T is a projective T -bimodule, as well as T ⊗_{A}T = T ⊗_{A}A ⊗_{A}T. We infer that the above exact sequence is a projective resolution of the T -bimodule T.

Example 2.13 Let Q be a quiver. A path of Q is a finite sequence of concatenated arrows, a trivial one is just a vertex. The path algebra kQ is the vector space which basis the set of paths, the multiplication of two paths is given by its concatenation if it can be done and 0 otherwise; the sum of all the vertices is the unit element of kQ. Actually it is easy to see that kQ is a tensor algebra: consider the separable sub-algebra kQ_{0} and the kQ_{0}-bimodule kQ_{1}. Then kQ = T_{kQ_{0}} (kQ_{1}) since we have an isomorphism of kQ_{0}-bimodules

kQ_{i} ⊗_{kQ_{0}}kQ_{j} ⟶ kQ_{i+j}

where Q_{i} is the set of paths having i arrows and kQ_{i} the corresponding vector space equipped with its natural kQ_{0}-bimodule structure.

3 Homology

Let k be a field and let Λ be a k-algebra; the Hochschild cohomology and homology vector spaces with coefficients in a Λ-bimodule X are given by

H^{i}(Λ, X) = Ext_{Λ⊗_{k}Λ^{op}}^{i}(Λ, X)
H_{i}(Λ, X) = Tor_{i}^{Λ⊗_{k}Λ^{op}}(X, Λ).

Recall that Λ-bimodules and left modules over the enveloping algebra Λ ⊗_{k}Λ^{op} coincide since Λ^{op} is the underlying vector space of Λ equipped with the reverse product.

We consider now a k-algebra A, an A-bimodule M and T = T_{A}(M) the tensor algebra. A two-sided ideal I of T is called positive if it is contained in the two-sided ideal of T generated by M. When I is positive, A is a sub-algebra of T/I.

Note that a T/I-bimodule X is also an A-bimodule by restriction. If Q_{0} is a central idempotent system of A, let X = X_{Δ} ⊕ X_{⊥} be the decomposition of X given by

X_{Δ} = ⨁_{u∈Q_{0}} uXu

X_{⊥} = ⨁_{{u,v∈Q_{0} \atop u≠v}} vXu

Theorem 3.1 Let A be a k-algebra, M be an A-bimodule and I a positive twosided ideal of T. Let X be a T/I-bimodule and assume there exist a system Q_{0} of central idempotent system of A such that the quiver of M ⊕ X_{⊥} has no oriented cycles.

Then

H_{∗}(T/I, X) = H_{∗}(A, X_{Δ}).

Before proving the Theorem, we infer some consequences and links to known results.

Corollary 3.2 Let T/I be a k-algebra as above, and assume there exist a central idempotent system of A such that the quiver of M has no oriented cycles.

Then

H_{∗}(T/I, T/I) = H_{∗}(A, A).

Proof of the Corollary. We consider the T/I-bimodule of coefficients T/I and we need to show two properties in order to apply the Theorem 3.1:

1) the quiver of M ⊕ (T/I)_{⊥} has no oriented cycles

2) (T/I)_{Δ} = A.

First note that the quiver of a direct sum is an appropriate union of quivers: the set of vertices do not change, and there is an arrow from s to t if there is already one at least for one of the quivers of the factors.

Secondly we remark that since I is positive, T/I = A ⊕ (T^{>0}/I), where T^{>0} = M ⊕ (M ⊗_{A}M) ⊕ …. Moreover (T/I)_{⊥} ⊂ (T^{>0}/I) since A = ⨁_{s∈Q_{0}} As = ⨁_{s∈Q_{0}} sAs.

Next we examine the quiver of T^{>0}/I which is obtained from the quiver of T^{>0} by deleting some arrows. We will show that the quiver of M ⊗_{A}M has no oriented cycles, the same property will hold for any A-tensor power of M by induction. Using the central idempotent system Q_{0}, we obtain a decomposition

M ⊗_{A}M = ⨁_{u_{1},u_{2},u_{3}∈Q_{0}} (u_{3}Mu_{2}) ⊗_{Au_{2}} (u_{2}Mu_{1}) .

Hence the quiver of M ⊗_{A}M is obtained by deleting some arrows from the square quiver of M that we define now: its set of vertices remains the same, and the set of arrows is the set of the original arrows added with a new arrow from a vertex s to a vertex t for each length two path from s to t in the original quiver. The square quiver of a quiver without oriented cycles still have this property, as well as quivers obtained by deleting some arrows.

These considerations shows that the existence of an oriented cycle in the quiver of M ⊕ (T/I)_{⊥} would imply the existence of a (perhaps longer) oriented cycle in the quiver of M.

Concerning the second property, we have to show that u (T^{>0}/I) u = 0 for each u ∈ Q_{0}; this is equivalent to the fact that the quiver of T^{>0}/I has no loops, and we already know that it has not even oriented cycles.

Example 3.3 If the quiver is just an arrow, we know by Proposition 2.10 that

T_{A}(M) = ( \begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix} ).

The Corollary reduces in this case to a result presented in [9].

In order to prove Theorem 3.1, we will use a projective resolution of T obtained through the separable sub-algebra E = ×_{u∈Q_{0}}ku of A provided by the central idempotent system Q_{0}.

Proposition 3.4 Let Λ be a k-algebra and E a separable sub-algebra. The following is a resolution of Λ by projective Λ-bimodules :

… ⟶ Λ^{⊗_{E} m} ⟶^{d} Λ^{⊗_{E} m−1} ⟶ … ⟶ Λ ⊗_{E}Λ ⟶ Λ ⟶ 0

where

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

Note that tensor products are considered over E and that the differentials are well defined; since E is separable, any E-bimodule Z is projective, hence Λ ⊗ EZ ⊗_{E}Λ is a projective Λ-bimodule.

Moreover t : Λ^{⊗_{E} n−1} ⟶ Λ^{⊗_{E} n} defined by t(x) = 1⊗x is an homotopy contraction, i.e. dt + td = 1; consequently Kerd = Imd.

For a tensor algebra Λ = T = T_{A}(M), there is a decomposition of the previous resolution. For this purpose we define below the trajectories of a quiver.

Definition 3.5 The set T R_{n}(Q) of n-trajectories of a quiver Q is the set of sequences (γ_{n}, …, γ_{1}) of n concatenated paths of the quiver, with no restriction on the length (i.e. number of arrows) of each γ_{i}. In particular 0-length paths (i.e. vertices) are allowed. Concatenated paths means that for each i ≥ 2 we have s(γ_{i}) = t(γ_{i−1}) where the source s (resp. target t) of a path is the source (resp. target) of its first (resp. last) arrow. If u is a vertex, s(u) = t(u) = u.

We set some notation. Let A be a k-algebra, Q_{0} a central idempotent system of A and let M be an A-bimodule with quiver Q with respect to Q_{0}.

For each vertex u we set M_{u} = Au.

Let a be an arrow; then M_{a} = t(a)Ms(a).

Let γ = a_{p} … a_{1} be a path of the quiver; then

M_{γ} = M_{a_{p}} ⊗_{As(a_{p})} … ⊗_{At(a_{1})}M_{a_{1}}.

Let τ = (γ_{n}, …, γ_{1}) be a trajectory; then

M_{τ} = M_{γ_{n}} ⊗_{k} … ⊗_{k}M_{γ_{1}}.

Proposition 3.6 Let T = T_{A}(M) be the tensor algebra of an A-bimodule M, and let Q be the quiver of M with respect to a central idempotent system Q_{0} of A. Let E = ×_{u∈Q_{0}}ku ⊂ A. Then

T^{⊗_{E}n} = ⨁_{τ∈T R_{n}(Q)} M_{τ} .

Proof. First note that the homogeneous component M^{⊗_{A}p} of T decomposes as follows

M^{⊗_{A}p} = ⨁_{γ∈Q_{p}} M_{γ}

where Q_{p} is the set of paths of length p of the quiver Q. Indeed, consider the decomposition of M as an A-bimodule

M = ⨁_{u,v∈Q_{0}} vMu

and

M ⊗_{A}M = ⨁_{u,v,u′,v′∈Q_{0}} (v′Mu′) ⊗_{A} (vMu) .

The terms with u′ ≠ v are 0, since u′ and v are orthogonal idempotents belonging to A. Moreover

v′Mv ⊗_{A} vMu = v′Mv ⊗_{Av} vMu

since the components of A different from Av acts as zero on the right of v′Mv and on the left of vMu.

These considerations shows that

M ⊗_{A}M = ⨁_{γ∈Q_{2}} M_{γ}.

Secondly note that

T ⊗_{E}T = ⨁ M_{γ′} ⊗_{E}M_{γ}

where γ′ and γ are paths of Q of any length, – recall that M_{u} = Au if u is a vertex –. Finally we have that

M_{γ′} ⊗_{E}M_{γ} = 0 if t(γ) ≠ s(γ′)

and

M_{γ′} ⊗_{E}M_{γ} = M_{γ′} ⊗_{k}M_{γ} if t(γ) = s(γ′).

We infer

T ⊗_{E}T = ⨁_{τ∈T R_{2}(Q)} M_{τ} .

Corollary 3.7 Considering the same setting and notation as before, let u and v be vertices, and let v[T R_{n}(Q)]u be the set of n-trajectories beginning at u and ending at v. Then

v (T^{⊗_{E} n}) u ≠ 0 ⇔ v[T R_{n}(Q)]u ≠ ∅.

We record now the following useful result for quotient algebras by positive ideals.

Proposition 3.8 Let A be a k-algebra, and let M be an A-bimodule having a quiver Q with respect to a central idempotent system. Let I be a positive ideal of T = T_{A}(M) and let u and v be two vertices of Q. Then

v ((T/I)^{⊗_{E}n}) u ≠ 0 ⇒ v[T R_{n}(Q)]u ≠ ∅.

Proof. There is a surjective morphism

T^{⊗_{E} n} ⟶ (T/I)^{⊗_{E} n}

which provides surjections

v (T^{⊗_{E} n}) u ⟶ v[(T/I)^{⊗_{E} n}]u

for each couple (v, u) of idempotents belonging to the system Q_{0}.

Proof of Theorem 3.1. In order to compute the Hochschild homology, we consider the projective resolution of Proposition 3.4 for the algebra T/I, and we tensor it by the bimodule X over the enveloping algebra of T/I .

For n ≥ 2 we obtain

(T/I)^{⊗_{E} n} ⨂_{T/I−T/I} X = (T/I)^{⊗_{E} n−2} ⨂_{E−E} X

and in degree 2

(T/I)^{⊗_{E} 2} ⨂_{T/I−T/I} X = X/⟨ux − xu | u ∈ Q_{0}⟩.

Actually we have for n ≥ 1

(T/I)^{⊗_{E} n} ⨂_{E−E} X = ⨁_{u∈Q_{0}}(Au)^{⊗_{k} n} ⊗_{k} uXu

and

X/⟨ux − xu⟩ = ⨁_{u∈Q_{0}} uXu = X_{Δ}.

We will show that these assertions follows directly from the hypothesis of the Theorem, namely that the quiver of M ⊕ X_{⊥} has no oriented cycles.

Indeed, first there is a decomposition which holds independently of the hypothesis:

(T/I)^{⊗_{E} n} = ⨁_{u,v∈Q_{0}} v(T/I)^{⊗_{E} n}u.

We know that if a piece of the above direct sum is non zero, then there is trajectory from u to v, and then a path from u to v. Note that the path can be trivial, in that case u = v.

It is useful to recall that if Y and Z are bimodules over a ring Λ (i.e. left Λ ⊗ Λ^{op}-modules), the tensor product Y ⊗_{Λ−Λ}Z (i.e. Y ⊗_{Λ⊗Λ^{op}}Z) affords the relations yλ ⊗ z = y ⊗ λz and λy ⊗ z = y ⊗ zλ for y ∈ Y, z ∈ Z and λ ∈ Λ. Consequently

\begin{aligned}(T/I)^{⊗_{E} n} ⨂_{E−E} X & = ⨁_{u,v,u′,v′∈Q_{0}} v[(T/I)^{⊗_{E} n}]u ⊗_{k}v′Xu′ \\ & = ⨁_{u,v∈Q_{0}} v[(T/I)^{⊗_{E} n}]u ⊗_{k}uXv. \end{aligned}

A non-zero component with u ≠ v in this direct sum provides two non-zero vector spaces v[(T/I)^{⊗_{E} n}]u and uXv. This gives an oriented cycle in the quiver of M ⊕ X_{⊥}, contradicting the hypothesis. Hence the only possibly non-zero components of the direct sum corresponds to u = v. Moreover u(T^{⊗_{E} n})u = Au^{⊗_{k}n} since we know that

u(T^{⊗_{E} n})u = ⨁_{τ∈(T R^{n}(Q))u} M_{τ}

and the trivial trajectory (u, u, …, u) is the only element in uT R^{n}(Q)u (recall that Q has no oriented cycles). Then

u((T/I)^{⊗_{E} n})u = Au^{⊗_{k}n}.

An easier consideration shows the equality

X/⟨ux − xu⟩ = X_{Δ}.

Actually these decompositions provide a decomposition of the complex of chains as a direct sum of sub-complexes indexed by u; each of them coincides precisely with the Hochschild complex of chains computing the Hochschild homology, (see for instance [3]).

4 Hochschild cohomology

In order to compute the Hochschild cohomology of a triangular algebra

( \begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix} )

we introduce a standard complex which has cohomology groups isomorphic to Ext_{A_{2}−A_{1}}^{∗}(M, Y) when M and Y are A_{2} − A_{1}-bimodules. Recall that an A_{2} −A_{1}-bimodule is an A_{2} ⊗_{k}A_{1}^{op}-left module and that Ext_{A_{2}−A_{1}}^{∗} is Ext_{A_{2}⊗A_{1}^{op}}^{∗}.

In the following we drop the the tensor product sign ⊗_{k} between vector spaces, and we replace the sign ⊗ between vectors by a comma.

Let A_{1} and A_{2} be k-algebras, and let M and Y be A_{2} − A_{1}-bimodules. Let

qM^{p} = A_{2} … A_{2}MA_{1} … A_{1}

where the number of vector spaces in this tensor product is p + q + 1 with q copies of A_{2} and p copies of A_{1}.

We consider the following complex of cochains

0 → Hom_{k}(M, Y) →^{∂} Hom_{k}(^{0}M^{1}, Y) ⊕ Hom_{k}(^{1}M^{0}, Y) →^{∂} …

→^{∂} ⨁_{q+p=n} Hom_{k}(^{q}M^{p}, Y) →^{∂} …

with coboundaries analogous to the Hochschild ones:

∂ : Hom_{k}(^{q}M^{p}, Y) ⟶ Hom_{k}(^{q+1}M^{p}, Y) ⊕ Hom_{k}(^{q}M^{p+1}, Y)

given by ∂f = ∂_{1}f + ∂_{2}f where

∂_{1}f(b_{1}, b_{2}, …, b_{q+1}, m, a_{1}, …, a_{p}) =
b_{1}f(b_{2}, …, b_{q+1}, ma_{1}, …, a_{p})
+Σ(−1)^{i}f(b_{1}, …, b_{i}b_{i+1}, …, b_{q+1}, m, a_{1}, …, a_{p}
+(−1)^{q+1}f(b_{1}, …, b_{q+1}m, a_{1}, …, a_{p}).

and

∂_{2}f(b_{1}, …, b_{q}, m, a_{1}, …, a_{p+1}) =
(−1)^{q+1}[f(b_{1}, …, b_{q}, ma_{1}, …, a_{p+1})
+Σ(−1)^{i}f(b_{1}, …, b_{q}, m, a_{1}, …, a_{i}a_{i+1}, … a_{p+1})
+(−1)^{p}f(b_{1}, …, b_{q}, m, a_{1}, …, a_{p})a_{p+1}]

Proposition 4.1 The cohomology of the preceding complex of cochains is Ext_{A_{2}−A_{1}}^{∗}(M, Y).

Remark 4.2 : Note that for f ∈ Hom_{k}(M, Y) we have (∂_{1}f)(b, m) = bf(m) − f(bm) and (∂_{2}f)(m, a) = −f(ma) + f(m)a. Consequently ker ∂ = Hom_{A_{2}−A_{1}}(M, Y).

Proof of Proposition 4.1. Consider the following complex of free A_{2} − A_{1}- bimodules

… →^{d} ⨁ ; \begin{matrix}p + q = n + 1 \\ p > 0, q > 0 \end{matrix} ^{q}M^{p} →^{d} … →^{d} ^{1}M^{2} ⊕^{2}M^{1} →^{d} ^{1}M^{1} →^{d} M → 0

The first differential is special, we have d(b, m, a) = bma. In larger degrees, the differential is the differential of the double complex which has ^{q}M^{p} at position (q, p) for positive p and q, with vertical and horizontal differentials qM^{p} → ^{q}M^{p−1} and ^{q}M^{p} → ^{q−1}M^{p} given respectively by

\begin{aligned}(b_{1}, …, b^{q}, m, a_{1}, …, a_{p}) ↦ & (−1)^{q+1}[(b_{1}, …, b_{q}, ma_{1}, …, a_{p})+ \\ & ∑(−1)^{i}(b_{1}, …, b_{q}, m, a_{1}, …, a_{i}a_{i+1}, …, a_{p})] \end{aligned}

\begin{aligned}(b_{1}, …, b^{q}, m, a_{1}, …, a_{p}) ↦ & ∑(−1)^{i}(b_{1}, …, b_{i}b_{i+1}, … b_{q}, m, a_{1}, …, a_{p})+ \\ & (−1)^{q}(b_{1}, …, b_{q}m, a_{1}, …, a_{p}) \end{aligned}

Claim. The above complex is acyclic, in other words it is a free resolution of the A_{2} − A_{1}-bimodule M. Note that this finish the proof: apply the functor Hom_{A_{2}−A_{1}}(−, M) to the above resolution, recall that

Hom_{A_{2}−A_{1}}(A_{2}ZA_{1}, M) = Hom_{k}(Z, M),

and finally note that through this identification the dual of the differential becomes the announced coboundary ∂.

In order to prove the claim, we first enlarge the above double complex allowing p = 0 or q = 0: actually we remark that the above formulas corresponds to bar resolutions, and we consider the double complex which has as bottom line the bar resolution of M as a left A_{2}-module

… → ^{q}M^{0} → …^{1}M^{0} → M → 0

while the q-column is the bar resolution of ^{q}M^{0} as a right A_{1}-module. Of course, the differentials at the p line corresponds to the bar resolution of the left A_{1}-module ^{0}M^{p}.

This acyclic double complex provides an acyclic total complex which is not exactly what we need:

… →^{d} ⨁_{p+q=n}^{q}M^{p} →^{d} … →^{d} ^{1}M^{0} ⊕^{0}M^{1} →^{d} M → 0

An elementary way of obtaining the required resolution is as follows: consider first the sub-complex S of the above bar-bar double complex B, provided by the bottom line and the last column of B. This subcomplex has just homology M in degree 0. Indeed, if f and g are two epimorphisms on M, the module Ker(f + g)/Kerf ⊕ Kerg is isomorphic to M, using the maps f or g.

Second consider the long exact sequence in homolgy associated to the exact sequence of complexes 0 → S → B → B/S → 0, which has zeros almost everywhere. We infer that

H_{2}(B/S) = H_{1}(S) = M.

Finally we record that the total complex associated to B/S, completed with the cokernel of the last arrow is the announced complex and is acyclic. This cokernel is M by the previous consideration.

We compute now the Hochschild cohomology of a triangular algebra. From the resolution of Proposition 3.4 of a k-algebra Λ with a given separable subalgebra E, we have that the Hochschild cohomology vector spaces H^{∗}(Λ, X) with coefficients in a Λ-bimodule X are given by the cohomology of the following complex of cochains:

H : 0 → X^{E} →^{δ} Hom_{E−E}(Λ, X) →^{δ} Hom_{E−E}(Λ⊗_{E}Λ, X) →^{δ} …

where X^{E} = {x ∈ X | ex = xe ∀e ∈ E} and

∂f(x_{1} ⊗ … ⊗ x_{n}) = x_{1}f(x_{2} ⊗ … ⊗ x_{n})
+Σ(−1)^{i}f(x_{1} ⊗ … ⊗ x_{i}x_{i+1} ⊗ … ⊗ x_{n})
+(−1)^{n−1}f(x_{1} ⊗ … ⊗ x_{n−1})x_{n}.

We restrict ourselves to the case where Λ is a tensor algebra T = T_{A}(M) such that there exists in A a central idempotent system of order two {e_{1}, e_{2}} verifying

e_{1}M_{1}e_{1} = e_{2}M_{2}e_{2} = e_{1}Me_{2} = 0.

We know from Proposition 2.10 that T = ( \begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix} ). Moreover E =

( \begin{matrix}k & 0 \\ 0 & k \end{matrix} ) for e_{1} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) and e_{2} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ).

Lemma 4.3 For T as above, we have

T^{⊗_{E} n} = A_{1}^{⊗_{E} n} ⊕ A_{2}^{⊗_{E} n} ⊕ ⨁_{p+q=n}^{q}M^{p}

Recall that ^{q}M^{p} = A_{2} … A_{2}MA_{1} … A_{1}, where the number of vector spaces in this tensor product is p + q + 1 with q copies of A_{2} and p copies of A_{1}.

Proof. Note that

0 = A_{1} ⊗_{E}A_{2} = A_{2} ⊗_{E}A_{1} = M ⊗_{E}A_{2} = A_{1} ⊗_{E}M.

Hence the “missing” terms in the decomposition of T^{⊗_{E} n} are actually zero.

Remark 4.4 We have proved at the preceding section a more general result, namely

T^{⊗_{E} n} = ⨁_{τ∈T R^{n}Q} M_{τ} .

Since the quiver is now only an arrow a, a trajectory of length n is either trivial staying in one vertex or it goes through the arrow, namely the trajectory is

(e_{2}, e_{2}, …, e_{2}, a, e_{1}, e_{1}, …, e_{1}).

This agrees to the decomposition of the preceding Lemma.

Theorem 4.5 Let T be as above, let X be a T -bimodule, _{2}X_{1} = e_{2}Xe_{1} and X_{0} = e_{1}Xe_{1} ⊕ e_{2}Xe_{2}. There is a long exact sequence

0  ⟶  H^{0}(T, X) ⟶ H^{0}(A, X_{0}) ⟶
⟶ Hom_{A_{2}−A_{1}}(M,_{2}X_{1}) ⟶  H^{1}(T, X) ⟶ H^{1}(A, X_{0}) ⟶
⟶ Ext_{A_{2}−A_{1}}^{1}(M,_{2}X_{1}) ⟶  H^{2}(T, X) ⟶ H^{2}(A, X_{0}) ⟶
…
⟶ Ext_{A_{2}−A_{1}}^{n}(M,_{2}X_{1}) ⟶  H^{n+1}(T, X) ⟶ H^{n+1}(A, X_{0}) ⟶
…

Remark 4.6 If A_{2} = k and M is a right A_{1}-module the triangular algebra T is a one-point extension. The above long exact sequence has been obtained by D. Happel [8] in this case.

Proof of Theorem 4.5. Note that the following vector spaces provides a subcomplex of cochains of H:

M^{n} = Hom_{E−E} ( ⨁_{p+q=n}^{q}M^{p}, X) = Hom_{k} ( ⨁_{p+q=n}^{q}M^{p}, e_{2}Xe_{1}) .

By contrast

Hom_{E−E} (A_{j}^{⊗_{E} n}, X) = Hom_{k}(A_{j}^{⊗_{E} n}, e_{j}Xe_{j})

for j = 1 or 2 do not give in general a sub-complex of H. However we obtain an exact sequence

0 ⟶ M ⟶ H ⟶ H/M ⟶ 0

of complexes of cochains and it is easy to make an identification of H/M with the usual complex of cochains computing the Hochschild cohomology of A with coefficients in X_{0} = e_{1}Xe_{1} ⊕ e_{2}Xe_{2}.

In order to finish the proof of the Theorem, note that the sub-complex M^{∗} is precisely the complex of cochains that we have considered in the Proposition above.

Corollary 4.7 Assume M is a projective A_{2} − A_{1}-bimodule, and let T be the

triangular algebra ( \begin{matrix}A_{2} & 0 \\ M & A_{1} \end{matrix} ). Then

H^{∗}(T, X) ≅ H^{∗}(A, X_{0})

for ∗ ≥ 2, where X is a T -bimodule, A = A_{1} × A_{2} and X_{0} = e_{1}Xe_{1} + e_{2}Xe_{2}.

Finally, we give an interpretation of the connecting homomorphism

δ : H^{n}(A, X_{0}) ⟶ Ext_{A_{2}−A_{1}}^{n+1}(M,_{2}X_{1})

using an analogous of the usual cup-product.

Proposition 4.8 Let T = ( \begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix} ) be the triangular algebra, where M is

an A_{2} −A_{1}-bimodule. A T -bimodule X is given by four A_{i} −A_{j}-bimodules _{i}X_{j} for all i, j ∈ {1, 2} and four maps

\begin{matrix}a : & M & ⊗_{A_{1}} & _{1}X_{1} & ⟶ _{2}X_{1} \\ b : & M & ⊗_{A_{1}} & _{1}X_{1} & ⟶ _{2}X_{2} \\ c : & _{1}X_{2} & ⊗_{A_{2}} & M & ⟶ _{1}X_{1} \\ d : & _{2}X_{2} & ⊗_{A_{2}} & M & ⟶ _{2}X_{1} \end{matrix}

of the corresponding bimodule structures, verifying

d(b ⊗ 1) = a(1 ⊗ c)

Proof : The decomposition

X = e_{1}Xe_{1} ⊕ e_{2}Xe_{2} ⊕ e_{1}Xe_{2} ⊕ e_{2}Xe_{1}

and the actions of M on X provides the morphisms a, b, c and d. The commutation formula reflects the bimodule requirement

(tx)t′ = t(xt′)

for t, t′ ∈ T and x ∈ X.

Consider the i-cochain vector space of the complex computing the Hochschild cohomology, f ∈ Hom_{k}(A_{1}^{⊗n},_{1}X_{1}) and g ∈ Hom(A_{2}^{⊗n}, X_{2}). Let also h be an element of End_{A_{2}−A_{1}}(M).

Note that

h ⌣ f : M ⊗ A_{1}^{⊗n} ⟶^{h⊗f} M ⊗_{1}X_{1} ⟶^{a} _{2}X_{1}

and

g ⌣ 1_{M} : A_{2}^{⊗n} ⊗ M ⟶^{g⊗h} _{2}X_{2} ⊗ M ⟶^{d} _{2}X_{1}

gives cochains of the complex computing Ext_{A_{2}−A_{1}}^{n+1}(M,_{2}X_{1}); actually this provides cup-products

End_{A_{2}−A_{1}}(M) ⊗ H^{n}(A_{1},_{1}X_{1}) ⟶ Ext_{A_{2}−A_{1}}^{n+1}(M,_{2}X_{1})
H^{n}(A_{2},_{2}X_{2}) ⊗ End_{A_{2}−A_{1}}(M) ⟶ Ext_{A_{2}−A_{1}}^{n+1}(M,_{2}X_{1}).

Proposition 4.9 The connecting homomorphism

∂ : H^{n}(A_{1},_{1}X_{1}) ⊕ H^{n}(A_{2},_{2}X_{2}) ⟶ Ext_{A_{2}−A_{1}}^{n}(M,_{2}X_{1})

of the long exact sequence of the Theorem above is given by the formula

∂(f + g) = 1_{M} ⌣ f + (−1)^{n+1}g ⌣ 1_{M}.

References

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[3] Cartan, H. Eilenberg, S.: Homological algebra. Princeton University Press, 1956.

[4] Cibils, C.: Hochschild homology of an algebra whose quiver has no oriented cycles. Representation Theory I - Finite Dimensional Algebras, Ottawa 1984, Springer Lect. Notes Math., 1177, 55–59, (1986).

[5] Cibils, C.: Rigidity of truncated quiver algebras, Adv. in Math 79, 18–42, (1990).

[6] Gabriel, P.: Finite representation type is open. Representations of algebras, Ottawa 1974, Lect. Notes Math. 488, 132–155 (1975).

[7] Gabriel P.: Indecomposable representations, II, Symposia Matematica 2 (Instituto Nazionale di Alta Matematica, Roma), 81–104 (1973).

[8] Happel, D.: Hochschild cohomology of finite-dimensional algebras. Séminaire d’algèbre Paul Dubreuil et Marie-Paule Malliavin, Lect. Notes Math. 1404, 108–126 (1989).

[9] Loday, J.-L.: Cyclic homology. Grundlehren der mathematischen Wissenschaften, Second edition 1998 (513 p.)

Département de Mathématiques, Université de Montpellier 2, F-34095 Montpellier cedex 5.

cibils@math.univ-montp2.fr

July 1998