The first cohomology group of an algebra with coefficients

Claude CIBILS  Manuel SAORIN

1 Introduction and preliminaries

In this paper we provide formulas for the dimension of the first Hochschild cohomology vector space H^{1}(Λ, M), where Λ is an algebra with a separable splitting subalgebra and M is a Λ − Λ-bimodule. We first consider a monomial non-commutative algebra, in order to provide a simpler specific formula in this case and to sketch the methods used in the sequel. Recall that such an algebra, Λ, is the quotient of the path algebra of a quiver Q by a two-sided ideal generated by oriented paths of length at least 2.. The important point is that the quiver of Λ can afford oriented cycles. The computation is based on a modelization of cohomological tools and on a computation of the center of those algebras. In Section 3, we tackle the most general situation, with different bimodules of coefficients and/or more general algebras. Examples of applications are given in the two final sections, where, among others, we apply our techniques to truncated algebras, recovering recent results obtained by A.C. Locateli ([7]) in characteristic zero, to commutative monomial algebras and to cancellative semigroup algebras.

All throughout the paper, our algebra Λ will have a decomposition Λ = E ⊕ r where E is a separable subalgebra and r is a two-sided ideal. An algebra is of that type if, and only if, it is isomorphic to T/I, where T = T_{E}(X) is the tensor ring of an E-E-bimodule X (see [4] for the definition) and I is a twosided ideal of T contained in the ideal r(X) generated by X. We shall identify in the sequel Λ with T/I and will fix a minimal set Z of generators of I as a twosided ideal (i.e., no proper subset of Z generates I as a twosided ideal). Observe also that we have a canonical decomposition T = E ⊕ r(X) inducing Λ = E ⊕ r in Λ = T/I. If now M is a Λ − Λ-bimodule, then M can be viewed as a T − T −bimodule such that IM = 0 = MI and conversely, a fact that we shall freely use.

A typical case of algebra Λ as above is anyone given by quiver and relations. A quiver Q is an oriented graph, with set of vertices Q_{0} and set of arrows Q_{1}, both of which we shall assume finite in the sequel. Each arrow a of Q has a source s(a) and a terminal vertex t(a). If they coincide a is a loop. A path of Q is a sequence of arrows, the source of each arrow has to be the terminus of the preceding one. The zero-length or trivial paths are the vertices. The linear span kQ of the set of paths has an algebra structure provided by the concatenation of paths, convening that the product of non-concatenable paths is zero. Let us consider an algebra of the form Λ = KQ/I, where Q is a finite quiver and I is a two-sided ideal of KQ, usually identified by a set Z of generators, called relations, which are linear combinations of paths of length ≥ 2. In this situation we have a decomposition Λ = E ⊕ r, where E = KQ_{0} ≅ K× ...^{n} ×K (n = |Q_{0}|) and r = J/I, where J is the ideal of KQ generated by the arrows.

The Hochschild cohomology groups H^{∗}(Λ, M) of an algebra Λ over a field k with coefficients in a Λ − Λ-bimodule M are the Ext groups between Λ and M over the enveloping algebra Λ^{e} = Λ ⊗ Λ^{op}. For the case of our type of algebras, a canonical reduced resolution of the bimodule_{Λ}Λ_{Λ} has been described for instance in [4], it uses tensor powers of r over E. This simplified resolution provides a complex of cochains whose cohomology is H^{∗}(Λ, M). In particular we have that

H^{0}(Λ, M) = C_{M}(Λ) = {m ∈ M | λm = mλ ∀λ ∈ Λ}

is the centralizer of Λ in M, also called the space of invariants, and

H^{1}(Λ, M) = Der_{E−E}(r, M)/Der^{0}_{E−E}(r, M)

where Der_{E−E}(r, M) is the vector space of E-bimodule derivations r ⟶ M, i.e., of E-bimodule maps ∂ : r → M verifying ∂(xy) = x∂(y)+∂(x)y. The denominator Der^{0}_{E−E}(r, M) consists of the inner derivations. ∂ is inner if there exists an element m_{0} ∈ M such that ∂(x) = xm_{0} − m_{0}x.

We also need to recall the following result from [3]:

Proposition 1. [3, p.98] Let A be a k-algebra and Λ = A/I where I is a twosided ideal of A. Let M be a Λ-bimodule, that is, M is an A-bimodule such that IM = MI = 0. Assume H^{2}(A, M) = 0. There is an exact sequence

0 → H^{1}(Λ, M) → H^{1}(A, M) → Hom_{A−A}(I/I^{2}, M) → H^{2}(Λ, M) → 0

The assumption H^{2}(A, M) = 0 is fulfilled whenever A = T = T_{E}(X) is the tensor ring of an E-bimodule X (cf. [4, Prop. 2.12]). The middle homomorphism restricts to I each derivation ∂ : A → M. The last homomorphism is provided by a connecting homomorphism of a long exact sequence.

2 Dimension of H^{1} for a monomial algebra

The purpose of this section is to provide a computation of the dimension of H^{1}(Λ, Λ) where Λ is a monomial noncommutative algebra, i.e., an algebra represented by quiver and relations Λ = KQ/I, where the ideal I is generated by a set Z of paths (of length ≥ 2). We shall assume that Z is minimal, i.e., no proper subpath of a path in Z is again in Z. Note that the computation for the monomial case provides a sample of subsequent developments in the paper.

Computations of the dimensions of the three last terms of the four-terms exact sequence have been worked out in [4], using a combinatorial approach to the middle restriction morphism. This enables to prove the following result, explanations and notations are given below.

Theorem 1. With the above assumptions, we have dim_{k} H^{1}(Λ, Λ) = dim_{k} Z(Λ) − |Q_{0}//B| + |Q_{1}//B| − |(Q_{1}//B)_{e}| − dim ImR_{g}

In the above formula, Z(Λ) denotes the center of Λ and its dimension can be computed through the number of connected components of an ad-hoc quiver Γ. The set of paths not belonging to I is denoted by B, i.e., B consists of those paths not containing an element of Z as a subpath. It is clear that the (classes modulo I of) elements of B form a basis of Λ. We shall denote by B_{n} the subset of B formed by the paths of length n. For X and Y sets of paths, the set of “parallel couples of paths” X//Y is a subset of X × Y: it consists of couples of paths sharing the same source vertex as well as the same terminus vertex. In particular Q_{0}//B is the set of oriented cycles in B, including the vertices. The set Q_{1}//B are parallel couples of an arrow and a path of B. A pair (a, ϵ) ∈ Q_{1}//B is called glued when ϵ ∈ B_{0} = Q_{0} or a is either the first or last arrow of ϵ. The subset of glued elements is denoted (Q_{1}//B)_{g}. We call (a, ϵ) admissible when, for every γ ∈ Z in which a appears, any one-replacement of a by ϵ yields a path not in B (equivalently, a path in I). The corresponding subset will be denoted (Q_{1}//B)_{a}. Finally, (a, ϵ) will be called effective when it is neither glued nor admissible. In the above formula, (Q_{1}//B)_{e} denotes the subset of effective elements of Q_{1}//B. The last term in the formula is the only one that remains to be explained. We define R : k(Q_{1}//B) → k(Z//B) as follows. Let (a, ε) be a parallel couple of Q_{1}//B. Let γ ∈ Z and consider the replacement by ε of each instance of a in γ. Such replacements provides couples of parallel paths which have first component γ ; we only keep couples such that the second component is still in B. Then R(a, ε) is the formal sum of these couples. Note that R(a, ε) is zero if either a does not belong to any path of Z, or if (a, ϵ) is admissible. We put R_{g} = R|_{K(Q_{1}//B)_{g}} and dim ImR_{g} is the last term of the formula.

Proof of the Theorem : In the four-terms exact sequence, we first examine H^{1}(kQ, kQ/⟨Z⟩). The length-one bimodule projective resolution of kQ (see for instance [4]) immediately shows that

dim_{k} H^{1}(kQ, kQ/⟨Z⟩) = dim Z(Λ) − Q_{0}//B + Q_{1}//B.

The next term is Hom_{kQ−kQ}(⟨Z⟩/⟨Z⟩^{2}, kQ/⟨Z⟩). In [3] a presentation of the kQ-bimodule ⟨Z⟩ is provided, as an explicit projective bimodule with relations. This enables to define a specific quiver called the parallel quiver, which has some particular connected components, the medals. We do not make precise these objects in this paper since the final result does not use those notions. The dimension of the Hom vector space we consider is the number of medals. Finally, the last term of the four-terms exact sequence is H^{2}(Λ, Λ). The paper [3] is concerned by the dimension of H^{2}(Λ, Λ) and determines when H^{2}(Λ, Λ) = 0. This is a key point for classifying the rigid monomial algebras. Actually dim_{k} H^{2}(Λ, Λ) is the number of medals, minus the number of effective elements in Q_{1}//B and minus the rank of R_{g}. Using that zero is the alternate sum of the dimensions in the four-terms exact sequence, we obtain the formula of the Theorem.

Concerning the center of Λ, we consider an auxiliary quiver Γ. Its set of vertices is the set of oriented cycles in B (hence, in particular, Q_{0} ⊆ Γ_{0}). If c, c′ ∈ Γ_{0} there will be one arrow c → c′ in Γ in case there is an arrow a ∈ Q_{1} such that ca = ac′ ∈ B. Observe, that, in case c, c′ have positive length, the existence of an arrow c → c′ in Γ implies that c′ is obtained from c by taking out the first arrow and adding it at its end, but the converse is not true in general. Some connected components of Γ contribute to the center of Λ through the sum of its vertices. More precisely, an oriented cycle c ∈ Γ_{0} is said to be closed if either c ∈ Q_{0} or, else, ca ∉ B for all a ∈ Q_{1} excepting perhaps the first arrow of c, and ac ∉ B for all arrows a ∈ Q_{1} excepting perhaps the last arrow of c. A connected component of Γ is closed if all its vertices are closed.

Proposition 2. Let Λ be a monomial algebra. Then dim Z(Λ) = n(Γ), where n(Γ) is the number of closed connected components of Γ

The proof is routine.

3 H^{1} with coefficients

We consider now the general situation described in the introduction. As mentioned at the end of Section 1, the middle homomorphism of the four-terms exact sequence of Proposition 1, with A = T, is the appropiate restriction of the K-linear map R_{M} : Der_{E−E}(r(X), M) ⟶ M^{Z} defined by the equality R_{M}(∂)(γ) = ∂(γ), for all ∂ ∈ Der_{E−E}(r(X), M) and γ ∈ Z. The following is the analogue of Theorem 1.

Theorem 2. In the above situation, we have the formulae:

dim H^{1}(Λ, M) = dim Ker(R_{M})− dim C_{M}(E)+ dim C_{M}(T) =
dim Hom_{E−E}(X, M)− dim C_{M}(E)+ dim C_{M}(T)− dim Im(R_{M})

and C_{M}(T) = C_{M}(E) ∩ C_{M}(X).

Proof. Since H^{2}(T, T) = 0 (cf. [4], Prop. 2.12), the four-terms exact sequence, with A = T, gives dim H^{1}(Λ, M) = dim H^{1}(T, M)− dim Im(R_{M}) = dim Der_{E−E}(r(X), M)− dim Der_{E−E}^{0}(r(X), M)− dim Im(R_{M}) = dim Ker(R_{M})− dim Der_{E−E}^{0}(r(X), M). Now we notice that Der_{E−E}(r(X), M) ≅ Hom_{E−E}(X, M) via the restriction map ∂ ⟶ ∂_{|X}. Finally, we have a canonical K-linear map C_{M}(E) ⟶ Der_{E−E}^{0}(r(X), M), m ⟶ ∂_{m} (∂_{m}(t) = tm − mt, for all t ∈ r(X)), which is surjective and has as kernel precisely C_{M}(E) ∩ C_{M}(X) = C_{M}(T).

The goal of the following is show that one can develop combinatorial techniques to identify all the dimensions that appear in the above formula, when restrictions are imposed on Λ and M. The restriction we need is that E is a (finite) direct product of (always finite-dimensional) central simple algebras over K and, via Morita equivalence, we can assume E = D_{1} × ⋯ × D_{t}, where each D_{i} is a central division algebra. We denote by E= {e_{1}, ..., e_{t}} the family of canonical (central) idempotents of E. Now, inspired by [5], we associate a quiver Δ^{N} to every E-E-bimodule N as follows. Δ_{0} = Δ_{0}^{N} = {1, ..., t} (it does not depend on N) and the number of arrows i → j in Δ^{N} is c_{ij}, the composition length of e_{i}Ne_{j} as a D_{i} − D_{j}−bimodule or, equivalently, as a left D_{i} ⊗_{K} D_{j}^{op}−module. It will be convenient to view the set Δ^{N}(i, j) of arrows i → j in Δ^{N} as a set of elements of e_{i}Ne_{j}, each one generating a simple D_{i} − D_{j}−subbimodule of e_{i}Ne_{j}, the direct sum of them being e_{i}Ne_{j} . The first two dimensions are now at hand.

Proposition 3. The number of loops in Δ^{M} equals dim C_{M}(E). In particular, if Λ = KQ/I is given by quiver and relations, then dim C_{M}(E)= ∑_{i∈Q_{0}} dim e_{i}Me_{i} Proof. The quiver with relations case is a straightforward consequence of the general one. In this latter one, it is clear that C_{M}(E) ⊆ C_{M}(E) = ⊕_{1≤i≤t}e_{i}Me_{i}. But then C_{M}(E) = C_{⊕_{1≤i≤t}e_{i}Me_{i}}(E) = ⊕_{1≤i≤t}C_{e_{i}Me_{i}}(D_{i}). Moreover, e_{i}Me_{i} ≅ D_{i}^{(c_{ii})} as a D_{i} − D_{i}−bimodule, because D_{i} ⊗ D_{i}^{op} is a simple Artinian algebra whose only simple left module, up to isomorphism, is D_{i}. As a consequence, C_{e_{i}Me_{i}}(D_{i}) ≅ Z(D_{i})^{(c_{ii})} = K^{(c_{ii})}, from which the result follows.

Proposition 4. There is an equality dim Hom_{E−E}(X, M) = ∑_{i,j∈Δ_{0}} \frac{|Δ^{X}(i,j)|·|Δ^{M} (i,j)|·[D_{i}:K]·[D_{j}:K]}{|Δ^{A^{⊗2}} (i,j)|^{2}}. In particular, when Λ = KQ/I is given by quiver and relations, dim Hom_{E−E}(X, M)= dim Hom_{KQ_{0}−KQ_{0}}(KQ_{1}, M)= ∑_{i,j∈Q_{0}} |Q(i, j)| · dim e_{i}Me_{j}, where Q(i, j) is the set of arrows i → j in Q.

Proof. One has Hom_{E−E}(X, M) = ⊕_{i,j=1,...,t}Hom_{D_{i}−D_{j}}(e_{i}Xe_{j}, e_{i}Me_{j}). By our interpretation of arrows in Δ^{X} and Δ^{M}, it is clear that Hom_{D_{i}−D_{j}}(e_{i}Xe_{j}, e_{i}Me_{j}) = ⊕_{x∈Δ^{X}(i,j),m∈Δ^{M} (i,j)}Hom_{D_{i}−D_{j}}(D_{i}xD_{j}, D_{i}mD_{j}). But, as a K-vector space, Hom_{D_{i}−D_{j}}(D_{i}xD_{j}, D_{i}mD_{j}) is isomorphic to the endomorphism algebra D_{ij} of the unique (up to iso) simple D_{i}−D_{j}−bimodule. Since e_{i}(E⊗E)e_{j} ≅ D_{i}⊗D_{j}^{op} ≅ M_{n_{ij}}(D_{ij}), where n_{ij} denotes the composition length of e_{i}(E ⊗ E)e_{j} = D_{i} ⊗ D_{j} as D_{i} − D_{j}−bimodules, we have | Δ^{E^{⊗2}}(i, j) |^{2} ·[D_{ij} : K] = [D_{i} : K] · [D_{j} : K]. The result clearly follows.

Computing the dimension of C_{M}(T) = C_{M}(E) ∩ C_{M}(X) is a more difficult task, and some restrictions will be needed. As suggested in the proof of Proposition 3, one can choose the loops in Δ^{M} to be a K-basis of C_{M}(E), so that C_{e_{i}Me_{i}}(D_{i}) is the K-span of Δ^{M}(i, i) . We shall assume that in the sequel. For x ∈ X, let us denote by l_{C_{M} (E)}(x) the left annihilator of x in C_{M}(E), i.e., the K-subspace of C_{M}(E) consisting of those m ∈ C_{M}(E) such that mx = 0 and, symmetrically, one defines the right annihilator r_{C_{M} (E)}(x).

Definition 1. In the above situation, we shall say that the set B of loops in Δ^{M} is X-good in case the following conditions hold:

1. For each arrow x ∈ Δ_{1}^{X}, l_{C_{M} (E)}(x) is the K-span of the loops b ∈B such that bx = 0, and symmetrically for r_{C_{M} (E)}(x)

2. For each arrow x ∈ Δ_{1}^{X}, C_{M}(E)x ∩ xC_{M}(E) is the K-span of Bx ∩ xB.

When one can choose the arrows in Δ^{M} so that the set of loops B is X −good, one can form a new quiver Γ = Γ(M), that we shall call the loop-quiver of M as follows. The vertices of Γ are just the elements of B. If b, b′ ∈B, there is one arrow b → b′ in Γ in case there exists a x ∈ Δ_{1}^{X} such that 0 ≠ bx = xb′. A connected component C of Γ will then be called closed in case Cx∪{0} = xC∪{0}, for every x ∈ Δ_{1}^{X}.

Proposition 5. Let us assume that Δ^{M} has a X-good set B of loops, let Γ = Γ(M) be the loop-quiver of M and, for each connected component C of Γ, let us put t_{C} = ∑_{b∈C} b. Then {t_{C}: C is a closed connected component of Γ} is a K-basis of C_{M}(E) ∩ C_{M}(X) = C_{M}(T).

Proof. From the fact that B is X−good and C is closed it follows that xt_{C} = t_{C}x, for each x ∈ Δ_{1}^{X}. From that one gets that xt_{C} = t_{C}x, for every x ∈ X, bearing in mind that t_{C} ∈ C_{M}(E) and X is generated as E-E-bimodule by Δ_{1}^{X}. Conversely, if c = ∑_{b∈B} λ_{b}b is an element of C_{M}(E) ∩ C_{M}(X), where λ_{b} ∈ K for every b, we claim that λ_{b} = λ_{b′} whenever b, b′ lie in the same connected component of Γ. Indeed, cx = xc ∈ C_{M}(E)x ∩ xC_{M}(E) and the fact that B is X−good tells us that cx is a K-linear combination of those bx, b ∈B, such that 0 ≠ bx ∈ xB. Without loss of generality, let us assume that there is an arrow b → b′ in Γ and pick up a x ∈ Δ_{1}^{X} such that 0 ≠ bx = xb′. Then bx appears with coefficient λ_{b} in cx and with coefficient λ_{b′} in xc. The X−good condition of B implies that {bx : b ∈B and bx ≠ 0} is a K-linearly independent set. We conclude that λ_{b} = λ_{b′} as desired.

Let us put now λ_{C} = λ_{b}, whenever b belongs to the connected component C of Γ. We have c = ∑_{C} λ_{C}t_{C}, where C runs through the set of connected components of Γ. We claim that if C is not closed, then λ_{C} = 0, which will end the proof. Indeed, suppose C is not closed and λ_{C} ≠ 0. Since we have some x ∈ Δ_{1}^{X} such that Cx ∪ {0} ≠ xC∪{0}, up to symmetry, we can assume that there are x ∈ (Δ^{X})_{1} and b ∈C such that 0 ≠ bx ∉ xC. But, from the fact that cx = xc ∈ C_{M}(E)x ∩ xC_{M}(E) and the 2nd condition in the definition of X−good set of loops, it follows that bx appears with coefficient λ_{b} ≠ 0 in cx = xc ∈ C_{M}(E)x ∩ xC_{M}(E), so that 0 ≠ bx = xb′, for some b′ ∈B which is necessarily in the same connected component C that b. But then bx ∈ xC, against our assumption.

The final dimensions in Corollary 2 that we need are either that of Ker(R_{M}) or that of Im(R_{M}). In general, they are the most difficult to identify and no recipe is valid everywhere. A strong simplification comes when Λ and M are graded. Indeed, consider the canonical (N-)grading on T_{E}(X) obtained by assigning degree 0 to the elements of E and degree 1 to those of X. When the ideal of relations I is generated by a set Z of homogeneous elements, A = T_{E}(X)/I inherites a canonical grading. In this situation, suppose also that M is a graded Λ − Λbimodule, i.e., it admits a decomposition M = ⊕_{n≥0}M_{n} as an E-E-bimodule such that Λ_{n}M_{p} + M_{p}Λ_{n} ⊆ M_{n+p}, for all n, p ≥ 0 (equivalently, XM_{p} + M_{p}X ⊆ M_{p+1}, for all p ≥ 0). When X has finite composition length as E-E-bimodule, since Der_{E−E}(T_{E}(X), M) ≅ Hom_{E−E}(X, M) = ⊕_{n≥0}Hom_{E−E}(X, M_{n}), we get a decompostion Der_{E−E}(T_{E}(X)M) = ⊕_{n≥0}Der_{E−E}(T_{E}(X), M)_{n}, where Der_{E−E}(T_{E}(X), M)_{n} = {∂ ∈ Der_{E−E}(T_{E}(X), M) such that ∂(X) ⊆ M_{n}}. We denote by R_{M}^{n} the restriction of R_{M} to Der_{E−E}(T_{E}(X), M)_{n}. The following is the main result in this context.

Proposition 6. Let Λ and M be graded in the above sense and put d_{γ} = deg(γ), for each γ ∈ Z. The following assertions hold:

1. Ker(R_{M}) = ⊕_{n≥0}Ker(R_{M}^{n})

2. Im(R_{M}) = ⊕_{n≥0}Im(R_{M}^{n})

3. For each n ≥ 0, Im(R_{M}^{n}) ⊆ (∑_{γ∈Z} M_{n+d_{γ}−1})^{Z}

Proof. If ∂ ∈ Der_{E−E}(T_{E}(X), M)_{n}, then ∂(γ) ∈ M_{n+d_{γ}−1}, for all γ ∈ Z, from which 3) follows. It is clear that Im(R_{M}) = ∑_{n≥0} Im(R_{M}^{n}). We prove that this sum is direct. Indeed, if R_{M}^{n}(∂) = ∑_{i≠n} R_{M}^{i}(∂_{i}), then we have ∂(γ) = ∑_{i≠n} ∂_{i}(γ), for every γ ∈ Z. By looking at the degrees on both members of the equality, we get ∂(γ) = 0 for all γ ∈ Z, so that R_{M}(∂) = 0 and 2) is proved. Finally, since the sum ∑ Ker(R_{M}^{n}) is clearly direct, we just need to check that Ker(R_{M}) ⊆ ∑ Ker(R_{M}^{n}). Indeed, if ∂ ∈ Ker(R_{M}) and ∂ = ∑ ∂_{n} is its decomposition, by comparing degrees in the equality 0 = ∂(γ) = ∑ ∂_{n}(γ), we get ∂_{n}(γ) = 0, for all n ≥ 0. Hence ∂_{n} ∈ Ker(R_{M}^{n}), for all n ≥ 0, as desired.

4 The monomial case revisited

In the particular case when E = D^{(t)} = D×D×...×D, for some central division Kalgebra D, T_{E}(X) is isomorphic to the path ”D-algebra” DΔ^{X} (i.e. the coefficients in D commute with the arrows). We are interested in the case when Λ = DQ/I, where Q is a finite quiver and I is a two-sided ideal generated by a set Z of paths of length ≥ 2. We shall say that Λ is a monomial D-algebra. In this situation, the sort of Λ−Λ-bimodules that we are interested in are those of the form M = H̄ = \frac{H}{I} or M = DQ/H ≅ \frac{A}{H̄}, where H is a two-sided ideal of DQ generated by paths of Q and containing Z. In that case, it is well-known that we can fix a D-basis B of M consisting of paths in Q. For that reason, such a bimodule will be called a monomial bimodule. The terminology of Section 2 concerning Q_{1}//B is translated mutatis mutandi to the present situation.

Remark 1. The assignment (a, ϵ) → f^{(a,ϵ)}, where f^{(a,ϵ)} ∈ Hom_{E−E}(DQ_{1}, M) = Hom_{E−E}(X, M) is the E − E−homomorphism which maps a onto ϵ and the remaining arrows onto 0, allows to identify Q_{1}//B with a K-basis of Hom_{E−E}(DQ_{1}, M) = Hom_{E−E}(X, M), actually identifying each (Q_{1}//B_{n}) with a K-basis of Hom_{E−E}(DQ_{1}, M_{n}) = Hom_{E−E}(X, M_{n}). On the other hand, to each (γ, ϵ) ∈ Z//B we can associate the map h_{(γ,ϵ)} ∈ M^{Z} which sends γ to ϵ and the remaining elements of Z to zero. In that way, Z//B gets identified with a basis of M^{Z} and we can interpret R_{M} as a linear map R_{M} : K(Q_{1}//B) ⟶ K(Z//B), which is just the map used in Section 2 (there with M =_{Λ} Λ_{Λ} and defined only on glued elements). With this interpretation, bearing in mind Proposition 6(3), we have R_{M}^{n} : K(Q_{1}//B_{n}) ⟶ K(Z// ⋃_{γ∈Z} B_{n+d_{γ}−1}), where d_{γ} is the length of γ.

All throughout this section, whenever needed, we shall see the above identifications as identities. Consider now, for every n ≥ 0, the K-vector subspace G_{n} of Hom_{E−E}(DQ_{1}, M_{n}) = Hom_{E−E}(X, M_{n}) generated by the glued elements of (Q_{1}//B_{n}) which are not admissible.

Lemma 1. In the situation described above, Ker(R_{M}^{n}) = K(Q_{1}//B_{n})_{a} ⊕ (G_{n} ∩ Ker(R_{M}^{n}), for every n ≥ 0, where (Q_{1}//B_{n})_{a} denotes the subset of (Q_{1}//B_{n}) consisting of admissible elements.

Proof. By the preceding comments, one can express every f ∈ Hom_{E−E}(X, M_{n}) as a K-linear combination f = ∑_{(a,ϵ)∈(Q_{1}//B_{n})} λ_{(a,ϵ)}f^{(a,ϵ)}. If now f ∈ Ker(R_{M}^{n}), since f^{(a,ϵ)} ∈ Ker(R_{M}^{n}) whenever (a, ϵ) is admissible, it is not restrictive to assume that λ_{(a,ϵ)} = 0 in this case. Then, it only remains to prove that if λ_{(a,ϵ)} ≠ 0, where (a, ϵ) is not admissible, then (a, ϵ) is glued. Indeed, pick up z = a_{1} · ·a_{i−1}aa_{i+1} · ·a_{t} ∈ Z such that a_{1} · ·a_{i−1}ϵa_{i+1} · ·a_{t} ∈ B. Since f(z) = 0, a_{1} · ·a_{i−1}ϵa_{i+1} · ·a_{t} ∈ B must be cancelled from f(z). That means that there is an index j ≠ i and a pair (a_{j}, ϵ^{∗}) which appears with nonzero coefficient in the expression of f, so that a_{1} · ·a_{i−1}ϵa_{i+1} · ·a_{t} = a_{1} · ·a_{j−1}ϵ^{∗}a_{j+1} · ·a_{t} in Q. After left and right cancellation if necessary, we get that either ϵ = ac or ϵ = ca, for some oriented cycle c, i.e., (a, ϵ) is glued as desired.

We are ready to give a formula for dim H^{1}(Λ, M). For that, one should observe that, in the present monomial situation, Δ^{M} can be identified with the quiver having Q_{0} as set of vertices and B as set of arrows, with as many arrows i → j in Δ^{M} as paths in B from i to j. The set B of loops in Δ^{M} is just the set of oriented cycles in B. One readily sees that B is X-good.

Remark 2. In the present situation, the loop-quiver Γ(M) admits a very handy description, entirely analogous to that given before Proposition 2 for M =_{Λ} Λ_{Λ}.

Theorem 3. Let D be a central division K-algebra, Λ a monomial D-algebra, M a monomial Λ − Λ-bimodule and Γ be the loop quiver of M. Then dim C_{M}(Λ) = n(Γ), the number of closed connected components of Γ, and dim H^{1}(Λ, M) = dimC_{M}(Λ) − | Q_{0}//B | + | Q_{1}//B | − | (Q//B)_{e} | − ∑_{n≥0}dim Im(R_{M} |_{G_{n}}). Moreover, once K is fixed, dim H^{1}(Λ, M) does not depend on the ground central division algebra D.

Proof. By Proposition 3, dim C_{M}(E) =| Q_{0}//B |. On the other hand, Δ^{X} = Q and one can identify X = DQ_{1}. As mentioned, the set of loops in Δ^{M} is X−good, so that Proposition 5 applies and dim C_{M}(Λ) = n(Γ). Since both Λ and M are canonically N-graded by the length of the paths, we can apply Proposition 6 and it only remains to calculate dim Ker(R_{M}^{n}), for every n ≥ 0. By the comments preceding the foregoing lemma, one can decompose Hom_{E−E}(X, M_{n}) = U_{n} ⊕ V_{n} ⊕ G_{n}, where U_{n} is the K-span of the admissible elements of (Q_{1} ∥ B_{n}) and V_{n} is that of the effective elements. The above lemma implies that dim Ker(R_{M}^{n}) = dim U_{n}+ dim G_{n}− dim Im(R_{M} |_{G_{n}})= | Q_{1}//B_{n} | − | (Q_{1}//B_{n})_{e} | - dim Im(R_{M} |_{G_{n}}). Hence dim Ker(R_{M}) = | Q_{1}//B | − | (Q_{1}//B)_{e} |-∑_{n≥0}dim Im(R_{M} |_{G_{n}}). From that the result follows.

Remark 3. Each dim Im(R_{M} |_{G_{n}}) is given by the rank of an integer matrix. These ranks are the only terms in the formula that depend on the ground field K, actually on its characteristic. Hence, if all glued elements of Q_{1}//B are admissible, dim H^{1}(Λ, M) is independent of K.

Example 1. Consider the (finite-dimensional) monomial D-algebra given by a quiver Q, with vertices Q_{0} = {1, 2} and arrows 1 ⟶^{u} 2, 2 ⟶^{v} 1 and 1 ⟶^{α} 1, and the set of relations Z = {vαu, α^{3}, uvu, vuv, αuvα}. We get the following relevant data:

1. dim Λ = | B | = 20

2. The loop quiver Γ of M =_{Λ} Λ_{Λ} has | Γ_{0} |=| Q_{0}//B |= 12 and has eight connected components, four of them closed. Hence n(Γ)= dim Z(Λ)=4.

3. (Q_{1}//B)_{e} = {(α, uv)}, so that | (Q_{1}//B)_{e} |=1 4. | Q_{1}//B |=17

5. One has G_{0} = K(α, e_{1}), G_{2} = K(α, α^{2}) + K(u, αu) + K(v, vα), G_{3} = K(u, α^{2}u) + K(v, vα^{2}) and G_{n} = 0, for the remaining natural numbers.

6. Since R(α, e_{1}) ≠ 0, one has dim Im(R_{|G_{0}}) = 1

7. By Remark 1, we know that Im(R_{|G_{2}}) ⊆ K(Z//(B_{4} ∪ B_{5})). By taking {(α, α^{2}), (u, αu), (v, vα)} and (Z/B_{4} ∪ B_{5}) as D-bases of the domain and codomain of R_{|G_{2}}, this map is represented by a 8 × 3−matrix with three identical columns. Hence, dim Im(R_{|G_{2}}) = 1.

8. By proceeding as in the previous case, R_{|G_{3}} is given by a 2 × 2−matrix with both columns equal to (1, 1). Therefore dim Im(R_{|G_{3}}) = 1.

By applying Theorem 3, we get dim H^{1}(Λ, Λ) = 4 − 12 + 17 − 1 − 3 = 5, independently of our choice of D and the ground field K in this case.

Remark 4. The reader should notice that, by using the formula of Theorem 1 in the above example, one would have to consider directly the linear map R_{g} : K(Q_{1}//B)_{g} ⟶ K(Z//B). In this case, | (Q_{1}//B)_{g} |=15 and | Z//B |=29. Hence, in general, it is adviseable to use the refined formula of Theorem 3

We shall end this section by applying our results to give the dimensions of the center and H^{1} of a truncated algebra over a central division algebra D. A monomial D−algebra DQ/⟨Z⟩, with Q connected, is a truncated D−algebra if Z is the set Q_{n} of all the paths of a given length n, with n ≥ 2.

In order to compute Z(Λ), we first identify the closed oriented cycles, i.e., the oriented cycles belonging to closed connected components of the loop-quiver Γ of M =_{Λ} Λ_{Λ}. Leaving aside the vertices of Q, which form always a closed connected component of Γ, clearly, an oriented cycle ε is closed if either its length is n − 1 or if the only arrows leaving or ending at the source-terminus vertex of ε are the first or the last arrow of ε. The connected components of Γ corresponds to orbits of cycles under the action of the appropriate cyclic group. Under this action the closed property of cycles is preserved for (n−1)-length cycles, whence the alternative condition is only preserved in case Q is a crown. If l ≥ n then we are in the trivial situation when Γ = Q and Z(Λ) = K has dimension 1. If l < n then the number of orbits is q for l = nq + r and 0 ≠ r < n, while this number is q − 1 if l = nq. This discussion shows the following

Proposition 7. Let Λ : DQ/⟨Q_{n}⟩ be a truncated D−algebra, where Q is a connected quiver, D is a central division K−algebra and n ≥ 2. Then dim_{k} Z(Λ) = 1 + c, where:

1. c is the number of orbits of (n − 1)-length cycles, if Q is not a crown

2. c = q, in case Q is a l-crown and l = nq + r, with 0 ≠ r < n

3. c = q − 1, in case Q is a l-crown and l = nq

We proceed to calculate the dimension of H^{1}(Λ, Λ). We need an auxiliary result.

Lemma 2. Let Λ = DQ/⟨Q_{n}⟩ be a truncated D−algebra, where Q is a connected quiver, D is a central simple algebra and n ≥ 2. The maps R_{k} : G_{k} ⟶ K(Z//B) are all zero, except perhaps R_{|G_{0}}, which is either zero or injective.

Proof. Bearing in mind that B_{m} = ∅, for all m ≥ n, Remark 1 tells us that R_{|G_{k}} = 0, for all k > 0. Assume, finally, that R_{|G_{0}} ≠ 0. In order to prove that it is injective, we show a stronger result: if (a, ε) and (a′, ε′) are different loops (which can be at the same vertex), the supports of R_{g}(a, ε) and R_{g}(a′, ε′) have an empty intersection. This is clear since if (γ, γ′) = (β, β′) then γ = β and deleting a or a′ provides the same path, which implies a = a′.

We are now in a position to give a precise formula for dim H^{1}(Λ, Λ), when Λ is a truncated algebra over a central division algebra. The formula was recently obtained by Locateli ((cf. [7, Thrms 1,2 and Prop. 9]) in case D = K and charK = 0. In that situation, our only new contribution is the explicit computation of dim Z(Λ) given by Proposition 7. Below, B_{≥1} is the set of all paths p in Q with 1 ≤ length(p) < n and B is the set of all paths in Q of length < n.

Theorem 4 (see [7]). Let Λ : DQ/⟨Q_{n}⟩ be a truncated D−algebra, where Q is a connected quiver, D is a central division algebra and n ≥ 2. Then dim H^{1}(Λ, Λ) = dim Z(Λ)+ |Q_{1}//B_{≥1}|- |Q_{0}//B|, except in case Λ ≅ K[X]/(X^{n}) and the characteristic of K divides n. In this latter case, dim H^{1}(Λ, Λ) = n.

Proof. We claim that the only case in which G_{0} ≠ 0 and R_{|G_{0}} : G_{0} = K(Q_{1}//Q_{0}) ⟶ K(Q_{n}//B_{n−1}) is the zero map is when Q consists on one vertex and one loop and charK divides n. Indeed, if R_{|G_{0}} ≡ 0 and (α, e) ∈ (Q_{1}//Q_{0}), then (α^{n}, α^{n−1}) ∈ (Q_{n}//B_{n−1}) and (α^{n}, α^{n−1}) appears in R_{|G_{0}}(α, e) with coefficient n. Hence charK divides n. On the other hand, if there were an arrow a ≠ α entering e, then (aα^{n−1}, aα^{n−2}) appears with coefficient n − 1 in R_{|G_{0}}(α, e). But then charK divides n − 1, which is a contradiction. By symmetry, there cannot be an arrow a′ ≠ α leaving e.

Hence, beside the case Λ ≅ K[X]/(X^{n}) and charK dividing n, in whose case dim H^{1}(Λ, Λ) = n using Theorem 3, we can assume R_{G_{0}} injective, whence dim Im(R_{|G_{0}}) = dim G_{0} = |Q_{1}//Q_{0}|. Since, by the previous lemma, dim Im(R_{G_{k}}) = 0 for all k > 0, the formula in Theorem 3 gives dim H^{1}(Λ, Λ) = dim Z(Λ)+ |Q_{1}//B|− |(Q_{1}//B)_{e}|− |Q_{0}//B|− |Q_{1}//Q_{0}|. But all elements of (Q_{1}//B) \ (Q_{1}//Q_{0}) are admissible in this case, so that (Q_{1}//B)_{e} = ∅. That ends the proof.

5 Further applications and examples

We come back to the general situation depicted in the introduction and in Section 3. The following is an immediate consequence of the proof of Theorem 2

Corollary 1. If C_{M}(E) ⊆ C_{M}(X), then dim H^{1}(Λ, M) = dim Ker(R_{M}) = ∑_{i,j∈Δ_{0}^{M}} \frac{|Δ^{X}(i,j)|·|Δ^{M} (i,j)|·[D_{i}:K]·[D_{j}:K]}{|Δ^{Λ^{⊗2}} (i,j)|^{2}}− dim Im(R_{M}).

Examples 1. The following are examples where the hypothesis of the above corollary is satisfied:

1. When M is a (semisimple) Λ − Λ-bimodule such that XM=MX=0. In this case, if the ideal of relations I is contained in the ideal of T_{E}(X) generated by X ⊗_{E} X, then R_{M} is the zero map, so that dim H^{1}(Λ, M) =

∑i,j∈ΔM 0 \frac{|Δ^{X}(i,j)|·|Δ^{M} (i,j)|·[D_{i}:K]·[D_{j}:K]}{|Δ^{Λ⊗2} (i,j)|^{2}}

2. When C_{M}(E)x = xC_{M}(E) = 0, for all x ∈ X. For instance, if Λ = M = DQ/I, where Q is a finite quiver, D is a central division algebra and I is a two-sided ideal of DQ containing all the products αc or cα, with α ∈ Q_{1} and c an oriented cycle in Q. Actually, in this latter case, one has D_{i} = D_{j} = D, for all i, j ∈ Δ_{0}^{M} = Q_{0}, Δ^{X}(i, j) = Q(i, j) and | Δ^{Λ^{⊗}2}(i, j) |= [D : K], where [−, K] denotes the K-dimension. On the other hand, | Δ^{M}(i, j) |=| Δ^{Λ}(i, j) |= \frac{[e_{i}Λe_{j}:K]}{[D:K]}. Hence, we get a simplified formula:

dim H^{1}(Λ, Λ) = \frac{1}{[D:K]} ∑_{i,j∈Q_{0}} | Q(i, j) | ·[e_{i}Λe_{j} : K] -dim Im(R_{Λ}).

A particular situation of the above corollary deserves a special treatment. It is when Λ is a commutative algebra and M is a commutative Λ − Λ-bimodule, i.e., the action of Λ is the same on the left and on the right. Here Λ is a finite direct product of factors of polynomial algebras over the ground field K. For simplification purposes, we shall assume that A = K[x_{1}, ..., x_{n}]/I, although the arguments work equally well for an infinite number of variables. Now X = ∑_{1≤i≤n} Kx_{i} and E=K. The fact that M is commutative implies that, for every ∂ ∈ Der_{E−E}(T_{E}(X), M), ∂(x_{i}x_{j} − x_{j}x_{i}) = 0. That means that if we consider a minimal set Z of generators of I containing {x_{i}x_{j} − x_{j}x_{i} : i, j = 1, ..., n}, in order to calculate Ker(R_{M}) we can forget these commuting relations. That is, if Z = Z_{0} ∪ {x_{i}x_{j} − x_{j}x_{i} : i, j = 1, ..., n} then R_{M} can be taken with arrival space M^{Z_{0}} instead of M^{Z}. That is particularly useful when Z_{0} consists of monomials, in whose case A will be called a commutative monomial algebra. One can essentially repeat the arguments of Section 4, when M = K[x_{1}, ..., x_{n}]/H or M = H/I, where H is a monomial two-sided ideal of K[x_{1}, ..., x_{n}] containing Z_{0} (in particular, when M = Λ). In this new situation, B is a basis of M consisting of (commutative!) monomials and (Q_{1} ∥ B) = V × B, where V = Q_{1} = {x_{1}, ..., x_{n}} is the set of variables. A glued element of Q_{1} ∥ B is either an element of Q_{1} ∥ B_{0} or a pair (x_{i}, b) such that x_{i} appears in b. The remaining terminology of Sections 2 and 4 is analogously adapted to the new commutative situation.

In order to give our result in this context, we shall denote by V_{0}(M) the subset of V = {x_{1}, ..., x_{n}} consisting of those variables x_{i} which appear as first component of some non-admissible pair in Q_{1}//B_{0} = V × B_{0}, i.e., x_{i} ∈ V_{0}(M) iff there is a γ = x_{1}^{m_{1}}..x_{i}^{m_{i}}..x_{n}^{m_{n}} ∈ Z_{0}, with m_{i} ≥ 1, such that γ′ = x_{1}^{m_{1}}..x_{i}^{m_{i}−1}..x_{n}^{m_{n}} ∈ B. In this situation, if whenever (γ, γ′) ∈ Z_{0}//B is a pair as stated, one has that γ ∈ K[x_{1}, ...x_{i−1}, x_{i}^{p}, x_{i+1}, ..., x_{n}], with p = charK, we shall put x_{i} ∈ V_{0}^{+}(M), thus defining a subset V_{0}^{+}(M) ⊆ V_{0}(M). The reader is referred to Example 2 to see the dependence of V_{0}(M) and V_{0}^{+}(M) on the bimodule M.

We are now ready to give our result.

Theorem 5. Let Λ = K[x_{1}, ..., x_{n}]/I be a monomial commutative algebra, M a monomial Λ−Λ−bimodule and B a basis of M consisting of monomials. Then dim H^{1}(Λ, M) =| (Q_{1} \ V_{0}^{+}(M)//B)_{a} | + | V_{0}^{+}(M) | ·dim M.

Proof. Since Λ and M are commutative, we have C_{M}(E) = C_{M}(X) = M and Corollary 1 applies. Hence dim H^{1}(Λ, M) = dimKer(R_{M}) = ∑_{n≥0}dim Ker(R_{M}^{n}). What changes now with respect to the situation of Section 4 is that Lemma 1 no longer holds. To use a suitable substitute, observe that, with the same terminology that in that lemma, if x_{i} ∈ V_{0}^{+}(M) then every pair (x_{i}, ϵ) ∈ (V_{0}^{+}(M)/B_{n}) yields a f^{(x_{i},ϵ)} ∈ Ker(R_{M}^{n}). On the other hand, in this case, if (x_{i}, ϵ) ∈ (Q_{1}//B) is a glued element which does not belong to Q_{1}//B_{0}, then it is admissible. Indeed, if x_{i} appears in γ ∈ Z_{0}, any replacement of x_{i} by ϵ in γ yields a monomial γ′ which is a multiple of γ, whence γ′ ∉ B. From those two observations, one deduces that Ker(R_{M}^{n}) = K(V_{0}^{+}(M)//B_{n})⊕K(Q_{1}\ V_{0}^{+}(M)//B_{n})_{a}, for all n ≥ 0, and the result follows.

Example 2. Let us put Λ = K[x, y, z]/(x^{3}, y^{4}, z^{5}). For every monomial Λ − Λ−bimodule M, we shall have a unique basis B consisting of (commutative) monomials x^{r}y^{s}z^{t}, where the triple (r, s, t) varies on a uniquely determined subset N_{M} of N^{3}, N being the set of natural numbers including zero. The pair (x, x^{r}y^{s}z^{t}) ∈ (Q_{1}//B) will be admissible iff (r + 2, s, t) ∉ N_{M}. Analogously, (y, x^{r}y^{s}z^{t}) ∈ (Q_{1}//B) will be admissible iff (r, s + 3, t) ∉ N_{M} and (z, x^{r}y^{s}z^{t}) ∈ (Q_{1}//B) will be admissible iff (r, s, t + 4) ∉ N_{M}. We consider three samples:

1. If M = rad(Λ)^{2} then N_{M} = {(r, s, t) ∈ N^{3} : 2 ≤ r + s + t, r < 3, s < 4, t < 5}. The reader can check that | (x//B)_{a} |= 39, | (y//B)_{a} |= 44 and | (z//B)_{a} |= 47.

2. If M =_{Λ} Λ_{Λ} then N_{M} = {(r, s, t) ∈ N^{3} : r < 3, s < 4, t < 5}. One then gets | (x//B)_{a} |= 40, | (y//B)_{a} |= 45 and | (z//B)_{a} |= 48.

3. If M = K[x, y, z]/(x^{2}, y^{2}, z^{5}) (viewed as a Λ−Λ−bimodule) then N_{M} = {(r, s, t) ∈ N^{3} : r < 2, s < 2, t < 5}. Here one has (x//B)_{a} = x//B, (y//B)_{a} = y//B and (z, x^{r}y^{s}z^{t}) ∈ (Q_{1}//B)_{a} iff (r, s, t) ∈ N_{M} and t ≥ 1.

We move now to identify V_{0}^{+}(M)

1. If M = rad(Λ)^{2} then B_{0} = ∅ and we have V_{0}(M) = V_{0}^{+}(M) = ∅

2. If M =_{Λ} Λ_{Λ} then V_{0}(M) = V = {x, y, z} and V_{0}^{+}(M) = {x}, {y}, {z} or ∅, depending on whether charK = 3, 2, 5 or none of them.

3. If M = K[x, y, z]/(x^{2}, y^{2}, z^{5}) then V_{0}(M) = {z} and V_{0}^{+}(M) = {z} or ∅, depending on charK = 5 or charK ≠ 5

Putting together all the above information, Theorem 5 yields:

1. dim H^{1}(Λ, rad(Λ)^{2}) = 130

2. dim H^{1}(Λ, Λ) = 153, 148, 145 or 133, depending on whether charK = 3,2,5 or none of them.

3. If M = K[x, y, z]/(x^{2}, y^{2}, z^{5}), dim H^{1}(Λ, M) = 60 or 56, depending on whether charK = 5 or ≠ 5.

Remark 5. The formula of Theorem 5 does not have much sense when dim M = ∝, for it is not difficult to see that dim H^{1}(Λ, M) is always infinite in that case. However, by tracing back the proof of the Theorem, one sees that, denoting by ∂^{(x_{i},ϵ)} the derivation corresponding to f^{(x_{i},ϵ)} (see Remark 1) via the isomorphism Der_{E−E}(r(X), M) ≅ Hom_{E−E}(X, M) = Hom_{K}(KV, M), we have that {∂^{(x_{i},ϵ)} : (x_{i}, ϵ) ∈ (Q_{1} \ V_{0}^{+}(M)//B)_{a} ∪ (V_{0}^{+}(M)//B)} is a K-basis of Der_{K}(Λ, M) = H^{1}(Λ, M), a fact that is also true when dim M =∝. In fact, the reader will have no difficulty in deducing the following corollary, which extends Theorem 2.2.1 in [1].

Corollary 2 (see [1]). With notation as in last theorem, we have a Λ-module decomposition

Der_{K}(Λ, M) = (⊕_{x_{i}∈V_{0}^{+}(M)}M \frac{∂}{∂x_{i}}) ⊕ (⊕_{x_{i}∈Q_{1}\V_{0}^{+}(M)}ann_{M}(I : x_{i}))\frac{∂}{∂x_{i}})

In particular, Der_{K}(Λ, M) = M^{(V_{0}^{+}(M))} ⊕ (⊕_{x_{i}∈Q_{1}\V_{0}^{+}(M)}ann_{M}(I : x_{i})) as a Λ-module, where ann_{M}(J) = {m ∈ M : Jm = 0} for every ideal J of Λ.

Suppose now that Q is a finite quiver and P(Q) is the path semigroup of Q, i.e. the elements of P(Q) are the paths in Q plus a zero element 0 and multiplication is just concatenation of paths. Consider any semigroup Σ obtained from P(Q) by factoring out by some congruencies p ≡ q or p ≡ 0, where p and q are paths in Q of lengths ≥ 2 sharing source and terminus (see [6] and [2] for more details). By abuse of notation, we say that an element of Σ is an arrow or an oriented cycle if it is the class, modulo the above congruencies, of an arrow or an oriented cycle in Q. We consider a central division algebra D and the (truncated) semigroup ”Dalgebra” Λ = DΣ (i.e., we convene that D0 = 0, a fact that will be also assumed for any of the appearing bimodules). Observe that DΣ = DQ/I, where I is the two-sided ideal of DQ generated by the set Z of defining relations of Σ. The sort of Λ − Λ−bimodule that we shall consider is either of the form M = Λ/H, where H is a two-sided ideal of Λ generated by elements of Σ, or of the form M = DΣ′, where Σ′ ⊆ Σ is a two-sided ideal (in the semigroup sense) of Σ. In both cases, M has a unique D-basis B consisting of elements of Σ, namely, B = Σ \ H in the first type and B = Σ′ \ {0} in the second type. For that reason, a Λ − Λ−bimodule of any of the two types will be called a semigroup bimodule. One can again extend to this situation the terminology used in Sections 2 and 4. Given a pair (a, ϵ) ∈ Q_{1}//B, we denote by ∂^{(a,ϵ)} : r(X) =< Q_{1} >⟶ M the derivation corresponding to f^{(a,ϵ)} (see Remark 1) via the isomorphism Der_{E−E}(r(X), M) ≅ Hom_{E−E}(X, M) = Hom_{DQ_{0}−DQ_{0}}(DQ_{1}, M). The pair (a, ϵ) will be called admissible if ∂^{(a,ϵ)}(γ) = 0 in M, for all relations γ ∈ Z in which a appears. The pair (a, ϵ) will be called glued if either ϵ = ē_{i} (=class of the vertex e_{i} in Σ) or there are σ, u, v ∈ Σ such that either σau = σϵv or uaσ = vϵσ. Finally, the pair (a, ϵ) will be effective if it is neither admissible nor glued. As in the monomial case, (Q_{1}//B)_{a}, (Q_{1}//B)_{g} and (Q_{1}//B)_{e} will denote the sets of admissible, glued and effective pairs of Q_{1}//B. We shall denote by G the K-vector subspace of K(Q_{1}//B) spanned by the glued elements which are not admissible. In case Z consists of homogeneous relations, the length of an element of Σ is well-defined. Then B_{n} stands for the subset of B consisting of elements of length n and G_{n} = G ∩ K(Q_{1}//B_{n}).

Definition 2. In the above situation, we shall say that the cycles of B satisfy the left (resp. right) cancellation property if ac = ac′ ∈ B (resp. ca = c′a ∈ B), with c, c′ oriented cycles in B and a arrow, implies c = c′. The cycles of B will be said to satisfy the cancellation property when they satisfy both the left and the right cancellation properties.

Proposition 8. Let D a central division K-algebra, Σ a semigroup as above, Λ = DΣ the corresponding (truncated) semigroup algebra and M a semigroup bimodule with D-basis B ⊆ Σ. Let us assume that the cycles of B satisfy the cancellation property. The following assertions hold:

1. The set B of loops in Δ^{M} (equivalently, the set of oriented cycles in B) is X − good.

2. If Γ is the loop-quiver of M, then dim C_{M}(Λ) = n(Γ) =number of closed connected components in Γ.

3. dim H^{1}(Λ, M) = dim C_{M}(Λ)- | Q_{0}//B |+ | Q_{1}//B |- | (Q_{1}//B)_{e} |- dim Im(R_{M}_{|G})

When, moreover, Σ is defined by homogeneous congruencies, dim Im(R_{M}_{|G}) can be replaced by ∑_{n≥0}dim Im(R_{M}_{|G_{n}}) in the last assertion.

Proof. If we prove 1), then 2) will follow from Proposition 5. Now an analogue of Lemma 1 also works here to show that Ker(R_{M}) = K(Q_{1}//B)_{a} ⊕ G and, in case Z consists of homogeneous relations, Ker(R_{M}^{n}) = K(Q_{1}//B_{n})_{a} ⊕ G_{n}, for all n ≥ 0. But then, 3) and its corresponding refinement in the graded case will follow from Theorem 2, with an argument similar to that in Theorem 3. Consequently, we only need to prove assertion 1).

Let us take a ∈ Q_{1} = (Δ_{1}^{X}) and m = λ_{1}c_{1}+...+λ_{s}c_{s} ∈ C_{M}(E) = C_{M}(DQ_{0}), where λ_{1}, ..., λ_{s} ∈ K∗ and c_{1}, ..., c_{s} are distint elements of B (i.e., oriented cycles in B). The cancellation property says that all the c_{i}a which are nonzero are different elements of B. But that implies that, if ma = 0 in M, then c_{i}a = 0 for i + 1, ..., s. By symmetry, one shows the corresponding property when am = 0 in M. Hence, condition 1) in Definition 1 holds.

Take now a and m as above and m′ = μ_{1}c′_{1} + ... + μ_{t}c′_{t} ∈ C_{M}(E), where μ_{j} ∈ K^{∗} and c′_{j} ∈B, for j = 1, ..., t. If ma = am′ in M and we delete any product c_{i}a or ac′_{j} which is zero in M, an argument similar to that of the above paragraph, shows that we should have, for each index i such that c_{i}a ≠ 0 in M, an index j such that c_{i}a = ac′_{j} in M, and viceversa. That proves condition 2) in Definition 1.

Example 3. Let D be a central division K-algebra and Λ = DQ/ < Z > be the algebra given by the quiver Q having vertices Q_{0} = {1, 2}, two arrows a_{1}, a_{2} : 1 ⟶ 2 and two arrows b_{1}, b_{2} : 2 ⟶ 1, with relations Z = {a_{1}b_{2}a_{2} − a_{2}b_{2}a_{1}, b_{1}a_{1}b_{2} − b_{2}a_{1}b_{1}, a_{i_{1}}b_{i_{2}}a_{i_{3}}b_{i_{4}} − a_{j_{1}}b_{j_{2}}a_{j_{3}}b_{j_{4}}, b_{i_{1}}a_{i_{2}}b_{i_{3}}a_{i_{4}} − b_{j_{1}}a_{j_{2}}b_{j_{3}}a_{j_{4}} (all choices of subindices) }∪{all paths of length 5}. We propose the reader to check that the cycles of B satisfy the cancellation property. In order to compute dim H^{1}(Λ, Λ), the following are the relevant data:

1. | Q_{1}//B |= 36

2. | Q_{0}//B |= 12

3. The loop-quiver Γ of M =_{Λ} Λ_{Λ} has as connected components: e_{1} ⇆ e_{2}, a_{i}b_{j} ⇆ b_{j}a_{i} (all choices of i, j ∈ {1, 2}) and isolated vertices abab and baba. The only closed connected components are e_{1} ⇆ e_{2}, a_{1}b_{2} ⇆ b_{2}a_{1}, {abab} and {baba}. Consequently, dim Z(Λ) = 4.

4. All elements in Q_{1}//B are admissible, except (a_{1}, a_{2}), (b_{2}, b_{1}).Hence, G_{n} = 0, for all n ≠ 1, G_{1} = K(a_{1}, a_{2}) + K(b_{2}, b_{1}) and (Q_{1}//B)_{e} = ∅. A very easy computation shows that Ker(R_{|G_{1}}) = 0, whence dim Im(R_{|G_{1}}) = 2.

Putting together all the above information, Proposition 8 yields:

dim H^{1}(Λ, Λ) = 4 − 12 + 36 − 2 = 26

Acknowledgements: The second named author has been partially supported by the D.G.E.S. of Spain (proyecto PB96-0961-C02-02) and the Fundación ’Séneca’ of the Comunicad Autónoma de Murcia (proyecto PB/16/FS/97)

References

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[2] BURGESS, W.D.; SAORIN, M.: Homological aspects of semigroup gradings on rings and algebras. To appear in Canad. J. Math.

[3] CIBILS, C.: Rigid monomial algebras. Math. Ann. 289 (1991), 95-109.

[4] CIBILS, C.: Tensor Hochschild homology and cohomology. To appear in Proc. of Euroconference on Interactions between Ring Theory and Repres. of Algebras (Murcia 1998).

[5] DLAB, V.; RINGEL, C.M.: ”Indecomposable representations of graphs and algebras”. Mem. Amer. Math. Soc. 173 (1976)

[6] FULLER, K.R.: Algebras from diagrams. J. Pure and App. Algebra 48 (1987), 23-37.

[7] LOCATELI, A.C.: Hochschild cohomology of truncated quiver algebras. Comm. Algebra 27(2) (1999), 645-664.

Claude Cibils Département de Mathématiques Université de Montpellier 2 F-34095 Montpellier cedex 5. FRANCE

cibils@math.univ-montp2.fr

Manuel Saorín Departamento de Matemáticas Universidad de Murcia. Aptdo. 4021 30100 Espinardo, Murcia SPAIN

msaorinc@fcu.um.es