Cohomologies of pullbacks

Juan Carlos Bustamante, Julie Dionne, and David Smith

Abstract. We study the Hochschild and simplicial (co)homology groups or a certain family of pullbacks of algebras, called oriented pullback. We prove that for such a pullback, there exists Mayer-Vietoris long exact sequences of Hochschild and simplicial (co)homology groups. Finally, we study the fundamental group, the simple connectedness and the strong simple connectedness of a particular family of pullbacks, the so-called articulated algebras.

Introduction

Let k be a field, and A be a finite dimensional k−algebra (with unit). Given an A−A−bimodule _{A}M_{A}, the Hochschild homology and cohomology groups of A with coefficients in _{A}M_{A} are the groups H_{i}(A, M) = Tor_{i}^{A⊗_{k}A^{op}}(M, A) and H^{i}(A, M) = Ext_{A⊗_{k}A^{op}}^{i}(A, M), respectively. In case _{A}M_{A} =_{A} A_{A}, we simply write H_{i}(A), or H^{i}(A). These groups are closely related: for every i ≥ 0 there exists an isomorphism ϕ_{i} : DH^{i}(A, DX) → H_{i}(A, X), where D = Hom_{k}(−, k) is the usual duality.

The cohomology groups corresponding to low degrees (i ≤ 2) have clear interpretations in terms of classical algebraic structures. Moreover, it was observed by Gerstenhaber in [20] that H^{2}(A) and H^{3}(A) are also related to the rigidity properties of A. From the representation theoretical point of view, several recent results show links between the vanishing of H^{1}(A) or H^{2}(A) and the simple connectedness, or representation directness of A (see [11, 4, 3]).

Beside this, the sum H^{∗}(A) = ∐_{i≥0} H^{i}(A) is a graded commutative ring under the Yoneda product, which coincides with a cup-product ∪ defined at the cochain level. In addition, there is another product on H^{∗}(A), namely the bracket [−, −] which makes of H^{∗}(A) a graded Lie algebra (see [21]). Moreover, these two structures are compatible, but very little is known about them.

In general, the Hochschild cohomology groups are difficult to compute. Direct formulae exist in some particular cases (see [25, 16, 29]). Moreover, some long exact sequences of cohomology groups have been obtained in case A admits an homological ideal (see [18]), or is a triangular matrix algebra (see [25, 17, 33, 23, 24, 8]). In case A is an incidence algebra, these sequences are the Mayer-Vietoris sequences associated to some simplical complex (see [22]). However, it is known that Hochschild cohomology does not satisfy the excision axiom, so Mayer-Vietoris sequences cannot be obtained in general contexts.

2000 Mathematics Subject Classification: 16E40

Keywords: Pullbacks, Hochschild cohomology, Simplicial cohomology, Fundamental group

The main purpose of this paper is to study the Hochschild cohomology groups of a particular family of pullbacks of algebras, which we call oriented pullbacks. It turns out that for this class of algebras there is a Mayer-Vietoris sequence of Hochschild cohomology groups. More precisely, we prove:

Theorem 1. Let R be the oriented pullback of the k-algebras projections f_{1} : A_{1} → C and f_{2} : A_{2} → C, and let X be a R-R-bimodule. Moreover, let e_{1}, e_{2} and e be the units of A_{1}, A_{2} and C respectively. Then there exists a long exact sequence of Hochschild cohomology groups :

[figure: 0; H^{0}(R, X); (α_{1}^{0},α_{2}^{0})^{t}; H^{0}(A_{1}, e_{1}Xe_{1}) ⊕ H^{0}(A_{2}, e_{2}Xe_{2}); (β_{1}^{0},β_{2}^{0}); H^{0}(C, eXe); δ^{0}; H^{1}(R, X); (α_{1}^{1},α_{2}^{1})^{t}; H^{1}(A_{1}, e_{1}Xe_{1}) ⊕ H^{1}(A_{2}, e_{2}Xe_{2}); (β_{1}^{1},β_{2}^{1}); H^{1}(C, eXe); ⋯]

Moreover, if X = R, then eXe = C, e_{i}Xe_{i} = A_{i} (i = 1, 2) and

(a) the maps α_{i}^{n} : H^{n}(R) → H^{n}(A_{i}) and β_{i}^{n} : H^{n}(A_{i}) → H^{n}(C) induce ring homomorphisms α_{i}^{∗} : H^{∗}(R) → H^{∗}(A_{i}) and β_{i}^{∗} : H^{∗}(A_{i}) → H^{∗}(C).

(b) if R is a triangular algebra, the maps α_{i}^{∗} : H^{∗}(R) → H^{∗}(A_{i}) and β_{i}^{∗} : H^{∗}(A_{i}) → H^{∗}(R) are also morphisms of Lie algebras.

Theorem 2. Let R be the oriented pullback of the k-algebras projections f_{1} : A_{1} → C and f_{2} : A_{2} → C, and let X be a R-R-bimodule. Moreover, let e_{1}, e_{2} and e be the units of A_{1}, A_{2} and C respectively. Then there exists a long exact sequence of Hochschild homology groups :

[figure: ⋯; H_{1}(C, eXe); H_{1}(A_{1}, e_{1}Xe_{1}) ⊕ H_{1}(A_{2}, e_{2}Xe_{2}); H_{1}(R, X); H_{0}(C, eXe); H_{0}(A_{1}, e_{1}Xe_{1}) ⊕ H_{0}(A_{2}, e_{2}Xe_{2}); H_{0}(R, X); 0.]

Moreover, in case where X = R, we have eXe = C and e_{i}Xe_{i} = A_{i} (i = 1, 2).

On the other hand, if an algebra A admits a semi-normed basis one can define its simplicial homology and cohomology groups ([10, 31]). In case k is algebraically closed, and A is basic, there exists a unique quiver Q_{A} and (several) surjective maps \xymatrix{ν : kQ_{A} \ar[r] & A} (see [9]) so that A ≃ kQ_{A}/Ker ν. To the pair (Q_{A}, Ker ν) one can associate its fundamental group π_{1}(Q_{A}, Ker ν) (see [32]). The strong simple connectedness of an algebra A = kQ/I can be defined in terms of some fundamental groups. There exist several links between the Hochschild (co)homology groups, the simplicial (co)homology groups, and the fundamental groups associated to an algebra A. In the same vein of Theorems 1 and 2, we establish analogous results concerning the simplicial (co)homology groups arising from oriented pullbacks.

The paper is organized as follows. In Section 1, we fix notation and terminology, and recall the definition of a pullback of algebras. In Section 2, we recall the relevant facts about Hochschild (co)homology, give the definition of oriented pullback of algebras and prove Theorem 1 and Theorem 2. Moreover, we give some consequences and examples, and compare the sequence of Theorem 1 with other existing long exact sequences. Finally, in Section 3, we show the existence of a long exact sequence relating the simplicial (co)homology groups of the algebras involved in a pullback. In addition, we study the fundamental group and the (strong) simple connectedness of a particular case of pullback, namely the articulated algebras.

1. Preliminaries

1.1. Quivers, algebras and categories. A quiver is a quadruple Q = (Q_{0}, Q_{1}, s, t), where Q_{0} and Q_{1} are two finite sets, respectively called the set of vertices and the set of arrows, and s and t are two maps from Q_{1} to Q_{0} associating to each arrow respectively, its source and its target.

A path is a sequence of arrows ω = α_{1}α_{2} ⋯ α_{n} such that t(α_{i}) = s(α_{i+1}) for each i, with 1 ≤ i ≤ n − 1. We denote by l(ω) the length of the path ω, that is, the number of arrows in ω. The source of such a path is s(α_{1}), and its target is t(α_{n}). Also, to each vertex x of Q, we associate the trivial path ε_{x}. Two paths ω and ω′ are parallel if s(ω) = s(ω′) and t(ω) = t(ω′). A vertex x of Q is a source of Q if there is no arrow α such that t(α) = x. The sinks are defined dually.

Given a commutative field k, we define the path algebra kQ as the k-vector space having as basis the set of all paths in Q endowed with the following multiplication: let ω_{1} and ω_{2} be two basis elements, then ω_{1}ω_{2} is their composition if t(ω_{1}) = s(ω_{2}), and 0 otherwise.

A relation from x to y, with x, y ∈ Q_{0}, is a linear combination ρ = ∑_{i=1}^{r} λ_{i}ω_{i} where, for each i, ω_{i} is a path of length at least two from x to y in Q and λ_{i} is a non-zero scalar in k. Let F be the two-sided ideal of kQ generated by all arrows of Q. An ideal I of kQ is admissible if there exists a m ≥ 2 such that F^{m} ⊆ I ⊆ F^{2}. In this case the pair (Q, I) is called a bound quiver and the algebra kQ/I is finite dimensional and basic. In this case, by abuse of notation we simply denote the primitive pairwise orthogonal idempotents ε_{x} + I by ε_{x}. If Q has no oriented cycles, kQ/I is said to be triangular.

Conversely, for every finite dimensional, connected and basic algebra A over an algebraically closed field k there exists a unique connected quiver Q and a surjective algebra morphism ν : kQ → A with I = Ker ν admissible. The map ν, or, equivalently, the pair (Q, I) is called a presentation of A. Any basic algebra A = kQ/I can equivalently be regarded as a locally bounded k-category [9] having as object class the set A_{0} = Q_{0} and as morphism set from x to y the k-vector space A(x, y) = ε_{x}Aε_{y}. A subcategory B of the category A is called full if for all x, y ∈ B_{0}, B(x, y) = A(x, y), and convex if any vertex z which lies on a path in Q from x ∈ B_{0} to y ∈ B_{0} also belongs to B_{0}.

In this paper, k will always denote a commutative field. Moreover, all the algebras considered will be of the form kQ/I, for a certain bound quiver (Q, I).

1.2. Pullbacks of algebras. Let A_{1} = kQ_{1}/I_{1}, A_{2} = kQ_{2}/I_{2} and C = kQ_{C}/I_{C} be algebras and f_{1} : A_{1} → C and f_{2} : A_{2} → C be two morphisms of algebras. The pullback R of the morphisms f_{1} and f_{2} is the set of pairs (a_{1}, a_{2}) in A_{1} × A_{2} such that f_{1}(a_{1}) = f_{2}(a_{2}). It is easy to see that R becomes an algebra with the natural operations.

In this paper, we consider the particular case where Q_{C} is a full and convex subquiver of Q_{1} and Q_{2} and I_{C} is the restriction of I_{1} and I_{2} to Q_{C}. In this case, let e be the unit element of C and let f_{i} : A_{i} → C be the canonical projections of algebras defined by f_{i}(a) = eae, for a ∈ A_{i}. Also, for i = 1, 2, let j_{i} : Q_{C} → Q_{i} be the natural inclusions of quivers. From [26, 27] (see also [30]), we have the following result:

Proposition 1.2.1. Let R be the pullback of f_{1} and f_{2}. Then R ≅ kQ_{R}/I_{R}, where Q_{1} ⨿_{Q_{C}} Q_{2} = Q_{R} is the pushout of j_{1} and j_{2} and I_{R} = I_{1} + I_{2}+ < \overline{ρ} >, with \overline{ρ} the set of paths linking (Q_{1})_{0} \ (Q_{C})_{0} and (Q_{2})_{0} \ (Q_{C})_{0} in Q_{R}. Moreover Q_{1} and Q_{2} are convex in Q_{R}. □

Example 1.2.2. Let R be the algebra given by the quiver

[figure: 2; β_{1}; 4; γ_{1}; 1; α_{1}; α_{2}; 6; 3; β_{2}; 5; γ_{2}]

bound by F^{3}. Then R is the pullback of the projections of A_{1} and A_{2} on C, where A_{1}, A_{2} and C are the full subcategories respectively generated by the set of objects {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6} and {2, 3, 4, 5}.

2. Hochschild (co)homology

2.1. Definitions and notations. Let R be a k-algebra and _{R}X_{R} be an R-R-bimodule. Following [14], the Hochschild cohomology groups of R with coefficients in X are the groups Ext_{R⊗_{k}R^{op}}^{i}(R, X) and we denote them as H^{i}(R, X). In the case where _{R}X_{R} = _{R}R_{R}, we simply write H^{i}(R).

Let J = radR and E = R/J. By J^{⊗n} we denote the n^{th} tensor power of J with itself over E. With these notations, and following [15, (1.2)], the Hochschild cohomology groups H^{i}(R, X) are the cohomology groups of the following sequence, denoted by C_{R,X}^{•} = (C_{R,X}^{i}, δ_{R,X}^{i}) in the sequel :

\xymatrix{0 \ar[r] & Hom_{E−E}(E, X) \ar[r]^{δ_{R,X}^{0}} & Hom_{E−E}(J, X) \ar[r]^{δ_{R,X}^{1}} & Hom_{E−E}(J^{⊗n}, X) \ar[r]^{δ_{R,X}^{2}} & ⋯}

where (δ_{R,X}^{0}f)(α) = αf(ε_{1}) − f(ε_{2})α, for α ∈ ε_{1}Jε_{2} and

(δ_{R,X}^{n}f)(x_{1} ⊗ ⋯ ⊗ x_{n}) = x_{1}f(x_{2} ⊗ ⋯ ⊗ x_{n}) + (−1)^{n}f(x_{1} ⊗ ⋯ ⊗ x_{n−1})x_{n}
+ ∑_{i=1}^{n−1}(−1)^{i}f(x_{1} ⊗ ⋯ ⊗ x_{i}x_{i+1} ⊗ ⋯ ⊗ x_{n}),

for n ≥ 1 and x_{1} ⊗ ⋯ ⊗ x_{n} ∈ J^{⊗n}.

In the case where R is the pullback of algebras as described in (1.2), we denote by J_{i} the Jacobson radical of A_{i} and by E_{i} the quotient A_{i}/J_{i}, for i = 1, 2. Also, J_{C} denotes the Jacobson radical of C and E_{C} = C/J_{C}. Finally, A′_{i} denotes the full subcategory generated by (A_{i})_{0} \ C_{0} and e_{i}, e′_{i} and e are the identities of the algebras A_{i}, A′_{i} and C, respectively. In this case e_{i} is the sum of the idempotents associated to the vertices of A_{i}, that is e_{i} = ∑_{x∈(Q_{i})_{0}} ε_{x}. Note that with these notations we have e′_{1}Re′_{2} = e′_{2}Re′_{1} = 0. This will be important in the sequel. Also, given τ_{1}, τ_{2} ∈ J_{C}, we can consider them as elements of J_{1}, J_{2} or J. Nevertheless, we have the following equalities:

τ_{1} ⊗_{E_{C}} τ_{2} = τ_{1} ⊗_{E_{1}} τ_{2} = τ_{1} ⊗_{E_{2}} τ_{2} = τ_{1} ⊗_{E} τ_{2}.

The Yoneda product in H^{∗}(R) = ⨿_{i>0}H^{i}(R) is induced by a cup-product defined at the cochain level as follows: Given f ∈ C_{R,R}^{n}, g ∈ C_{R,R}^{m} we define f ∪ g ∈ C_{R,R}^{n+m} by the rule:

(f ∪ g)(r_{1} ⊗ ⋯ ⊗ r_{n+m}) = f(r_{1} ⊗ ⋯ ⊗ r_{n})g(r_{n+1} ⊗ ⋯ ⊗ r_{n+m}).

On the other hand, if R is a triangular algebra, the elements f ∈ C_{R,R}^{n} take values in the radical of R, so that, given i ∈ {1, …, n}, we can define f ∘_{i} g by the rule :

(f ∘_{i} g)(r_{1} ⊗ ⋯ ⊗ r_{n+m−1}) = f(r_{1} ⊗ ⋯ ⊗ g(r_{i} ⊗ ⋯ ⊗ r_{i+m−1}) ⊗ ⋯ ⊗ r_{n+m−1}).

Then we define the composition f ∘ g = \{ \begin{matrix}0, & if n = 0; \\ ∑_{i=1}^{n}(−1)^{(i−1)(m−1)}f ∘_{i} g, & if n > 0. \end{matrix}

and finally the bracket of f and g as [f, g] = f ∘ g − (−1)^{(n−1)(m−1)}g ∘ f.

These two structures are related. In fact H^{∗}(R) is a Gerstenhaber algebra (see [21]), that is a graded k-vector space endowed with a product ∪ which makes of H^{∗}(R) a graded commutative algebra, and a Lie bracket [−, −] of degree −1 that makes of H^{∗}(R) a graded Lie algebra, and such that [x, yz] = [x, y]z + (−1)^{(|x|−1)|y|}y[x, z].

On the other hand, following [14], the Hochschild homology groups of R with coefficients in X are the groups Tor_{i}^{R⊗_{k}R^{op}}(X, R) and we denote them as H_{i}(R, X). Consequently, there exists an isomorphism ϕ_{i} : DH^{i}(R, DX) → H_{i}(R, X), where D = Hom_{k}(−, k) is the usual duality [14].

2.2. Main results. In this section, we give the proof of Theorem 1 and Theorem 2. We begin with some definitions and preliminary results.

Definition 2.2.1. Let R = kQ_{R}/I_{R} be the pullback of the k-algebra projections f_{1} : A_{1} → C and f_{2} : A_{2} → C as defined in (1.2), where A_{i} ≅ kQ_{i}/I_{i} for i ∈ {1, 2} and C ≅ kQ_{C}/I_{C}. Then R is called oriented pullback if one of the following conditions is satisfied:

1. eRe′_{1} = 0 and eRe′_{2} = 0;

2. e′_{1}Re = 0 and e′_{2}Re = 0;

3. e′_{1}Re = 0 and eRe′_{2} = 0;

Using the decomposition 1_{R} = e′_{1} + e + e′_{2}, one can see that the oriented pullback algebras respectively have one of the following form:

\begin{matrix}1. R = ( \begin{matrix}A′_{1} & e′_{1}Re & 0 \\ 0 & C & 0 \\ 0 & e′_{2}Re & A′_{2} \end{matrix} ); & 2. R = ( \begin{matrix}A′_{1} & 0 & 0 \\ eRe′_{1} & C & eRe′_{2} \\ 0 & 0 & A′_{2} \end{matrix} ); \\ 3. R = ( \begin{matrix}A′_{1} & 0 & 0 \\ eRe′_{1} & C & 0 \\ 0 & e′_{2}Re & A′_{2} \end{matrix} ) \end{matrix}

Clearly, this definition generalizes the definition of "Nakayama oriented pullback" introduced in [30]. Moreover, in this case, one easily see that if _{R}X_{R} = _{R}R_{R} then eXe = C and e_{i}Xe_{i} = A_{i}, for i = 1, 2. For the remainder of this section, we suppose that R is an oriented pullback of projections f_{1} : A_{1} → C and f_{2} : A_{2} → C, and we use the notations of (2.2.1). Also, note that oriented pullbacks are particular cases of extensions of algebras (compare with [17, 33, 23, 8]).

Lemma 2.2.2. Under the above assumptions, there is no non zero path from A′_{2} to A′_{1} or from A′_{1} to A′_{2} in (Q_{R}, I_{R}). Moreover,

1. If eRe′_{i} = 0, then there is no non zero path from C to A′_{i};

2. If e′_{i}Re = 0, then there is no non zero path from A′_{i} to C;

3. The categories A_{1} and A_{2} are full and convex subcategories of R.

Proof : (1). Let eRe′_{1} = 0 and suppose that α_{1}α_{2} ⋯ α_{n} is a path from a vertex x of C to a vertex y of A′_{1}. Let j =min{i | s(α_{i}) ∉ (A′_{1})_{0} and t(α_{i}) ∈ (A′_{1})_{0}}. Then α_{j} ∈ ε_{s(α_{j})}Rε_{t(α_{j})}, thus eRe′_{1} ≠ 0 or e′_{2}Re′_{1} ≠ 0, a contradiction. Statement (2) is shown in the same way. For statement (3), fullness follows from the fact that e_{j}Re′_{i} = eRe′_{i} = 0, or e′_{i}Re_{j} = e′_{i}Re = 0 when 1 ≤ i ≠ j ≤ 2. The convexity follows from (1) and (2). □

Lemma 2.2.3. Under the above assumptions, J_{i} = e_{i}Je_{i} for i = 1, 2.

Proof : Since J is the k-vector space generated by {ω + I_{R} | ω is a path in Q and l(ω) ≥ 1}, e_{i}Je_{i} is generated by

{e_{i}ωe_{i} + I_{R} | ω is a path in Q and l(ω) ≥ 1}
= {e_{i}ωe_{i} + I_{R} | ω is a path in Q and l(e_{i}ωe_{i}) ≥ 1}
= {ω′ + I_{R} | ω′ is a path in Q_{i} and l(ω′) ≥ 1}
= {ω′ + I_{i} | ω′ is a path in Q_{i} and l(ω′) ≥ 1}
= J_{i},

where the second equality follows from the convexity of A_{i} and the third one follows from the fact that I_{i} is the restriction of I_{R} to A_{i}. □

Lemma 2.2.4. Under the above assumptions, e_{i}J^{⊗n}e_{i} = (e_{i}Je_{i})^{⊗n}, for i = 1, 2.

Proof : First, we prove that (e_{i}Je_{i})^{⊗n} ⊆ e_{i}J^{⊗n}e_{i}. Clearly, e_{i}Je_{i} is included in J, and hence, (e_{i}Je_{i})^{⊗n} ⊆ J^{⊗n}. We only have to multiply by e_{i} on each side to obtain the result.

On the other hand, let τ = e_{i}x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n}e_{i} be a non-zero element of e_{i}J^{⊗n}e_{i}. We can assume without loss of generality that x_{j} is a path in Q for each j. Thus e_{i}x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n}e_{i} = e_{1}ε_{s(x_{1})}x_{1}ε_{t(x_{1})} ⊗ ⋯ ⊗ ε_{s(x_{n})}x_{n}ε_{t(x_{n})}e_{i} and it follows from the convexity of A_{i} that x_{j} = ε_{s(x_{j})}x_{j}ε_{t(x_{j})} = e_{i}x_{j}e_{i} for each j. Then, τ ∈ (e_{i}Je_{i})^{⊗n}. □

Lemma 2.2.5. Under the above assumptions, the following hold, for i = 1, 2 :

1. If eRe′_{i} = 0, then J_{i}^{⊗n} = e′_{i}J^{⊗n}e′_{i} ⊕ e′_{i}J^{⊗n}e ⊕ eJ^{⊗n}e, and;

2. If e′_{i}Re = 0, then J_{i}^{⊗n} = e′_{i}J^{⊗n}e′_{i} ⊕ eJ^{⊗n}e′_{i} ⊕ eJ^{⊗n}e;

Proof : Keeping in mind that e′_{1}, e and e′_{2} are pairwise orthogonal idempotents, this follows from the equality e_{i} = e′_{i} + e and the fact that J_{i}^{⊗n} = (e_{i}Je_{i})^{⊗n} = e_{i}J^{⊗n}e_{i} = (e′_{i} + e)J^{⊗n}(e′_{i} + e). □

This allows us to prove Theorem 1.

PROOF OF THEOREM 1 : We only prove the existence of the long exact sequence in case (1) of the definition of oriented pullbacks (2.2.1), since the proof in the other cases is similar.

We first observe that, since 1_{R} = e′_{1} + e + e′_{2}, we have

J^{⊗n} = e′_{1}J^{⊗n}e′_{1} ⊕ e′_{1}J^{⊗n}e ⊕ e′_{1}J^{⊗n}e′_{2} ⊕ eJ^{⊗n}e′_{1} ⊕ eJ^{⊗n}e
⊕eJ^{⊗n}e′_{2} ⊕ e′_{2}J^{⊗n}e′_{1} ⊕ e′_{2}J^{⊗n}e ⊕ e′_{2}J^{⊗n}e′_{2}
= e′_{1}J^{⊗n}e′_{1} ⊕ e′_{1}J^{⊗n}e ⊕ eJ^{⊗n}e ⊕ e′_{2}J^{⊗n}e ⊕ e′_{2}J^{⊗n}e′_{2},
⊆ J_{1}^{⊗n} ⊕ J_{2}^{⊗n}.

and consequently,

Hom_{E−E}(J^{⊗n}, X) = Hom_{E−E}(e′_{1}J^{⊗n}e′_{1}, X) ⊕ Hom_{E−E}(e′_{1}J^{⊗n}e, X)
⊕ Hom_{E−E}(eJ^{⊗n}e, X)
⊕ Hom_{E−E}(e′_{2}J^{⊗n}e′_{2}, X) ⊕ Hom_{E−E}(e′_{2}J^{⊗n}e, X)
≅ Hom_{E_{1}−E_{1}}(e′_{1}J^{⊗n}e′_{1}, e_{1}Xe_{1}) ⊕ Hom_{E_{1}−E_{1}}(e′_{1}J^{⊗n}e, e_{1}Xe_{1})
⊕ Hom_{E_{C} −E_{C}} (eJ^{⊗n}e, eXe)

⊕ Hom_{E_{2}−E_{2}}(e′_{2}J^{⊗n}e′_{2}, e_{2}Xe_{2}) ⊕ Hom_{E_{2}−E_{2}}(e′_{2}J^{⊗n}e, e_{2}Xe_{2})

where the isomorphism follows from immediate identifications. Similarly, for i = 1, 2, we obtain, from Lemma (2.2.5), that

Hom_{E_{i}−E_{i}}(J_{i}^{⊗n}, e_{i}Xe_{i}) ≅ Hom_{E_{i}−E_{i}}(e′_{i}J^{⊗n}e′_{i}, e_{i}Xe_{i}) ⊕ Hom_{E_{i}−E_{i}}(e′_{i}J^{⊗n}e, e_{i}Xe_{i})
⊕ Hom_{E_{i}−E_{i}}(eJ^{⊗n}e, e_{i}Xe_{i})

With these equalities in mind define, for i = 1, 2, the natural projection

\begin{aligned}α_{i}^{n} : & Hom_{E−E}(J^{⊗n}, X) & ⟶ & Hom_{E_{i}−E_{i}}(J_{i}^{⊗n}, e_{i}Xe_{i}) \\ & f & ⟼ & e_{i}fe_{i}, \end{aligned}

that is α_{i}^{n}(f)(x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n}) = f(e_{i}x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n}e_{i}), for x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n} ∈ J_{i}^{⊗n}. Clearly, α^{n} = (α_{1}^{n}, α_{2}^{n})^{t} is injective for each n. Moreover, α^{•} = (α_{1}^{•}, α_{2}^{•})^{t} is a morphism of complexes, as a straightforward computation shows.

On the other hand, it follows from the above decompositions that the cokernel of α^{•} is β^{•}, where, for each n, β^{n} is defined by the rule: β^{n}(f, g) = efe − ege.

Therefore, with the notation of section 2.1, we obtain the following short exact sequence of complexes :

\xymatrix{0 \ar[r] & C_{R,X}^{•} \ar[r]^{α^{•}} & C_{A_{1},e_{1}Xe_{1}}^{•} ⊕ C_{A_{2},e_{2}Xe_{2}}^{•} \ar[r]^{β^{•}} & C_{C,eXe}^{•} \ar[r] & 0,}

and the result follows. In order to prove statement a), we show that α_{i}^{∗} is compatible with the cup-product. But this follows from the fact that if x ∈ J_{i}^{⊗(n+m)}, then x = e_{i}xe_{i}. So (α_{i}^{n}(f)) ∪ (α_{i}^{m}(g))(x) = (f ∪ g)(x) = α_{i}^{n+m}(f ∪ g)(x), for f ∈ C_{R,R}^{n} and g ∈ C_{R,R}^{m}. A similar argument applies to β_{i}^{∗}. In order to prove statement b), we show that α_{i}^{∗} is compatible with the bracket. Let f ∈ C_{R,R}^{n}, g ∈ C_{R,R}^{m} and x_{1} ⊗ ⋯ ⊗ x_{n+m−1} ∈ J_{i}^{⊗n+m−1}. Then

α_{i}(f ∘_{j} g)(x_{1} ⊗ ⋯ ⊗ x_{n+m−1})

= (f ∘_{j} g)(e_{i}x_{1} ⊗ ⋯ ⊗ x_{n+m−1}e_{i})
= f(e_{i}x_{1} ⊗ ⋯ ⊗ g(x_{j} ⊗ ⋯ ⊗ x_{j+m−1}) ⊗ ⋯ ⊗ x_{n+m−1}e_{i})
= f(e_{i}x_{1} ⊗ ⋯ ⊗ g(e_{i}x_{j} ⊗ ⋯ ⊗ x_{j+m−1}e_{i}) ⊗ ⋯ ⊗ x_{n+m−1}e_{i})
= (α_{i}(f)) ∘_{j} (α_{i}(g))(x_{1} ⊗ ⋯ ⊗ x_{n+m−1}).

The result follows by linearity. Similarly, β_{i}^{∗} is compatible with the bracket. □

We now give an example showing that Theorem 1 does not hold true when we remove the assumption of oriented pullback.

Example 2.2.6. Let R be the algebra given by the quiver \xymatrix{1 \ar[r]^{α} & 2 \ar[l]^{β}} bound by the

relations αβ = 0 and βα = 0. Then R is a pullback of the projections of A_{1} and A_{2} on C, where A_{1} and A_{2} are the subcategories respectively generated by the arrows α and β and C is the semisimple algebra generated by the vertices. However, R is not an oriented pullback and H^{i}(R) ≠ 0 for each i ≥ 0, but H^{i}(A_{1}) = H^{i}(A_{2}) = H^{i}(C) = 0 for each i > 0.

We now use the isomorphism ϕ_{i} : DH^{i}(R, DX) → H_{i}(R, X) mentioned in section 2.1 and prove Theorem 2.

PROOF OF THEOREM 2 : The first observation to make is that, for i = 1, 2, we clearly have an isomorphism e_{i}Hom_{k}(X, k)e_{i} ≅ Hom_{k}(e_{i}Xe_{i}, k). Similarly, eHom_{k}(X, k)e ≅ Hom_{k}(eXe, k).

On the other hand, it follows from Theorem 1 that there exists a long exact sequence of cohomology groups

⋯ → H^{i}(R, DX) → H^{i}(A_{1}, e_{1}(DX)e_{1}) ⊕ H^{i}(A_{2}, e_{2}(DX)e_{2}) → H^{i}(C, e(DX)e) → ⋯ and consequently a long exact sequence of homology groups

[figure: ⋯; DH^{i}(R,DX); ≅; DH^{i}(A_{1},e_{1}(DX)e_{1})⊕DH^{i}(A_{2},e_{2}(DX)e_{2}); ≅; DH^{i}(C,e(DX)e); ≅; ⋯; ⋯; DH^{i}(R,DX); ϕ_{i}; ≅; DH^{i}(A_{1},D(e_{1}Xe_{1}))⊕DH^{i}(A_{2},D(e_{2}Xe_{2})); ϕ_{i}; ≅; DH^{i}(C,D(eXe)); ϕ_{i}; ≅; ⋯; ⋯; H_{i}(R,X); H_{i}(A_{1},e_{1}Xe_{1})⊕H_{i}(A_{2},e_{2}Xe_{2}); H_{i}(C,eXe); ⋯]

which proves the statement. □

2.3. The case where C is semisimple. We have a better description of the Hochschild cohomology groups in the case where R is an oriented pullback as above and C is a semisimple algebra. Observe that this particular case of oriented pullback corresponds exactly with the "unidirectional articulated algebras" of [19], see (3.2.1) for more details. We begin with the following result on the center of an algebra.

Lemma 2.3.1. Let A = kQ/I be a connected algebra, ε_{i} be the idempotent at the vertex i for each i ∈ Q_{0} and ω belongs to the center Z(A) of A. Then, ω = ∑_{i∈Q_{0}} ω_{i}+λ1_{A}, where λ ∈ k and ω_{i} ∈ ε_{i}radAε_{i} for each i.

Proof : Since classes modulo I of the paths in Q form a basis of kQ/I, ω can be written as ω = ∑_{i,j∈Q_{0}} ω_{i,j}, where ω_{i,j} ∈ ε_{i}Aε_{j} for all i, j. For each vertex l in Q_{0}, the centrality of ω gives

∑_{j≠l} ω_{l,j} + ω_{l,l} = ε_{l}ω = ωε_{l} = ∑_{j≠l} ω_{j,l} + ω_{l,l}.

Hence, ∑_{j≠l} ω_{l,j} = 0 = ∑_{j≠l} ω_{j,l}. Therefore, we have ω = ∑_{i∈Q_{0}} ω_{i} + ∑_{i∈Q_{0}} λ_{i}ε_{i}, where ω_{i} ∈ ε_{i}radAε_{i} and λ_{i} ∈ k for each i ∈ Q_{0}. To end the proof, it remains to show that λ_{i} = λ_{j} for all i, j ∈ Q_{0}. Let α : l → m be an arrow in Q. Then, αω_{m} + λ_{m}α = αω = ωα = ω_{l}α + λ_{l}α. The admissibility of the ideal I gives αω_{m} = ω_{l}α and λ_{l}α = λ_{m}α. In particular, λ_{l} = λ_{m}. Since Q is connected, λ_{i} = λ_{j} for all i, j ∈ Q_{0}. □

Proposition 2.3.2. Let R be an oriented pullback of k-algebra projections f_{i} : A_{i} → C, i = 1, 2, and assume that C is a semisimple algebra with m vertices. Then:

(a) dim_{k}H^{0}(R) = dim_{k}H^{0}(A_{1}) + dim_{k}H^{0}(A_{2}) − 1;

(b) dim_{k}H^{1}(R) = dim_{k}H^{1}(A_{1}) + dim_{k}H^{1}(A_{2}) + (m − 1);

(c) H^{i}(R) = H^{i}(A_{1}) ⊕ H^{i}(A_{2}) for i ≥ 2.

Proof : (a). Since the 0-Hochschild cohomology group H^{0}(A) of an algebra A is isomorphic to its center Z(A), it is sufficient to show that dim_{k}Z(R) = dim_{k}Z(A_{1})+dim_{k}Z(A_{2})− 1. For j = 1, 2, let 𝔹_{j} be a k-basis of Z(A_{j}) containing 1_{A_{j}}, the identity of A_{j}. We claim that 𝔹 = (1_{R} ∪ 𝔹_{1} ∪ 𝔹_{2}) \ {1_{A_{1}}} ∪ {1_{A_{2}}} is a k-basis of Z(R). Since this set is clearly linearly independent, it remains to show that it is a generating set in Z(R). Let ω be an element of Z(R). By (2.3.1), we have ω = ∑_{i} ω_{i} + λ1_{R}, where λ ∈ k and ω_{i} ∈ e_{i}radRe_{i} for each i. Moreover, since C is semisimple, we have ω = ∑_{i∉Q_{C}} ω_{i} + λ1_{R}. Therefore, we can write without ambiguity ω = ρ_{1} + ρ_{2} + λ1_{R}, where ρ_{j} = ∑_{i∈Q_{A′_{j}}} ω_{i}, j = 1, 2. It then

remains to show that ρ_{j} ∈ Z(A_{j}) for j = 1, 2 since, by construction, ρ_{j} ∈ Z(A_{j}) if and only if ρ_{j} belongs to the k-vector space generated by 𝔹_{j} \ {1_{A_{j}} }. Let η_{1} be a path in Q_{1}. We have :

ρ_{1}η_{1} + η_{1} = ρ_{1}η_{1} + 0 + η_{1} = ρ_{1}η_{1} + ρ_{2}η_{1} + 1_{r}η_{1} = ωη_{1}
= η_{1}ω = η_{1}ρ_{1} + η_{1}ρ_{2} + η_{1}1_{R} = η_{1}ρ_{1} + 0 + η_{1}
= η_{1}ρ_{1} + η_{1}

and hence ρ_{1}η_{1} = η_{1}ρ_{1}. Therefore, ρ_{1} ∈ Z(A_{1}). Since, similarly, ρ_{2} ∈ Z(A_{2}), ω belongs to the k-vector space generated by 𝔹 and 𝔹 is a k-basis of Z(R). In particular, dim_{k}Z(R) = dim_{k}Z(A_{1}) + dim_{k}Z(A_{2}) − 1, and this proves (a).

(b). Since C is a semisimple algebra, we have H^{i}(C) = 0 for each i ≥ 1 and thus we have the following exact sequence :

\xymatrix{0 \ar[r] & H^{0}(R) \ar[r] & H^{0}(A_{1}) ⊕ H^{0}(A_{2}) \ar[r] & H^{0}(C) \ar[r] & H^{1}(R) \ar[r] & H^{1}(A_{1}) ⊕ H^{1}(A_{2}) \ar[r] & 0}

and hence,

dim_{k}H^{1}(R) = dim_{k}H^{1}(A_{1}) + dim_{k}H^{1}(A_{2})
+ dim_{k}H^{0}(C) + dim_{k}H^{0}(R) − dim_{k}H^{0}(A_{1}) − dim_{k}H^{0}(A_{2})

Since C is semisimple with m vertices, we have dim_{k}H^{0}(C) = m. Moreover dim_{k}H^{0}(R) − dim_{k}H^{0}(A_{1}) − dim_{k}H^{0}(A_{2}) = −1 by (a). (c). Follows directly from Theorem 1. □

2.4. Examples and comparison. We now compare the sequence of Theorem 1 with the existing ones. First of all, let

R = ( \begin{matrix}A & 0 \\ M & B \end{matrix} )

and X is a R-R-bimodule, where A and B are arbitrary finitely generated k-algebras and M a finitely generated B-A-bimodule. Moreover, let e_{A} and e_{B} be the unit elements of A and B. By [17, 33, 23, 24], we have the following result.

Theorem 2.4.1. Let R be as above and let X be a R-R-bimodule. Then, there exists a long exact sequence

[figure: 0; H^{0}(R, X); H^{0}(A, e_{A}Xe_{A}) ⊕ H^{0}(B, e_{B}Xe_{B}); Ext_{B⊗_{k}A^{op}}^{0}(M, e_{B}Xe_{A}); H^{1}(R, X); H^{1}(A, e_{A}Xe_{A}) ⊕ H^{1}(B, e_{B}Xe_{B}); Ext_{B⊗_{k}A^{op}}^{1}(M, e_{B}Xe_{A}); ⋯]

□

If B = k, the algebra R is a one-point extension. The above long exact sequence has been obtained by D. Happel [25] in this case. The following examples show the difference between the exact sequence of Theorem 1 and the exact sequence recalled above.

Examples 2.4.2.

1. The following example shows that the long exact sequences of Theorem 1 and (2.4.1) may differ from each other. Let R be the radical square zero algebra given by the quiver

\xymatrix{5 \ar[d] \ar[dr] \ar[drr] &  & 6 \ar[d] \ar[dl] \ar[dll] \\ 2 \ar[dr] & 3 \ar[d] & 4 \ar[dl] \\  & 1}

Then R is the oriented pullback (satisfying the condition (1) of (2.2.1)) of the k-algebras projections f_{1} : A_{1} → C and f_{2} : A_{2} → C, where A_{1}, A_{2} and C are the full subcategories generated by the sets of vertices {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6} and {1, 2, 3, 4} respectively. By [16, (p. 96)],

H^{i}(A_{1}) = H^{i}(A_{2}) = \{ \begin{matrix}k, & if i = 0; \\ k^{2}, & if i = 1; \\ 0, & otherwise; \end{matrix} and H^{i}(R) = \{ \begin{matrix}k, & if i = 0; \\ k^{4}, & if i = 1; \\ 0, & otherwise. \end{matrix}

Moreover, since the ordinary quiver of C is a tree, H^{0}(C) = k and H^{i}(C) = 0 for i ≥ 1. By Theorem 1, there exists an exact sequence :

[figure: 0; H^{0}(R); H^{0}(A_{1})⊕H^{0}(A_{2}); H^{0}(C); H^{1}(R); H^{1}(A_{1})⊕H^{1}(A_{2}); H^{1}(C); ···; 0; k; k⊕k; k; k^{4}; k^{2}⊕k^{2}; 0; ···]

On the other hand, if we respectively denote by S_{x} and P_{x} the simple module and the projective module associated to the vertez x in the quiver, one can see R as the one-point extension

A_{1}[M] = ( \begin{matrix}A_{1} & 0 \\ M & k \end{matrix} ) ,

where M = radP_{6} = S_{2} ⊕ S_{3} ⊕ S_{4} is a semisimple module without self-extension. In this case, the long exact sequence of (2.4.1) corresponds with the Happel’s sequence [25], that is :

[figure: 0; H^{0}(R); H^{0}(A_{1}); (Hom_{A_{1}} (M,M))/k; H^{1}(R); H^{1}(A_{1}); Ext_{A_{1}}^{1} (M,M); ···; 0; k; k; k^{2}; k^{4}; k^{2}; 0; ···]

We observe that the two long exact sequences are different.

2. The following example shows that the long exact sequences of Theorem 1 can be more appropriate than the sequence of (2.4.1) in some circumstances. Let R be the algebra given by the quiver

[figure: 1; ε; 2; δ; 3; γ; 4; β; α]

bound by the relations α^{2} = αβ = βγ = δε = ε^{2} = 0. Then R is the oriented pullback (satisfying the condition (3) of (2.2.1)) of the k-algebras projections f_{1} : A_{1} → C and f_{2} : A_{2} → C, where A_{1}, A_{2} and C are the full subcategories generated by the sets of vertices {1, 2, 3}, {3, 4} and {3} respectively. A quick calculation gives:

H^{i}(A_{1}) = H^{i}(A_{2}) = \{ \begin{matrix}k^{2}, & if i = 0; \\ k^{2}, & if i = 1; \\ k, & otherwise. \end{matrix}

Moreover, since the C is semisimple, H^{0}(C) = k and H^{i}(C) = 0 for i ≥ 1. By Theorem 1, there exists an exact sequence

[figure: 0; H^{0}(R); H^{0}(A_{1})⊕H^{0}(A_{2}); H^{0}(C); H^{1}(R); H^{1}(A_{1})⊕H^{1}(A_{2}); H^{1}(C); ···; 0; k^{3}; k^{2}⊕k^{2}; k; k^{4}; k^{2}⊕k^{2}; 0; ···]

from which we obtain that H^{1}(R) ≅ k^{4} and H^{i}(R) ≅ H^{i}(A_{1}) ⊕ H^{i}(A_{2}) ≅ k ⊕ k for i ≥ 2. On the other hand, one can see R as given by the matrix algebra

R = ( \begin{matrix}A_{1} & 0 \\ M & A′_{2} \end{matrix} ) ,

where A′_{2} is the full subcategory generated by the vertex 4 and M is the A′_{2}-A_{1}- bimodule given by the arrow β. In order to apply the sequence of (2.4.1), one needs to compute Ext_{A′_{2}⊗A_{1}^{op}}^{i}(M, M) for i ≥ 0. A direct calculation leads to the fact that A′_{2} ⊗ A_{1}^{op} is given by the quiver

[figure: 4 ⊗ 1; 4⊗ε; α⊗1; 4⊗δ; 4 ⊗ 2; α⊗2; 4⊗γ; 4 ⊗ 3; α⊗3]

bound by the relations induced by the ordinary quiver of R, that is (α ⊗ 1)^{2} = (α ⊗ 2)^{2} = (α ⊗ 3)^{2} = (4 ⊗ ε)^{2} = (4 ⊗ ε)(4 ⊗ δ) = 0, (α ⊗ 1)(4 ⊗ δ) = (4 ⊗ δ)(α ⊗ 2) and (α ⊗ 2)(4 ⊗ γ) = (4 ⊗ γ)(α ⊗ 3). Moreover, the A′_{2} ⊗ A_{1}^{op}-module M is given by the simple module S_{4⊗3}. However, since (α⊗3)^{2} = 0, it follows from [28, (1.4)] that Ext_{A′_{2}⊗A_{1}^{op}}^{i}(M, M) ≠ 0 for i ≥ 1. Hence, the long exact sequence of (2.4.1) cannot efficiently be used to compute the Hochschild cohomology groups of R.

3. Simplicial (co)homology, fundamental group and simple connectedness

As mentionned in the introduction, close links exist between the (simplicial and Hochschild) (co)homology groups, the fundamental group and the simple connectedness of an algebra. In this section, we study the simplicial (co)homology, the fundamental group and the simple connectedness of a pullback of algebras. In particular, we prove the existence of a long exact sequence of simplicial (co)homology groups relating the algebras involved in a pullback of algebras.

3.1. Simplicial (co)homology. Given an algebra R = kQ/I and a pair (x, y) of vertices of Q, let _{y}𝔹_{x} be a basis of the k-space R(x, y) and 𝔹 = ⋃_{(x,y)∈Q_{0}×Q_{0}} _{y}𝔹_{x}. Then we say that 𝔹 is a semi-normed basis [31] if:

1. ε_{x} ∈ _{x}𝔹_{x} for every vertex x ∈ Q_{0};

2. α + I ∈ _{y}𝔹_{x} for every arrow α : x → y;

3. For σ and σ′ in 𝔹, either σσ′ = 0 or there exist λ_{σ,σ′} ∈ k, and b(σ, σ′) ∈ 𝔹 such that σσ′ = λ_{σ,σ′}b(σ, σ′).

To an algebra R = kQ/I which admits a semi-normed basis, we can associate the chain complex (SC_{•}(R), d) in the following way: SC_{0}(R) and SC_{1}(R) are the free abelian groups with basis Q_{0} and 𝔹, respectively. For n ≥ 2, let SC_{n}(R) be the free abelian group with basis the set of n-tuples (σ_{1}, σ_{2}, …, σ_{n}) of 𝔹^{n} such that σ_{1}σ_{2} ⋯ σ_{n} ≠ 0, and σ_{i} ≠ e_{j} for all i, j ∈ Q_{0}. The differential d_{n} is induced by the rules d_{1}(σ) = y − x for σ ∈ _{y}𝔹_{x}, and, for n ≥ 1:

d_{n}(σ_{1}, σ_{2}, …, σ_{n}) = (σ_{2}, …, σ_{n}) + ∑_{j=1}^{n−1}(−1)^{j}(σ_{1}, …, b(σ_{j}, σ_{j+1}), …, σ_{n})
+(−1)^{n}(σ_{1}, …, σ_{n−1})

We define the i^{th}-simplicial homology group of R (with respect to the basis 𝔹) as the i^{th}-homology group of the chain complex (SC_{•}(R), d), and we denote it by SH_{i}(R). Naturally, given an abelian group G, the ith-simplicial cohomology group SH^{i}(R, G) of R with coefficients in G is defined by applying Hom_{ℤ}(−, G) on the chain complex and by taking the ith-cohomology group of the complex obtained.

The reader can easily see that if R is a pullback of the k-algebras projections f_{1} : A_{1} → C and f_{2} : A_{2} → C, as defined is (1.2), with 𝔹_{1} and 𝔹_{2} the semi-normed basis of A_{1} and A_{2}, respectively, then 𝔹 = 𝔹_{1} ∪ 𝔹_{2} and 𝔹_{C} = 𝔹_{1} ∩ 𝔹_{2} are semi-normed basis of R and C respectively. This leads to the following results.

Proposition 3.1.1. Let R be the pullback of the k-algebra projections f_{1} : A_{1} → C and f_{2} : A_{2} → C as defined in (1.2). Moreover, assume that 𝔹_{1} and 𝔹_{2} are semi-normed basis of A_{1} and A_{2}. Then there exists a long exact sequence of simplicial homology groups:

\xymatrix{⋯ \ar[r] & SH_{n}(C) \ar[r] & SH_{n}(A_{1}) ⊕ SH_{n}(A_{2}) \ar[r] & SH_{n}(R) \ar[r] & SH_{n−1}(C) \ar[r] & ⋯}

Proof : We define

\begin{aligned}p_{i} : & SC_{i}(A_{1}) ⊕ SC_{i}(A_{2}) & ⟶ & SC_{i}(R) \\ & ((β_{1}, …, β_{i}), (γ_{1}, …, γ_{i})) & ⟼ & (β_{1}, …, β_{i}) − (γ_{1}, …, γ_{i}) \end{aligned}

The map p_{·} is a morphism of complexes. Moreover, let α = (α_{1}, …, α_{n}) be a generator of SC_{n}(R). Since e′_{1}Re′_{2} = 0 = e′_{2}Re′_{1} by asumption, it is easy to see that, for each j, α_{j} ∈ 𝔹_{1} or α_{j} ∈ 𝔹_{2}. Consequently, α ∈ SC_{i}(A_{1}) or α ∈ SC_{i}(A_{2}), and hence p_{i} is surjective for each i.

Moreover, the kernel of p_{i} is exactly SC_{i}(C). We obtain a short exact sequence of complexes 0 → SC_{•}(C) → SC_{•}(A_{1}) ⊕ SC_{•}(A_{2}) → SC_{•}(R) → 0, from which we deduce the result. □

Proposition 3.1.2. Let R be the pullback of the k-algebra projections f_{1} : A_{1} → C and f_{2} : A_{2} → C as defined in (1.2). Moreover, assume that 𝔹_{1} and 𝔹_{2} are semi-normed basis of A_{1} and A_{2}. If G is a divisible abelian group, then there exists a long exact sequence of simplicial cohomology groups :

\xymatrix{⋯ \ar[r] & SH^{n}(R, G) \ar[r] & SH^{n}(A_{1}, G) ⊕ SH^{n}(A_{2}, G) \ar[r] & SH^{n}(C, G) \ar[r] & SH^{n+1}(R, G) \ar[r] & ⋯}

Proof : By the proof of (3.1.1), we have an exact sequence of complexes (∗) : 0 → SC_{•}(C) → SC_{•}(A_{1}) ⊕ SC_{•}(A_{2}) → SC_{•}(R) → 0. Since G is divisible, the functor Hom_{ℤ}(−, G) is exact. Therefore the short sequence of complexes obtained by applying Hom_{ℤ}(−, G) to (∗) is exact and the result follows at once. □

3.2. Fundamental group. In this section, we study the fundamental group of a particular class of pullback of algebras, that of articulated algebras, see below for definition.

Let (Q, I) be a bound quiver. Recall from [32] that given two vertices x, y ∈ Q_{0}, a relation ρ = ∑_{i=1}^{n} λ_{i}ω_{i} ∈ I ∩ kQ(x, y) is called minimal provided m ≥ 2 and ∑_{i∈J} λ_{i}ω_{i} ∉ I ∩ kQ(x, y) for every proper subset J of {1, 2, …, n}. We define the homotopy relation in the set of walks on Q as the smallest equivalence relation satisfying:

1. For each arrow α ∈ Q_{1}, αα^{−1} ∼ ϵ_{x} and α^{−1}α ∼ ϵ_{y};

2. For each minimal relation ∑_{i=1}^{n} λ_{i}ω_{i}, ω_{i} ∼ ω_{j}.

3. If u, v, w, w′ are walks such that w ∼ w′, then uwv ∼ uw′v, whenever these compositions are defined.

Let π_{1}(Q, x_{0}) be the fundamental group of the underlying graph of Q at the vertex x_{0}, that is the free group on χ(Q) = |Q_{1}| − |Q_{0}| + 1 generators. Moreover, let N(Q, I, x_{0}) be the normal subgroup of π_{1}(Q, x_{0}) generated by the elements of the form wuv^{−1}w^{−1} where w is a path beginning at x_{0} and u and v are two homotopic paths. The fundamental group of the bound quiver (Q, I) is defined to be π_{1}(Q, I, x) = π_{1}(Q, x)/N(Q, I, x).

Since we assume Q to be connected, this definition does not depend on the choice of x_{0}. Accordingly, we write π_{1}(Q, I).

On other hand, following [12], to a bound quiver (Q, I), we associate a C.W. complex B(Q, I) (called its classifying space) in the following way: the 0-cells are given by Q_{0} and the n-cells are given by n-tuples (σ_{1}, …, σ_{n}) of homotopy classes of non trivial paths of (Q, I) such that the composition σ_{1}σ_{2} ⋯ σ_{n} is a path of (Q, I) not in I. Recall (see [12]) that the groups π_{1}(Q, I) and π_{1}(B(Q, I)) are isomorphic. We shall use this fact, together with Van Kampen’s theorem, in the proof of Proposition 3.2.3 below.

However, we first recall the following definition from [19].

Definition 3.2.1. Let (Q_{1}, I_{1}) and (Q_{2}, I_{2}) be two connected bound quivers containing at least one arrow. A bound quiver (Q, I) is called articulated along (Q_{1}, I_{1}) and (Q_{2}, I_{2}), or simply articulated, if

1. Q_{1} and Q_{2} are subquivers of Q_{R};

2. (Q_{R})_{0} = (Q_{1})_{0} ∪ (Q_{2})_{0} and (Q_{R})_{1} = (Q_{1})_{1} ∪ (Q_{2})_{1};
3. (Q_{1})_{0} ∩ (Q_{2})_{0} ≠ ∅ and for each x ∈ (Q_{1})_{0} ∩ (Q_{2})_{0}, x is either:

(a) a source of Q_{1} and a sink of Q_{2} or;

(b) a source of Q_{2} and a sink Q_{1};

4. I_{R} = I_{1}+I_{2}+ < \overline{ρ} >, with \overline{ρ} the set of paths linking (Q_{1})_{0}\(Q_{2})_{0} and (Q_{2})_{0}\(Q_{1})_{0}

in Q_{R}. Under these conditions, the algebra R = kQ/I is called articulated along A_{1} = kQ_{1}/I_{1} and A_{2} = kQ_{2}/I_{2}.

In this case R is the pullback of the projections of A_{1} and A_{2} to the non-empty semisimple intersection C = A_{1} ∩ A_{2}. Moreover, any minimal relation in I_{R} is either a minimal relation in I_{1} or in I_{2} and conversely.

Example 3.2.2.

1. Let R, A_{1}, A_{2} and C be the algebras of Example 2.2.6. Then R is articulated along A_{1} and A_{2}, with C = A_{1} ∩ A_{2}.

2. Let R be the algebra of Example 2.4.2(1). Then R is an articulated algebra along the algebras B_{1} and B_{2}, where B_{1} and B_{2} are given by the full subcategories generated by the set of vertices {1, 2, 3, 4} and {2, 3, 4, 5, 6} respectively.

3. Let R, A_{1}, A_{2} and C be the algebras of Example 2.4.2(2). Then R is articulated along A_{1} and A_{2}, with C = A_{1} ∩ A_{2}.

Proposition 3.2.3. Let R = kQ_{R}/I_{R} be an articulated algebra along A_{1} = kQ_{1}/I_{1} and A_{2} = kQ_{2}/I_{2} such that |(Q_{1})_{0} ∩ (Q_{2})_{0}| = m. Then π_{1}(Q_{R}, I_{R}) ≅ π_{1}(Q_{1}, I_{1}) ∗ π_{1}(Q_{2}, I_{2}) ∗ L_{m−1}, where L_{m−1} is the free group generated in m − 1 generators.

Proof : Let Q_{C} = Q_{1} ∩ Q_{2}, and n_{1}, n_{2}, n_{R} and n denote the number of vertices of Q_{1}, Q_{2}, Q_{R} and Q_{C}, respectively. In particular, we have n_{R} = n_{1} + n_{2} − n. For i ∈ {1, 2}, let T_{i} be a maximal tree in Q_{i} and set

Q̂_{i} = Q_{i} ∐_{Q_{C}} T_{i} and Î_{i} = I_{R} ∩ kQ̂_{i}.

Note that Î_{i} = I_{i} + X_{i}, where X_{i} are monomial ideals. It is easily seen that |(Q̂_{i})_{0}| = n_{R}, for i ∈ {1, 2}. In addition we have:

|(Q̂_{1})_{1}| = |(Q_{1})_{1}| + |(T_{2})_{1}| = |(Q_{1})_{1}| + n_{2} − 1.

and a similar expression for |(Q̂_{2})_{1}|.

But π_{1}(Q̂_{1}) is the free group in χ(Q̂_{1}) generators, and χ(Q̂_{1}) = χ(Q_{1}) + (n − 1). Thus, π_{1}(Q̂_{1}) ≅ π_{1}(Q_{1}) ∗ L_{n−1}. It follows from the construction of Î_{i} that the homotopy relation on (Q̂_{i}, Î_{i}) coincides with that of (Q_{i}, I_{i}) so that :

π_{1}(Q̂_{i}, Î_{i}) ≅ π_{1}(Q̂_{i})/N(Q̂_{i}, Î_{i})
≅ (π_{1}(Q_{i}) ∗ L_{n−1})/N(Q̂_{i}, Î_{i})
≅ π_{1}(Q_{i})/N(Q_{i}, I_{i}) ∗ L_{n−1}
≅ π_{1}(Q_{i}, I_{i}) ∗ L_{n−1}.

Now set Q = Q̂_{1} ∩ Q̂_{2}, or, equivalently Q = T_{1} ⨿_{Q_{C}} T_{2} and I = kQ ∩ I_{R}, which is a monomial ideal, so that there are no minimal relations in I. Consequently, the fundamental group π_{1}(Q, I) is the free group on χ(Q) = n − 1 generators, and therefore π_{1}(Q, I) ≅ L_{n−1}.

Keeping in mind the expressions of π_{1}(Q̂_{i}, Î_{i}) and π_{1}(Q, I) given above, one can see that the morphisms q_{i} : π_{1}(Q, I) → π_{1}(Q̂_{i}, Î_{i}) given by the rules q_{i}(x) = 1∗x are injective, having L_{n−1} as image.

The next step is to use Van Kampen’s theorem. For this sake, let B̂_{1}, B̂_{2}, B̂_{R} and B̂ be the classifying spaces of (Q̂_{1}, Î_{1}), (Q̂_{2}, Î_{2}), (Q̂_{R}, Î_{R}), (Q̂_{C}, Î_{C}), respectively.

By construction, B_{R} = B̂_{1} ∪ B̂_{2} and the intersection B̂_{1} ∩ B̂_{2} = B is connected. The result then follows from Van Kampen’s theorem and the isomorphism π_{1}(B_{R}) ≅ π_{1}(Q_{R}, I_{R}). □

3.3. Simple connectedness. The result of the preceding section give rise to immediate consequences on simple connectedness and strong simple connectedness of articulated algebras. For further details about (strong) simply connected algebras, we refer the reader to [7, 5, 2, 1]. We first recall the relevant definitions.

Definition 3.3.1. A connected and triangular algebra A is called simply connected if, for any presentation (Q, I) of A, the group π_{1}(Q, I) is trivial. Moreover, it is called strongly simply connected if every full convex subcategory of A is simply connected.

In general, the above condition may be complicated to verify since, even if an algebra A determines a unique ordinary quiver Q, the fundamental groups of two presentations (Q, I_{1}) and (Q, I_{2}) of an algebra A can be completely different (see [13, 6], for instance).

Lemma 3.3.2. Let R = kQ/I be an articulated algebra along A_{1} = kQ_{1}/I_{1} and A_{2} = kQ_{2}/I_{2} such that |(Q_{1})_{0} ∩ (Q_{2})_{0}| = 1. Then A_{1} and A_{2} are full and convex subcategories of R.

Proof : We only proof the statement for A_{1}, the proof for A_{2} being similar. Since I_{1} is the restriction of I to Q_{1}, if A_{1} is not a full and convex subcategory of R, then there

exists a path x_{1} →^{f_{1}} x_{2} →^{f_{2}} ⋯ →^{f_{n−1}} x_{n} in Q with x_{1}, x_{n} ∈ (Q_{1})_{0} but f_{i} ∉ (Q_{1})_{1} for a

certain i ∈ {1, …, n}. Consequently, f_{i} ∈ (Q_{2})_{1} and since R is articulated there exist x_{j}, x_{k} ∈ (Q_{1})_{0} ∩ (Q_{2})_{0}, a contradiction. □

Corollary 3.3.3. Let R be a connected articulated algebra along A_{1} and A_{2}. Let C = A_{1} ∩ A_{2}. Then

1. R is simply connected if and only if A_{1} and A_{2} are simply connected and |C_{0}| = 1;

2. R is strongly simply connected if and only if A_{1} and A_{2} are strongly simply connected and |C_{0}| = 1.

Proof : 1. Necessity. First, if |C_{0}| > 1, then it follows from (3.2.3) that π_{1}(Q_{R}, I_{R}) is not trivial, a contradiction, and hence |C_{0}| = 1 since R is connected. On the other hand, let (Q_{1}, I_{1}) and (Q_{2}, I_{2}) be presentations of A_{1} and A_{2} respectively. By [19], there exists a presentation (Q, I) of R such that (Q, I) is articulated along (Q_{1}, I_{1}) and (Q_{2}, I_{2}). It then follows from (3.2.3) that the groups π(Q_{1}, I_{1}) and π(Q_{2}, I_{2}) are trivial, and consequently A_{1} and A_{2} are simply connected. Sufficiency. Let (Q, I) be a presentation of R. By [19], (Q, I) is articulated along presentations (Q_{1}, I_{1}) and (Q_{2}, I_{2}) of A_{1} and A_{2} respectively. Since A_{1} and A_{2} are simply connected, π_{1}(Q, I) is trivial by (3.2.3) and R is simply connected.

2. Necessity. Assume that R is strongly simply connected. In particular, R is simply connected and it follows from the first part that so are A_{1} and A_{2}. Let i ∈ {1, 2} and B be a full convex subcategory of A_{i}. By (3.3.2), A_{i} is a full convex subcategory of R, and hence so is B. Since R is strongly simply connected, B is simply connected. Therefore, A_{i} is strongly simply connected. Sufficiency. Let R′ be a full convex subcategory of R. If R′ is subcategory of A_{1} or A_{2}, then R′ is simply connected and this finish the proof. Otherwise, since R′ is convex, one has R′_{0} ∩ C_{0} ≠ ∅, and hence |R′_{0} ∩ C_{0}| = 1. For i = 1, 2, let A′_{i} be the restriction of R′ to A_{i}. In particular, A′_{i} is a full convex subcategory of A_{i} and, by construction, R′ is articulated along A′_{1} and A′_{2}. Since A′_{1} and A′_{2} are both simply connected, and |A′_{1} ∩ A′_{2}| = 1, it follows from the first part that R′ is simply connected. Consequently, R is strongly simply connected. □

ACKNOWLEDGEMENTS. This paper was written while the first author visited the Universities of Sherbrooke and Bishop’s under a local post-doctoral fellowship, and while the second and the third author respectively had an ISM and NSERC graduate studies fellowship. The authors warmly thank Ibrahim Assem for his precious help, and Guillermo Cortiñas for pertinent questions during the XVI^{to} Coloquio Latinoamericano de Álgebra in Uruguay.

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J. C. Bustamante; Departamento de Matemáticas, Universidad San Francisco de Quito, Quito, Ecuador. E-mail address: juan.bustamante@usfq.edu.ec

J. Dionne; Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada. E-mail address: julie.dionne2@usherbrooke.ca

D. Smith; Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada. E-mail address: david.smith@usherbrooke.ca