A NOTE ON THE FUNDAMENTAL GROUP OF A ONE-POINT EXTENSION

In this note, we consider an algebra A which is a one-point extension of another algebra B and we study the morphism of fundamental groups induced by the inclusion of (the bound quiver of) B into (that of) A . Our main result says that the cokernel of this morphism is a free group and we prove some consequences from this fact.


Introduction
In the representation theory of finite dimensional algebras over algebraically closed fields, the use of covering techniques, initiated by Gabriel, Riedtmann and others (see, for instance [3]) has shown the importance of the fundamental groups of bound quivers (see [7] for the definition). In this note, we consider an algebra A which is a one-point extension of another algebra B and the morphism of fundamental groups induced by the inclusion of (the bound quiver of) B into (that of) A. Our main result says that the cokernel of this morphism is a free group. We then deduce various consequences from this fact.

Algebras and quivers.
By an algebra is always meant a basic and connected finite dimensional algebra over an algebraically closed field k, and by module is meant a finitely generated right module. For every algebra A there exists a (unique) quiver Q A and (at least) a surjective morphism of algebras ν : kQ A → A so that, setting I ν = Kerν, we have A ' kQ A /I ν (see [3]). The bound quiver (Q A , I ν ) is then called a presentation of A. An algebra A is said to be triangular whenever Q A has no oriented cycles. If x is a point in Q A , we denote by e x the corresponding idempotent and by P x = e x A the corresponding indecomposable projective module.

Fundamental groups.
Let (Q, I) be a connected bound quiver. For the definitions and properties of the fundamental group π 1 (Q, I), we refer to [7,2]. A triangular algebra is called simply connected if, for every presentation (Q, I) of A, we have π 1 (Q, I) = 1. On the other hand, to a given connected bound quiver (Q, I), one can associate a CW complex B = B(Q, I) called the classifying space of (Q, I), and this construction behaves well with respect to homotopy, in particular one has π 1 (Q, I) ' π 1 (B) (see [6]). The fundamental group of a bound quiver (Q, I) affords the following description: let T be a maximal tree in Q, F be the free group with basis the set of all arrows of Q, and K be the normal subgroup of F generated by 1. all the arrows in T ; and 2. all expressions of the form (β 1 β 2 · · · β p )(γ 1 γ 2 · · · γ q ) −1 where β 1 β 2 · · · β p and γ 1 γ 2 · · · γ q are two paths appearing in a minimal relation (in the sense of [7]).

One-point extensions.
Let A be an algebra, x be a source in Q A , and B = A/Ae x A. Letting M = rad x , the algebra A can be written in matrix form Our aim is to compute the cokernel of the induced morphism Following [2] (2.1), we denote by x → the set of all the arrows of Q A starting in x. Let ≈ be the least equivalence on x → such that α ≈ β whenever there exists a minimal relation We denote by [α] ν A the equivalence class of α, and by t(ν A ) the number of equivalence classes [α] ν A in x → . Finally, we denote by s = s(x) the number of indecomposable direct summands of M . It is shown in [

Theorem
Let A = B [M ] be given an arbitrary presentation (Q A , I vA ). Then the cokernel of the morphism η :

Proof :
Clearly, if δ 1 is an arrow from x to a point in Q (1) , and δ 2 is an arrow from x to a point in Q (2) , then δ 1 6 ≈ δ 2 . We may then assume, without loss of generality that B is connected, that is, c = 1. We set t = t(ν A ), and let {α 1 , α 2 , . . . α t } be a complete set of representatives of the classes [α] ν . For i ∈ {1, 2, . . . t}, we set [α i ] ν = a i . Our aim is to show that the cokernel of η is the free group L t−1 having {a 2 , . . . a t } as set of (free) generators.
Clearly, we have ϕη = 1. Now, assume there exist a group morphism ϕ 0 : F A /K A → G such that ϕ 0 η = 1. We define ϕ : L t−1 → G by a i 7 → ϕ 0 (αK A ), where [α] v = a i . We must verify that ϕ 0 (αK A ) does not depend on the choice of the representative α. Indeed, if [β] v = [γ] v , we may assume without loss of generality that there exist two paths ββ 2 · · · β p , and γγ 2 · · · γ q in Q A appearing in the same minimal relation. But then Thus, ϕ is well-defined. Clearly ϕϕ = ϕ 0 , and ϕ is uniquely determined (because ϕ is an epimorphism). This completes the proof. A = B [M ] be a given by an arbitrary prsentation (Q A , I vA ), and Z be any abelian group. There exists an exact sequence of abelian groups such that the cokernel of the morphism η :

Corollary [2] (2.3). Let
is the free group L s−c in s − c generators.

Corollary [2] (2.6).
Let A be simply connected. Then all sources of Q A are separating (see for instance [2]).

Proof :
Since A is simply connected, it follows from the previous result that s = c.

Proof:
If this is not the case, then L s−c 6 = 1 and, since the short exact sequence 1 → Kerϕ → π 1 (Q A , I µA ) ϕ → L s−c → 1 splits (because L s−c is a free group), we get π 1 (Q A , I µA ) ' L s−c 1 Kerϕ, which is infinite.  Note that the sufficiency of b) was proven in [2] (2.5), [8] (2.3). We now suppose A to be triangular and schurian (that is, for every two vertices x, y of Q A , we have dim k e x Ae y ≤ 1). In this case, the fundamental group does not depend on the presentation (see [4]), so it can unambiguously be denoted by π 1 (A). Moreover, the classifying space B of (Q A , I ν A ) is a simplicial complex [5]. We denote by SH 1 (A) the first simplicial homology group of A, and recall that SH 1 (A) is the abelianisation of π 1 (A). Hom Z (π 1 (B j ) , Z)

Corollary. Let
By [1] (3.8), the absence of quasi-crowns forces Hom Z (η, Z) to be an epimorphism. Since the above sequence may be rewritten as 0 → Hom Z (π 1 (A) , Z) → Hom Z (SH 1 (A) , Z) where η * is induced from η by abelianising, replacing Z by the injective cogenerator Q/Z yields the result.