Citable reference for structure of extension quiver of path algebra

Question

Let $Q$ be a finite quiver such that $A = kQ$ is a finite dimensional path algebra, where $k$ is a field. Then let $1, \dots n$ be the vertices of $Q$ and $S_1, \dots, S_n$ the corresponding simple modules. I learned in a lecture that

dim$_k$Ext$^{1}_{A}(S_{i}, S_{j}) = |\{\alpha: i \to j\}|$

is just the number of arrows from $i$ to $j$ in $Q$. Do you know a good reference for this?

Answer 1

Let $\left\langle -,- \right\rangle : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}^n$ be the homological bilinear form, given on dimension vectors of modules over a hereditary finite-dimensional algebra $A$ by

$$\left\langle \underline{\dim} X, \underline{\dim}Y\right\rangle = \dim_k \text{Hom}_A(X, Y) - \dim_k\text{Ext}^1_A (X, Y).$$

Then if $A = kQ$ is the finite-dimensional path algebra of a finite connected quiver $Q$ we get the following description of the homological bilinear form according to chapter 7.2 in Representations of finite-dimensional algebras by Gabriel and Rolter:

$$\left\langle x, y\right\rangle = \sum_{i \in Q_0} x_i y_i - \sum_{\alpha \in Q_1} x_{s(\alpha)}y_{t(\alpha)},$$

where $s(\alpha)$ is the starting vertex of $\alpha$ and $t(\alpha)$ is the terminal vertex. Accordingly we get

\begin{align*}\delta_{ij} - \dim_k \text{Ext}^1_A(S_i, S_j) & = \dim_k \text{Hom}_A(S_i, S_j) - \dim_k \text{Ext}^1_A(S_i, S_j) \\ & = \left\langle \underline{\dim}S_i, \underline{\dim}S_j \right\rangle \\ & = \left\langle e(i), e(j) \right\rangle \\ & = \sum_{k \in Q_0} e(i)_k e(j)_k - \sum_{\alpha \in Q_1} e(i)_{s(\alpha)}e(j)_{t(\alpha)} \\ & = \delta_{ij} - | \{\alpha : i \to j\}|, \end{align*}

where $e(i), e(j)$ are the standard basis vectors in $\mathbb{Z}^n$. The result follows.

Answer 2

There are many references but here is one:

Representations of quivers by Michel Brion (page 21).

See the statement immediately after Corollary 1.4.3.