= Ext quiver
{c}
{title2=$a_{ij}=\dim_k\operatorname{Ext}^1_A(S_i,S_j)$}
For a finite-dimensional <associative algebra> over an <algebraically closed field>, the <Ext quiver> has a vertex for each <simple module> isomorphism class and $a_{ij}$ arrows $i\to j$. For a <basic algebra> with $P_i=Ae_i$, applying $\operatorname{Hom}_A(-,S_j)$ to the projective-cover sequence gives $a_{ij}=\dim e_j(J/J^2)e_i$. Our <path algebra> convention multiplies consecutive arrows $\alpha:i\to j$, $\beta:j\to l$ as $\beta\alpha$, making left modules have arrow maps $V_i\to V_j$. The orientation convention must be stated when comparing other definitions.
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