Ext quiver ($a_{ij}={\dim }_k{\mathrm{Ext}}_A^1(S_i,S_j)$)

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 $P_i=A{e}_i$, applying ${\mathrm{Hom}}_A(-,S_j)$ 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.