Extension complex of quiver representations
= Extension complex of quiver representations
{title2=$C^0\xrightarrow{\delta}C^1$}
Put $C^0=\bigoplus_i\operatorname{Hom}(X_i,Y_i)$ and $C^1=\bigoplus_{\rho:i\to j}\operatorname{Hom}(X_i,Y_j)$. The differential has components $g_\rho h_i-h_jf_\rho$. Its kernel is $\operatorname{Hom}_Q(X,Y)$ and its cokernel is $\operatorname{Ext}^1_Q(X,Y)$. In a vertexwise splitting of an extension, the off-diagonal arrow blocks change by these coboundaries when the splitting changes.