Kernel-cokernel obstruction to splitting a quiver extension
= Kernel-cokernel obstruction to splitting a quiver extension
Restriction at one arrow induces a surjection $\operatorname{Ext}^1_Q(X,Y)\twoheadrightarrow\operatorname{Hom}_k(\ker f_\rho,\operatorname{coker}g_\rho)$. Coboundaries vanish on the kernel after quotienting by the image, while every map between these spaces can be lifted and extended. Nonzero kernel and cokernel therefore force a nonsplit extension. For $X=Y$, rigidity forces every arrow to have maximal rank.