Kernel-cokernel obstruction to splitting a quiver extension

Restriction at one arrow induces a surjection ${\mathrm{Ext}}_Q^1(X,Y)\twoheadrightarrow {\mathrm{Hom}}_k(\ker f_{\rho },\mathrm{coker}{g}_{\rho })$. 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.