Path algebras are hereditary (${\mathrm{Ext}}_{kQ}^n(V,W)=0(n\ge 2)$)

For a finite quiver $Q$ over a field $k$, the standard projective resolution of a quiver representation has length one, even for quivers with oriented cycles. Thus each such path algebra is a hereditary ring, and higher extension groups vanish. Applying a long exact sequence of Ext groups to $0\to I\to K\to C\to 0$ gives a surjection ${\mathrm{Ext}}^1(K,W)\twoheadrightarrow {\mathrm{Ext}}^1(I,W)$, since the next term ${\mathrm{Ext}}^2(C,W)$ is zero. This is the hereditary step in the Ringel lemma on bricks.