= Path algebras are hereditary
{title2=$\operatorname{Ext}_{kQ}^{n}(V,W)=0\quad(n\ge2)$}
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 $0\to I\to K\to C\to0$ gives a <surjection> $\operatorname{Ext}^1(K,W)\twoheadrightarrow\operatorname{Ext}^1(I,W)$, since the next term $\operatorname{Ext}^2(C,W)$ is zero. This is the hereditary step in the <Ringel lemma on bricks>.
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