Vertex projective module of a path algebra ($P(i)=kQ{e}_i$)

The vertex projective module of a path algebra $P(i)=kQ{e}_i$ has at vertex $j$ the span of paths from $i$ to $j$. Each arrow appends itself to such paths. It is a projective module because it is a summand of the free module $kQ$. Its endomorphism ring is $(e_{i}kQ{e}_i{)}^{\mathrm{op}}$, acting by right multiplication. The closed-path corner of a path algebra is a domain, so this ring has no nontrivial idempotents and $P(i)$ is indecomposable. Directed cycles may make these projectives infinite-dimensional.

Evaluation at $e_i$ identifies a module homomorphism $h:kQ{e}_i\to X$ with $h(e_i)\in e_{i}X$. Given $x\in e_{i}X$, the inverse sends $p$ to $px$. The isomorphism is natural and shows that Hom functor evaluation on a vertex projective is exact.