= Vertex projective module of a path algebra
{title2=$P(i)=kQe_i$}
The <vertex projective module of a path algebra> $P(i)=kQe_i$ 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 $(e_ikQe_i)^{\mathrm{op}}$, 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.
Back to article page
