Exercise 7, § 1

Suppose that $k$ is an algebraically closed field. For every $s\in$ Som(Q), we put $P(s) =x_sA_Q$.

a) Prove that $P(s)$ is a $A_Q$-module projective and indecomposable, and that its socle is isomorphic to the $A_Q$-module $k(s)$. Also prove that $P(s)/P(s)J_Q$ is isomorphic to $k(s)$.

b) Suppose that every loop of Q is of length zero. Prove that, for every $s\in$ Som(Q), the canonical homomorphism from $k$ into End(P($s$)) is an isomorphism. Prove that, for every $A_Q$-module P which is projective and indecomposable, there exists a unique vertex $s$ of Q such that P is isomorphic to $P(s)$.