Exercise 8, § 1

For $s\in$ Som(Q), put $P(s) =x_sA_Q$. Let M be an $A_Q$-module.

a) Let $f\in$ Fl(Q), denote $u=o(f)$ and $v=t(f)$. Prove that there exists a unique pair $(\varphi '_f, \varphi ''_f)$ of homomorphisms of $A_Q$-modules

such that $\varphi '_f(m\otimes p) =m\otimes x_fp$ and $\varphi ''_f(m\otimes p) =mx_f\otimes p$ for all $m\in Mx_u$ and all $p\in P(v)$.

b) Let $s\in$ Som(Q). Prove that there exists a unique homomorphism of $A_Q$-modules $\psi_s: Mx_s\otimes_kP(s)\rightarrow M$ such that $\psi_s(m\otimes p) =mp$ for $p\in P(s)$ and $m\in Mx_s$. c) Put $\varphi =\bigoplus\varphi '_f-\bigoplus\varphi ''_f$ and $\psi =\sum\psi_s$. Prove that one has an exact sequence of $A_Q$-modules

$f\in$Fl(Q) $s\in$Som(Q)

d) Prove that every right ideal of the algebra $A_Q$ is projective.