Exercise 9, § 1

a) Let M and N be $A_Q$-modules; for $s\in$ Som(Q), we put $M_s=$ $Mx_s$ and $N_s= Nx_s$. For $\varphi = (\varphi_s)_{s\in Som(Q)}$ and $f\in$ Fl(Q), define a mapping $\theta_f(\varphi ): Mx_{o(f)}\rightarrow Nx_{t(f)}$ by putting

Prove that the mapping

$\theta :\prod$ Hom$_k(M_s,N_s)\rightarrow \prod$ Hom(M$_{o(f)},N_{t(f)}$)

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

given by $\varphi \mapsto (\theta_f(\varphi ))_{f\in Fl(Q)}$ is linear.

b) Prove that Ker($\theta$ ) is isomorphic to Hom$_{A_Q}(M,N)$ and that Coker($\theta$ ) is isomorphic to Ext$^1_{A_Q}(M,N)$. Prove that Ext$^p_{A_Q}(M,N) = 0$ for every integer $p\geqslant 2$.

c) Suppose that $k$ is a commutative field and that the quiver Q is finite If M and N are finite-dimensional $k$-vector spaces, then Ext$^p_{A_Q}(M,N)$ is a finite-dimensional $k$-vector space for every integer $p\geqslant 0$, zero for $p\geqslant 2$, and

dim$_k$(Hom$_{A_Q}(M,N)$)$-$ dim$_k$(Ext$^1_{A_Q}(M,N)$)

= $\sum$ dim(M$_s$) dim(N$_s$)$-\sum$ dim(M$_{o(f)}$) dim(N$_{t(f)}$).

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

d) Suppose that $k$ is a commutative field. Prove that for every pair $(u, v)$ of vertices of Q, the dimension of the vector space Ext$^1_{A_Q}(k(u), k(v))$ is equal to the cardinal of the set of arrows of Q with origin $v$ and term $u$.