I am providing the context. You can skip to the actual doubt below. My doubt doesnt need knowledge of quivers or even almost split sequences.
I am reading Theorem 5.4.10 from the book An introduction to Quiver Representations by Derksen.
The claim is the existence of almost split sequence 'ending' at $V$ for any non-projective $A$ module where $A$ is a finite dimensional $\mathbb{C}$ algebra and $V$ is a finite module over $A$.
Formally:
Theorem 5.4.10
- If $N$ is an indecomposable finite dimensional $A$-module which is not projective, then there exists an almost split sequence
$0 \to L \xrightarrow{f} M \xrightarrow{g} N \to 0$
The Proof begins with : 'Let M_1, \cdots M_r' be all the indecomposable modules for which there exists an irreducible morphism $M_i \to N$.
My doubt is how can we assume that there are only finitely many such modules upto isomorphism? There is no mention of $A$ being of finite type and I know for a fact that such sequences exist for any finite type algebra $A$.
Please help me.