Let $A$ be a finite dimensional algebra over a field $k$, and let $\tau$ be the Auslander-Reiten translation functor vanishing on projective $A$-modules. I'm studing the book "An introduction to Quiver Representations" of Derksen and Weyman and I don't understand the following fact:
Suppose that A is of inite representation type, and $N$ is a non-projective indecomposable A-module, then there exists a positive $n$ such that $\tau^n(N)=0$.
Can anyone help me to understand this fact please? I'm also curious about the next 2 questions that naturally arises in trying to understand the above sentence
- Is the above sentence still true without the assumption that $A$ is of finite representation type?
- Is it true (always without the assumption that $A$ is of finite representation type) that if $M$ if an $A$-module (not necessarily indecomposable) such that $\tau^m(M)=M$ for $m>0$ then $M=0$.
Thank's for your help.
The statement appears to be false: take the algebra $A = k[x]/(x^2)$. Then there are precisely two indecomposable $A$-modules, up to isomorphism: $A$ itself, which is projective and injective, and the simple module $S = Ax$, which is neither. Then $\tau S = S$, so $\tau^n S \neq 0$ for all positive $n$.