# Injective morphisms between preprojective representations

Let $$Q$$ be an acyclic quiver. Is it true that if $$P$$ is a preprojective representation of $$Q$$ and $$r\geq 0$$, there exists $$s\geq r$$ and a preprojective $$P'$$ with an injective morphism $$P\rightarrow \tau^{-s}P'$$ where $$\tau^-,\tau$$ are the Auslander-Reiten translates?

In wild type, it follows frop Proposition 5.3 in this book, and in type $$A_n^{(1)}$$, it follows from this paper, Corollary 2.2.

I am not able to prove it for other affine quivers.

• Maybe a related question is the following one (it implies the previous one): Let $P$ be an indecomposable preprojective. Is there an injective morphism to some other indecomposable preprojective $P'$ ($P'$ not isomorphic to $P$) ? Nov 18, 2019 at 16:21
• In fact, my first question is an easy consequence of Auslander-Reiten theory. Nov 18, 2019 at 22:39