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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.

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  • $\begingroup$ 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 $P$) ? $\endgroup$
    – hennlu
    Nov 18, 2019 at 16:21
  • $\begingroup$ In fact, my first question is an easy consequence of Auslander-Reiten theory. $\endgroup$
    – hennlu
    Nov 18, 2019 at 22:39

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