Let $A$ be a quiver algebra over a field $K$ (maybe we need algebraically closed?). Then the following is two statements are well known:

  1. In case $A$ is representation-infinite, every Auslander-Reiten component contains indecomposable modules of arbitrary large dimension.

  2. In case there is an indecomposable modules of dimension larger than $\max(2 \dim(A),30)$, $A$ is representation-infinite.

Question: Can those two statements be generalised from the category $\text{mod-}A$ to certain subcategories of $\text{mod-}A$ that have Auslander-Reiten sequences, maybe when replaying the bound $\max(2 \dim(A),30)$ by some other finite bound? I'm especially interested in the full subcategory of Gorenstein projective $A$-modules in case $A$ is Gorenstein or some other functiorially finite resolving subcategories.

  • 1
    $\begingroup$ An analogue of the first statement is contained in the PhD thesis of Matthias Krebs entitled "Auslander-Reiten theory in functorially finite resolving subcategories", Theorem 2.2.9 with no explicit bound. $\endgroup$ – Julian Kuelshammer Jan 26 '18 at 7:57

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