Auslander-Reiten components containing modules with bounded Betti numbers

Green, Edward; Zacharia, Dan

doi:10.1090/S0002-9947-09-04782-5

Auslander-Reiten components containing modules with bounded Betti numbers
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by Edward L. Green and Dan Zacharia PDF

Trans. Amer. Math. Soc. 361 (2009), 4195-4214 Request permission

Abstract:

Let $R$ be a connected selfinjective Artin algebra, and $M$ an indecomposable nonprojective $R$-module with bounded Betti numbers lying in a regular component of the Auslander-Reiten quiver of $R$. We prove that the Auslander-Reiten sequence ending at $M$ has at most two indecomposable summands in the middle term. Furthermore we show that the component of the Auslander-Reiten quiver containing $M$ is either a stable tube or of type $\mathbb ZA_{\infty }$. We use these results to study modules with eventually constant Betti numbers, and modules with eventually periodic Betti numbers.

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