Module categories without short cycles are of finite type

Happel, Dieter; Liu, Shi Ping

doi:10.1090/S0002-9939-1994-1164144-9

Module categories without short cycles are of finite type
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by Dieter Happel and Shi Ping Liu PDF

Proc. Amer. Math. Soc. 120 (1994), 371-375 Request permission

Abstract:

Let $A$ be an artin algebra. An indecomposable finitely generated $A$-module $X$ is said to be on a short cycle if there exists an indecomposable finitely generated $A$-module $Y$ and two nonzero noninvertible maps $f:X \to Y$ and $g:Y \to X$. If there are no short cycles we show that there exist only finitely many indecomposable $A$-modules up to isomorphism.

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