Short chains and short cycles of modules

Reiten, I.; Skowroński, A.; Smalø, S. O.

doi:10.1090/S0002-9939-1993-1136238-4

Short chains and short cycles of modules
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by I. Reiten, A. Skowroński and S. O. Smalø

Proc. Amer. Math. Soc. 117 (1993), 343-354

DOI: https://doi.org/10.1090/S0002-9939-1993-1136238-4

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Abstract:

We show that for a large class of artin algebras including the algebras of finite representation type, an indecomposable module $M$ is not the middle of a short chain if and only if there is no short cycle $M \to N \to M$ of nonzero nonisomorphisms between indecomposable modules. We apply this to get sufficient conditions for modules to be determined by their composition factors. We also show that if for an algebra of finite representation type there is a sincere indecomposable $\Lambda$-module that is not the middle of a short chain, then $\Lambda$ is a tilted algebra.

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