Finite cycles of indecomposable modules

Abstract

We solve a long standing open problem concerning the structure of finite cycles in the category $\mo A$ of finitely generated modules over an arbitrary artin algebra $A$, that is, the chains of homomorphisms $M_0 \buildrel {f_1}\over {\hbox to 6mm{\rightarrowfill}} M_1 \to \cdots \to M_{r-1} \buildrel {f_r}\over {\hbox to 6mm{\rightarrowfill}} M_r=M_0$ between indecomposable modules in $\mo A$ which do not belong to the infinite radical of $\mo A$. In particular, we describe completely the structure of an arbitrary module category $\mo A$ whose all cycles are finite. The main structural results of the paper allow to derive several interesting combinatorial and homological properties of indecomposable modules lying on finite cycles. For example, we prove that for all but finitely many isomorphism classes of indecomposable modules $M$ lying on finite cycles of a module category $\mo A$ the Euler characteristic of $M$ is well defined and nonnegative. Moreover, new types of examples illustrating the main results of the paper are presented.