Finite cycles of indecomposable modules

TL;DR

This paper characterizes the structure of finite cycles in module categories over artin algebras, revealing their properties and implications for module theory and homological algebra.

Contribution

It provides a complete description of finite cycles in mod A and explores their structural and combinatorial properties, addressing a long-standing open problem.

Findings

Finite cycles in mod A are fully characterized.

Most modules on finite cycles have nonnegative Euler characteristic.

Cycle-finite module categories with finitely many torsion classes are classified.

Abstract

We solve a long standing open problem concerning the structure of finite cycles in the category mod A of finitely generated modules over an arbitrary artin algebra A, that is, the chains of homomorphisms $M_0 \stackrel{f_1}{\rightarrow} M_1 \to \cdots \to M_{r-1} \stackrel{f_r}{\rightarrow} M_r=M_0$ between indecomposable modules in mod A which do not belong to the infinite radical of mod A. In particular, we describe completely the structure of an arbitrary module category mod 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 mod A the Euler characteristic of M is well defined and nonnegative. As an another application of these results we obtain a characterization of all cycle-finite module categories mod A having only a finite number of functorially finite torsion classes. Moreover, new types of examples illustrating the main results of the paper are presented.