Representation-theoretic properties of balanced big Cohen-Macaulay modules

TL;DR

This paper introduces a numerical invariant called $ngth$ for balanced big Cohen-Macaulay modules over complete Cohen-Macaulay local rings, establishing new results on module decomposability, the Brauer-Thrall conjecture, and Cohen-Macaulay algebra classifications using Gabriel-Roiter measures.

Contribution

It defines the $ngth$ invariant and proves its implications for module decomposability, the Brauer-Thrall conjecture, and Cohen-Macaulay algebra types, extending known results and applying Gabriel-Roiter measures.

Findings

Bounded $ngth$ implies full decomposability of modules.

The Brauer-Thrall conjecture holds for complete Cohen-Macaulay local rings.

Characterization of finite $ ext{CM}$-type via closure properties of decomposable modules.

Abstract

Let $(R, \m, k)$ be a complete Cohen-Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen-Macaulay module, called $\uh$-length. Among other results, it is proved that, for a given balanced big Cohen-Macaulay $R$-module $M$ with an $\m$-primary cohomological annihilator, if there is a bound on the $\uh$-length of all modules appearing in $\CM$-support of $M$, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer-Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen-Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen-Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen-Macaulay modules is settled. Namely, it is shown that $R$ is of finite $\CM$-type if and only if the category of all fully decomposable balanced big Cohen-Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen-Macaulay artin algebras admitting a dualizing bimodule $\omega$, as defined by Auslander and Reiten. It will turn out that, $\omega$-Gorenstein projective modules with bounded $\CM$-support are fully decomposable. In particular, a Cohen-Macaulay algebra $\Lambda$ is of finite $\CM$-type if and only if every $\omega$-Gorenstein projective module is of finite $\CM$-type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel-Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel-Roiter (co)measure in the category of maximal Cohen-Macaulay modules.