Representation-theoretic properties of balanced big Cohen–Macaulay modules

Abstract

Let $$(R, {\mathfrak {m}}, k)$$ be a complete Cohen–Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen–Macaulay module, called $$\underline{\mathsf {h}}$$ -length. Among other results, it is proved that, for a given balanced big Cohen–Macaulay R-module M with an $${\mathfrak {m}}$$ -primary cohomological annihilator, if there is a bound on the $$\underline{\mathsf {h}}$$ -length of all modules appearing in $$\mathsf {CM}$$ -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 $$\mathsf {CM}$$ -type if and only if R is an isolated singularity and 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 $$\mathsf {CM}$$ -support are fully decomposable. In particular, a Cohen–Macaulay algebra $$\Lambda $$ is of finite $$\mathsf {CM}$$ -type if and only if every $$\omega $$ -Gorenstein projective module is of finite $$\mathsf {CM}$$ -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.