Brauer-Thrall theory for maximal Cohen-Macaulay modules

TL;DR

This paper surveys progress on the Brauer-Thrall conjectures, originally for finite-dimensional algebras, now interpreted for maximal Cohen-Macaulay modules over Cohen-Macaulay local rings, highlighting their implications for module theory.

Contribution

It provides a comprehensive overview of recent developments and results related to the Brauer-Thrall conjectures in the context of Cohen-Macaulay modules.

Findings

Progress in understanding the structure of maximal Cohen-Macaulay modules

Extensions of Brauer-Thrall conjectures to local rings

New classifications of modules based on size and complexity

Abstract

The Brauer-Thrall Conjectures, now theorems, were originally stated for finitely generated modules over a finite-dimensional $\sk$-algebra. They say, roughly speaking, that infinite representation type implies the existence of lots of indecomposable modules of arbitrarily large $\sk$-dimension. These conjectures have natural interpretations in the context of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings. This is a survey of progress on these transplanted conjectures.