# Two definable subcategories of maximal Cohen-Macaulay modules

## TL;DR

This paper investigates two definable subcategories of maximal Cohen-Macaulay modules over Cohen-Macaulay rings, analyzing their properties, relationships, and how they relate to the broader module category.

## Contribution

It introduces and compares two new definable subcategories of maximal Cohen-Macaulay modules, exploring their properties and interactions within the module category.

## Key findings

- The two subcategories are closed under direct limits, products, and pure submodules.
- Comparison of properties inherited from maximal Cohen-Macaulay modules.
- Analysis of how these subcategories interact with the entire module category.

## Abstract

Over a Cohen-Macaulay ring we consider two extensions of the maximal Cohen-Macaulay modules from the viewpoint of definable subcategories, which are closed under direct limits, direct products and pure submodules. After describing these categories, we compare them and consider which properties they inherit from the maximal Cohen-Macaulay modules. We then consider some further properties of these classes and how they interact with the entire module category.