# A topology on the set of isomorphism classes of maximal Cohen--Macaulay modules

## TL;DR

This paper introduces a new topology on the set of isomorphism classes of finitely generated modules, focusing on maximal Cohen--Macaulay modules over Cohen--Macaulay local rings, and explores their irreducible components.

## Contribution

It defines a topology on module classes and analyzes the structure of maximal Cohen--Macaulay modules, especially over hypersurfaces, providing new insights into their classification.

## Key findings

- Topology on module classes is well-defined.
- Irreducible components of modules over hypersurfaces are characterized.
- Framework for studying module classification via topology.

## Abstract

In this paper, we introduce a topology on the set of isomorphism classes of finitely generated modules over an associative algebra. Then we focus on the relative topology on the set of isomorphism classes of maximal Cohen--Macaulay modules over a Cohen--Macaulay local ring. We discuss the irreducible components over certain hypersurfaces.