The MCM-approximation of the trivial module over a category algebra

TL;DR

This paper constructs an explicit Gorenstein-projective module over the category algebra of a finite free EI category, providing conditions for when the trivial module is Gorenstein-projective, and introduces a Cohen-Macaulay approximation.

Contribution

It introduces a method to explicitly construct a Gorenstein-projective module over the category algebra of finite free EI categories and characterizes when the trivial module is Gorenstein-projective.

Findings

Constructs explicit Gorenstein-projective modules for certain categories

Provides conditions for the trivial module to be Gorenstein-projective

Establishes a Cohen-Macaulay approximation framework

Abstract

For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is Gorenstein-projective and is a maximal Cohen-Macaulay approximation of the trivial module. We give conditions on when the trivial module is Gorenstein-projective.