Gorenstein triangular matrix rings and category algebras

TL;DR

This paper investigates conditions under which triangular matrix rings are Gorenstein with specific self-injective dimensions and applies these results to category algebras of finite EI categories, characterizing when they are 1-Gorenstein.

Contribution

It provides new criteria for Gorenstein properties of triangular matrix rings and characterizes 1-Gorenstein category algebras of finite EI categories.

Findings

Triangular matrix rings are Gorenstein under specific conditions.

Category algebra of a finite EI category is 1-Gorenstein iff the category is free and projective.

The paper links algebraic properties to categorical structures.

Abstract

We give conditions on when a triangular matrix ring is Gorenstein of a given selfinjective dimension. We apply the result to the category algebra of a finite EI category. In particular, we prove that for a finite EI category, its category algebra is 1-Gorenstein if and only if the given category is free and projective.