Homological conditions on locally gentle algebras

TL;DR

This paper explores the homological properties of locally gentle algebras, extending the well-understood gentle algebras to infinite-dimensional cases with new combinatorial and classification results.

Contribution

It provides combinatorial descriptions of key algebraic invariants and classifies when these algebras are Artin-Schelter Gorenstein, regular, or Cohen-Macaulay.

Findings

Explicit injective resolutions for locally gentle algebras

Classification of when these algebras are Artin-Schelter Gorenstein, regular, or Cohen-Macaulay

An analogue of Stanley's theorem for locally gentle algebras

Abstract

Gentle algebras are a class of special biserial algebra whose representation theory has been thoroughly described. In this paper, we consider the infinite dimensional generalizations of gentle algebras, referred to as locally gentle algebras. We give combinatorial descriptions of the center, prime spectrum, and homological dimensions of a locally gentle algebra, including an explicit injective resolution. We classify when these algebras are Artin-Schelter Gorenstein, Artin-Schelter regular, and Cohen-Macaulay, and provide an analogue of Stanley's theorem for locally gentle algebras.