The Auslander-Gorenstein condition for monomial algebras

TL;DR

This paper characterizes Auslander-Gorenstein monomial algebras, showing they are string algebras, and links their properties to the Auslander-Reiten map, providing new classification and homological insights.

Contribution

It provides a combinatorial classification of Auslander-Gorenstein gentle algebras and establishes a homological characterization via the Auslander-Reiten map for monomial algebras.

Findings

Auslander-Gorenstein monomial algebras are string algebras

A procedure reduces 2-Gorenstein monomial algebras to Nakayama algebras

Monomial algebras satisfy a stronger form of the Auslander-Reiten Conjecture

Abstract

This paper investigates the Auslander-Gorenstein property for monomial algebras. First, we prove that every Auslander-Gorenstein monomial algebra is a string algebra and present a simple combinatorial classification of Auslander-Gorenstein gentle algebras. Furthermore, we describe a procedure to transform any 2-Gorenstein monomial algebra into a Nakayama algebra, thereby reducing the classification of Auslander-Gorenstein monomial algebras to that of Auslander-Gorenstein Nakayama algebras. As an application of this reduction method, we prove that every monomial algebra satisfies a stronger version of the Auslander-Reiten Conjecture. Our second main result establishes that a monomial algebra is Auslander-Gorenstein if and only if it has a well-defined, bijective Auslander-Reiten map, confirming a conjecture of Marczinzik for monomial algebras. This yields a new homological characterisation of the Auslander-Gorenstein property. Additionally, we provide an explicit description of the Auslander-Reiten bijection in the case of gentle algebras. Along the way, we also generalise a result of Iwanaga and Fuller: We show that every $2n$-Gorenstein monomial algebra is also $(2n+1)$-Gorenstein for every $n\ge 1.$