Monomial Gorenstein algebras and the stably Calabi--Yau property

TL;DR

This paper establishes a converse relationship for monomial algebras, showing that certain 1-Gorenstein monomial algebras with 3-Calabi--Yau singularity categories are actually 2-Calabi--Yau tilted, expanding understanding of their Calabi--Yau properties.

Contribution

It proves that 1-Gorenstein monomial algebras with 3-Calabi--Yau singularity categories are 2-Calabi--Yau tilted, providing a new characterization in this algebraic setting.

Findings

Converse holds for monomial algebras with specific Gorenstein and Calabi--Yau properties.

Characterization of Gorenstein monomial algebras with stably Calabi--Yau categories.

Extension of Keller--Reiten results to the monomial algebra case.

Abstract

A celebrated result by Keller--Reiten says that $2$-Calabi--Yau tilted algebras are Gorenstein and stably $3$-Calabi--Yau. This note shows that the converse holds in the monomial case: a $1$-Gorenstein monomial algebra with a $3$-Calabi--Yau singularity category is $2$-Calabi--Yau tilted. We study the case of other Goresntein monomial algebras with stably Calabi--Yau singularity categories.