Monomial Gorenstein algebras and the stably Calabi--Yau property

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.