A Criterion for Gorenstein Algebras to Be Regular

In this paper we give a criterion for a left Gorenstein algebra to be AS-regular. Let A be a left Gorenstein algebra such that the trivial module Ak admits a finitely generated minimal free resolution. Then A is AS-regular if and only if its left Gorenstein index is equal to− inf{i |ExtA A (k, k)i = 0}. Furthermore, A is Koszul AS-regular if and only if its left Gorenstein index is depthAA = − inf{i |Ext depthAA A (k, k)i = 0}. As applications, we prove that the category of AS-regular algebras is a tensor category and that a left Noetherian p-Koszul, left Gorenstein algebra is AS-regular if and only if it is p-standard. This generalizes a result of Dong and the second author.