Gorenstein Local Rings Whose Cohomological Annihilator Is the Maximal Ideal

Abstract

This work concerns the cohomological annihilators introduced by Iyengar and Takahashi. Let $R$ be a Gorenstein local ring whose cohomological annihilator is the maximal ideal. We prove that $R$ has finite Cohen–Macaulay type if, in addition, $R$ is either of minimal multiplicity, one-dimensional, or an equicharacteristic Artinian algebra over an algebraically closed field of characteristic not two. Under mild assumptions on residue fields, we classify the equicharacteristic complete Gorenstein local rings with finite Cohen–Macaulay type whose cohomological annihilator is the maximal ideal. For an Artinian Gorenstein local ring but not a field, we prove that the nilpotency degree of the cohomological annihilator is at most three.