Annihilators and dimensions of the singularity category

TL;DR

This paper investigates the relationship between the annihilator of the singularity category of certain commutative noetherian rings and the Jacobian ideal, providing new bounds on the category's dimension especially for non Cohen-Macaulay rings.

Contribution

It establishes that the annihilator of the singularity category coincides with the radical of the Jacobian ideal for specific classes of rings and extends dimension bounds to non Cohen-Macaulay rings.

Findings

Annihilator of singularity category equals Jacobian ideal up to radical.

Provides an upper bound for the dimension of the singularity category.

Extends previous results to non Cohen-Macaulay rings.

Abstract

Let R be a commutative noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relation between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non Cohen-Macaulay rings.