Stability of annihilators of cohomology and closed subsets defined by Jacobian ideals

TL;DR

This paper investigates the stability of cohomology annihilators in Noetherian rings, showing their powers can annihilate higher Ext modules, and characterizes ring equidimensionality via Jacobian ideals and singular loci.

Contribution

It establishes the annihilation of higher Ext modules by powers of cohomology annihilators and links ring equidimensionality to Jacobian ideals and singular loci.

Findings

A power of the cohomology annihilator annihilates certain Ext modules.

Characterization of equidimensionality via Jacobian ideals.

Relationship between singular locus and cohomology annihilators.

Abstract

Let $R$ be a commutative Noetherian ring of dimension $d$. In this paper, we first show that some power of the cohomology annihilator annihilates the $(d+1)$-th Ext modules for all finitely generated modules when either $R$ admits a dualizing complex or $R$ is local. Next, we study the Jacobian ideal of affine algebras over a field and equicharacteristic complete local rings, and characterize the equidimensionality of the ring in terms of the singular locus and the closed subsets defined by the cohomology annihilator and the Jacobian ideal.