The Jacobian ideal of a commutative ring and annihilators of cohomology

TL;DR

This paper demonstrates that under certain conditions, powers of the Jacobian ideal of specific rings annihilate higher Ext modules, introducing a derived Noether different concept to establish these results.

Contribution

It establishes conditions under which the Jacobian ideal or its powers annihilate Ext modules, and introduces a derived Noether different for this purpose.

Findings

Some powers of the Jacobian ideal annihilate Ext^{d+1}_R(-,-).

Conditions are identified when the Jacobian ideal itself annihilates Ext modules.

Examples show the Jacobian ideal does not always annihilate Ext modules.

Abstract

It is proved that for a ring $R$ that is either an affine algebra over a field, or an equicharacteristic complete local ring, some power of the Jacobian ideal of $R$ annihilates $\mathrm{Ext}^{d+1}_{R}(-,-)$, where $d$ is the Krull dimension of $R$. Sufficient conditions are identified under which the Jacobian ideal itself annihilates these Ext-modules, and examples are provided that show that this is not always the case. A crucial new idea is to consider a derived version of the Noether different of an algebra.