The Cohomology Annihilator of a Curve Singularity

TL;DR

This paper explores the cohomology annihilator in Gorenstein rings, establishing its properties, relations with other ideals, and applications to algebraic curves and their covers, using a triangulated category approach.

Contribution

It introduces new connections between cohomology annihilators, conductor ideals, and noncommutative resolutions in the context of Gorenstein rings and algebraic curves.

Findings

Cohomology annihilator and conductor ideal coincide in dimension one.

A ring homomorphism condition transfers cohomology annihilators between Gorenstein rings.

Generalization of the Milnor-Jung formula to double branched covers.

Abstract

The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to their double branched covers. We also show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization and in dimension one the cohomology annihilator of its completion. Finally, we investigate a relation between the cohomology annihilator of a Gorenstein ring and stable annihilators of its noncommutative resolutions.