Annihilation of cohomology over one dimensional almost Gorenstein rings

TL;DR

This work analyzes the relationship between the cohomology annihilator $\\mathrm{ca}(R)$ and the stable annihilator of maximal Cohen–Macaulay modules, providing a concrete sufficiency condition for their equality. The authors show that if $R$ is Cohen–Macaulay with canonical module $\\omega$ and $\\Omega \\mathrm{CM}^{\\times}(R)$ is closed under the duality $D(-)$, then $\\mathrm{ca}(R) = \\underline{\\mathrm{ann}}_R(\\mathrm{CM}(R))$, and they apply this to one-dimensional analytically unramified almost Gorenstein complete local rings to conclude $\\mathrm{ca}(R) = \\mathrm{co}(R)$. This result, together with known inclusions between the conductor and cohomology annihilators, extends the equality $\\mathrm{ca}(R) = \\mathrm{co}(R)$ beyond Gorenstein rings and highlights the role of trace ideals and $\\omega$-Ulrich modules in mediating duality properties. Ultimately, the paper provides structural criteria and concrete consequences for singularity invariants in dimension one, linking homological and birational concepts through precise annihilator equalities.

Abstract

Given a Cohen-Macaulay local ring, the cohomology annihilator ideal and the annihilator of the stable category of maximal Cohen-Macaulay modules are two ideals closely related both with each other and the singularities of the ring. Kimura recently showed that the two ideals agree up to radicals. In this article, we give a sufficient condition for the two ideals to be equal. As an application, we show that the cohomology annihilator ideal of a one dimensional analytically unramified almost Gorenstein complete local ring agrees with the conductor ideal.

Theorems & Definitions (41)

• Theorem A: Theorem \ref{['main-theorem']}
• Theorem B: Theorem \ref{['theorem-cohomology-annihilator-of-an-almost-Gorenstein-ring']}
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Theorem 2.6