Annihilators of (co)homology and their influence on the trace Ideal

2024/09/07 by Justin Lyle, Lyle, Justin, Sarasij Maitra +1 · 2 citations
Chemistry · #13C13 #13D02 #13D07 #Axial and Atropisomeric Chirality Synthesis #Commutative Algebra (math.AC) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2409.04686

openalex publication_date 2024/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let (R,\mathfrakm) be a commutative Noetherian local ring, and suppose R is Cohen-Macaulay with canonical module ω_R. We develop new tools for analyzing questions involving annihilators of several homologically defined objects. Using these, we study a generalization introduced by Dao-Kobayashi-Takahashi of the famous Tachikawa conjecture, asking in particular whether the vanishing of \mathfrakm Ext_Rⁱ(ω_R,R) should force the trace ideal of ω_R to contain \mathfrakm, i.e., for R to be nearly Gorenstein. We show this question has an affirmative answer for numerical semigroup rings of minimal multiplicity, but that the answer is negative in general. Our proofs involve a technical analysis of homogeneous ideals in a numerical semigroup ring, and exploit the behavior of Ulrich modules in this setting.

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