Trace ideal and annihilator of Ext and Tor of regular fractional ideals, and some applications

TL;DR

This paper investigates the relationships between trace ideals, annihilators of Ext and Tor modules of fractional ideals in Noetherian rings, providing new equalities, answering open questions, and offering alternative proofs for known results.

Contribution

It establishes an equality between trace ideals and certain annihilators of Ext and Tor for fractional ideals, and applies this to answer open questions and reprove recent results.

Findings

Proved an equality between trace ideal and annihilators of Ext and Tor.

Answered an open question in one-dimensional local analytically unramified rings.

Provided an alternative proof that the cohomology annihilator equals the conductor in certain rings.

Abstract

Given a commutative Noetherian ring $R$ with total ring of fractions $Q(R)$, and a finitely generated $R$-submodule $M$ of $Q(R)$, we prove an equality between trace ideal, and certain annihilator of Ext and Tor of $M$. As a consequence, we answer in one-dimensional local analytically unramified case, a question raised by the present author and R. Takahashi. As another application, we give an alternative proof of a recent result of \"O. Esentepe that for one-dimensional analytically unramified Gorenstein local rings, the cohomology annihilator of Iyengar and Takahashi coincides with the conductor ideal.