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Centers of Endomorphism Rings and Reflexivity

Souvik Dey, Justin Lyle

Abstract

Let $R$ be a local ring and let $M$ be a finitely generated $R$-module. Appealing to the natural left module structure of $M$ over its endomorphism ring and corresponding center $Z(\operatorname{End}_R(M))$, we study when various homological properties of $M$ are sufficient to force $M$ to have a nonzero free summand. Consequences of our work include a partial converse to a well-known result of Lindo describing $Z(\operatorname{End}_R(M))$ when $M$ is faithful and reflexive, as well as some applications to the famous Huneke-Wiegand conjecture.

Centers of Endomorphism Rings and Reflexivity

Abstract

Let be a local ring and let be a finitely generated -module. Appealing to the natural left module structure of over its endomorphism ring and corresponding center , we study when various homological properties of are sufficient to force to have a nonzero free summand. Consequences of our work include a partial converse to a well-known result of Lindo describing when is faithful and reflexive, as well as some applications to the famous Huneke-Wiegand conjecture.

Paper Structure

This paper contains 4 sections, 12 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Partial Converse to A Result of Lindo
  4. When is $Z(E) \cong R$?

Key Result

Theorem 1.1

Suppose $M$ is a faithful reflexive $R$-module. Then there is an isomorphism of $R$-algebras $Z(\operatorname{End}_R(M)) \cong \operatorname{End}_R(\operatorname{tr}_R(M))$.

Theorems & Definitions (21)

  • Theorem 1.1: Li17
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1: see Ly23
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 11 more