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A generalization of the dimension and radius of a subcategory of modules and its applications

Yuki Mifune

Abstract

Let $R$ be a commutative noetherian local ring and denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. In this paper, we give some evaluations of the singular locus of $R$ and annihilators of Tor and Ext from a viewpoint of the finiteness of dimensions/radii of full subcategories of $\operatorname{mod} R$. As an application, we recover a theorem of Dey and Takahashi when $R$ is Cohen--Macaulay. Moreover, we obtain the divergence of the dimensions of specific full subcategories of $\operatorname{mod} R$ in non-Cohen--Macaulay case.

A generalization of the dimension and radius of a subcategory of modules and its applications

Abstract

Let be a commutative noetherian local ring and denote by the category of finitely generated -modules. In this paper, we give some evaluations of the singular locus of and annihilators of Tor and Ext from a viewpoint of the finiteness of dimensions/radii of full subcategories of . As an application, we recover a theorem of Dey and Takahashi when is Cohen--Macaulay. Moreover, we obtain the divergence of the dimensions of specific full subcategories of in non-Cohen--Macaulay case.
Paper Structure (3 sections, 10 theorems, 3 equations)

This paper contains 3 sections, 10 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of main theorem

Key Result

Theorem 1.1

Let $R$ be a noetherian local ring with residue field $k$ and $\mathcal{X}, \mathcal{Y}$ be full subcategories of $\operatorname{mod} R$ with $\operatorname{radius}(\mathcal{X},\mathcal{Y}) < \infty$. Assume that $\mathcal{X}$ is closed under extensions and contains $\Omega_R ^n k$ for some $n\geq0$

Theorems & Definitions (32)

  • Theorem 1.1: Theorem \ref{['main thm']}
  • Corollary 1.2: Corollary \ref{['cor1 of thm']}
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 22 more