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Contravariantly infinite resolving subcategories

Gen Tanigawa

Abstract

Let $R$ be a commutative Noetherian ring. Denote by $\textrm{mod}R$ the category of finitely generated $R$-modules. In this paper, a contravariantly infinite subcategory of $\textrm{mod}R$ is defined as a full subcategory $\mathscr{X}$ of $\textrm{mod}R$ such that no module outside $\mathscr{X}$ admits a right $\mathscr{X}$-approximation. This paper provides several criteria for contravariant infiniteness in the case where $R$ is a local complete intersection.

Contravariantly infinite resolving subcategories

Abstract

Let be a commutative Noetherian ring. Denote by the category of finitely generated -modules. In this paper, a contravariantly infinite subcategory of is defined as a full subcategory of such that no module outside admits a right -approximation. This paper provides several criteria for contravariant infiniteness in the case where is a local complete intersection.
Paper Structure (4 sections, 24 theorems, 9 equations)

This paper contains 4 sections, 24 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and properties
  3. Proof of the main theorem
  4. A question related to the main theorem

Key Result

Theorem 1.1

Let $R$ be a henselian local complete intersection ring with positive dimension. Let $\mathscr{X}$ be a resolving subcategory of $\operatorname{mod}{R}$. Then the following are equivalent. If in addition $R$ is quasi-excellent, then the above three conditions are equivalent to the following condition as well.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8: T21
  • Definition 2.9
  • ...and 40 more