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Supports of intersections of thick subcategories and proxy smallness

Kiriko Kato, Ryo Takahashi

Abstract

Let R be a commutative noetherian ring. Let D^b(R) be the bounded derived category of finitely generated R-modules. Let X and Y be thick subcategories of D^b(R). In this paper, we consider the question asking when the equality Supp(X\cap Y)=Supp X\cap Supp Y holds, and give several answers. As applications, we obtain a characterization of the proxy small subcategories, and classifications of certain thick subcategories.

Supports of intersections of thick subcategories and proxy smallness

Abstract

Let R be a commutative noetherian ring. Let D^b(R) be the bounded derived category of finitely generated R-modules. Let X and Y be thick subcategories of D^b(R). In this paper, we consider the question asking when the equality Supp(X\cap Y)=Supp X\cap Supp Y holds, and give several answers. As applications, we obtain a characterization of the proxy small subcategories, and classifications of certain thick subcategories.

Paper Structure

This paper contains 5 sections, 19 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cases where the support equality holds or not
  4. Characterizing thick subcategories satisfying the support equality
  5. Applications of Corollary \ref{['2']}

Key Result

Theorem 1.2

Theorems & Definitions (42)

  • Theorem 1.2: Theorem \ref{['25']}
  • Theorem 1.3: Theorem \ref{['21']}
  • Corollary 1.4: Corollary \ref{['2']}
  • Corollary 1.5: Theorems \ref{['3']} and \ref{['6']}
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Hopkins--Neeman
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7: Pollitz
  • ...and 32 more