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Classifying localizing subcategories of a Grothendieck category

Reza Sazeedeh

TL;DR

This work develops a spectrum-based framework to classify localizing subcategories of finite type in locally coherent Grothendieck categories by translating subcategory data into topological data on atom-related spectra. It introduces the Zariski-atom spectrum $\mathrm{ZASpec}\,\mathcal{A}$ and the Ziegler spectrum $\mathrm{Zg}\,\mathcal{A}$, establishing bijections between Serre subcategories of $\mathrm{fp-}\mathcal{A}$ and open sets in $\mathrm{ZASpec}\,\mathcal{A}$, and between open sets in $\mathrm{ZASpec}\,\mathcal{A}$ and localizing subcategories of finite type in $\mathcal{A}$. The results connect atom-theoretic data with injective envelopes and show when $\mathcal{A}$ is locally noetherian via a Cohen-type criterion. For commutative coherent rings, the paper provides a complete correspondence among open subsets of Hochster-dual Spec, closed sets in the Ziegler spectrum, Gabriel topologies with finitely generated bases, and finite-type localizing subcategories of $\mathrm{Mod}\,A$, thereby giving a concrete classification in this important special case.

Abstract

Let $\cA$ be a locally coherent Grothendieck category, $\fp\cA$ be the full subcategory of $\cA$ consisting of finitely presented objects and $\ASpec\cA$ be the atom spectrum of $\cA$. In this paper, we classify localizing subcategories of finite type of $\cA$ via open subsets of $\ASpec\cA$. We investigate $\ASpec\fp\cA$ and show that if $\ASpec\fp\cA=\ASpec\cA$, then $\cA$ is locally noetherian. As an application, we specialize our investigation to the case of commutative coherent rings.

Classifying localizing subcategories of a Grothendieck category

TL;DR

This work develops a spectrum-based framework to classify localizing subcategories of finite type in locally coherent Grothendieck categories by translating subcategory data into topological data on atom-related spectra. It introduces the Zariski-atom spectrum and the Ziegler spectrum , establishing bijections between Serre subcategories of and open sets in , and between open sets in and localizing subcategories of finite type in . The results connect atom-theoretic data with injective envelopes and show when is locally noetherian via a Cohen-type criterion. For commutative coherent rings, the paper provides a complete correspondence among open subsets of Hochster-dual Spec, closed sets in the Ziegler spectrum, Gabriel topologies with finitely generated bases, and finite-type localizing subcategories of , thereby giving a concrete classification in this important special case.

Abstract

Let be a locally coherent Grothendieck category, be the full subcategory of consisting of finitely presented objects and be the atom spectrum of . In this paper, we classify localizing subcategories of finite type of via open subsets of . We investigate and show that if , then is locally noetherian. As an application, we specialize our investigation to the case of commutative coherent rings.

Paper Structure

This paper contains 5 sections, 36 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Categories with enough atoms
  3. The atom spectrum of a locally coherent category
  4. Ziegler spectrum of a locally coherent category
  5. In the case of commutative coherent rings

Key Result

Theorem 1

Let $\mathcal{A}$ be a locally coherent category. Then the map $\mathcal{X}\mapsto \mathop{\mathrm{ASupp}}\nolimits\mathcal{X}$ provides a bijective correspondence between the lattice of Serre subcategories of $\mathop{\mathrm{fp-}}\nolimits\mathcal{A}$ and the lattice of open subsets of $\mathop{\m

Theorems & Definitions (80)

  • Theorem 1: \ref{['corpropp']}
  • Theorem 2: \ref{['noeth']}
  • Theorem 3: \ref{['fgftg']}
  • Theorem 4: \ref{['flfg']}
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 70 more