Classifying localizing subcategories of a Grothendieck category

TL;DR

This paper classifies certain subcategories of a Grothendieck category using its atom spectrum, linking the structure of finitely presented objects to the overall category, with applications to commutative coherent rings.

Contribution

It provides a classification of localizing subcategories of finite type in Grothendieck categories via open subsets of the atom spectrum, and explores the relationship between the atom spectra of the category and its finitely presented objects.

Findings

Classification of localizing subcategories via atom spectrum

Condition for a category to be locally noetherian

Application to commutative coherent rings

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.