Integrality of noetherian Grothendieck categories

TL;DR

This paper introduces a new notion of integrality for Grothendieck categories, generalizing concepts from noncommutative ring theory and algebraic geometry, and establishes foundational correspondences and classifications within this framework.

Contribution

It defines and proves the equivalence of two notions of integrality in Grothendieck categories, extending Gabriel's correspondence and classifying localizing subcategories.

Findings

Equivalence of two definitions of integrality in Grothendieck categories

Classification of locally closed localizing subcategories

Development of a theory of singular objects and proof of Goldie's theorem

Abstract

We introduce the notion of integrality of Grothendieck categories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of locally noetherian schemes. Two different spaces associated to a Grothendieck category yield respective definitions of integrality, and we prove the equivalence of these definitions using a Grothendieck-categorical version of Gabriel's correspondence, which originally related indecomposable injective modules and prime two-sided ideals for noetherian rings. The generalization of prime two-sided ideals is also used to classify locally closed localizing subcategories. As an application of the main results, we develop a theory of singular objects in a Grothendieck category and deduce Goldie's theorem on the existence of the quotient ring as its consequence.