Classifying Serre subcategories via atom spectrum

TL;DR

This paper introduces the atom spectrum of an abelian category and uses it to classify Serre subcategories in noetherian cases, also relating it to the Ziegler spectrum in Grothendieck categories.

Contribution

It defines the atom spectrum as a topological space and applies it to classify Serre subcategories, connecting it with the Ziegler spectrum.

Findings

Atom spectrum is a topological space of monoform objects.

Classifies Serre subcategories in noetherian abelian categories.

Shows homeomorphism between atom spectrum and Ziegler spectrum in locally noetherian Grothendieck categories.

Abstract

In this paper, we introduce the atom spectrum of an abelian category as a topological space consisting of all the equivalence classes of monoform objects. In terms of the atom spectrum, we give a classification of Serre subcategories of an arbitrary noetherian abelian category. Moreover we show that the atom spectrum of a locally noetherian Grothendieck category is homeomorphic to its Ziegler spectrum.