Spectrum of an abelian category via premonoform objects

TL;DR

This paper introduces a new topological framework for classifying subcategories of an abelian category using premonoform objects, connecting it to classical spectra and providing a unified perspective.

Contribution

It defines a new topology on the spectrum of an abelian category and classifies torsion, Serre, and localizing subcategories via this topology, extending classical results to a broader context.

Findings

$ PSpec A$ is homeomorphic to $ ext{Spec } A$ for commutative noetherian rings.

Classifies torsion, Serre, and localizing subcategories using the new topology.

Establishes correspondences between closed subsets of $ PSpec A$ and open subsets of $ ext{ASpec } A$.

Abstract

Let $\cA$ be an abelian category. In this paper, we study ${\rm (n)PSpec}\cA$, a topological space formed by equivalence classes derived from an equivalence relation on (noetherian) premonoform objects. We classify torsion classes of $\cA$ via closed subclasses of $\nPSpec\cA$. We introduce a new topology on $\PSpec\cA$ and we classify Serre subcategories of $\noeth A$ and localizing subcategories of $\cA$ using this topology. If $A$ is a commutative noetherian ring, we show that $\nPSpec A$ is homeomorphic to $\Spec A$. Moreover, there is a one-to-one correspondence between the closed subsets of $\nPSpec A$ and the open subsets of $\ASpec A$, the atom spectrum of $A$. Finally, we explore the relationships between the new subctegories of $\Mod A$ and subsets of $\nPSpec A$ introduced in this paper, and the known subcategories of $\Mod A$ and subsets of other spectra of $A$.