Spectrum of an abelian category via premonoform objects

Reza Sazeedeh

Spectrum of an abelian category via premonoform objects

Reza Sazeedeh

TL;DR

The paper introduces the pspectrum $\PSpec\mathcal{A}$ of an abelian category and its noetherian subspace $\nPSpec\mathcal{A}$ to classify subcategories via closed/extension-closed subsets, establishing a bridge between torsion classes and spectral data. It develops prem monoform/premoform objects, connects them to extreme/prime ideals in noncommutative and commutative rings, and demonstrates how these objects yield structural classifications of Serre, localizing, and torsion subcategories. A new Serre-based topology on $\PSpec\mathcal{A}$ (via Serre psupports) provides bijections between Serre subcategories of noetherian objects and Serre closed/extension-closed subsets of $\nPSpec\mathcal{A}$, and, in the locally noetherian setting, a correspondence between localizing subcategories and Serre-closed, Serre-extension-closed subsets. In the commutative noetherian case, $\nPSpec A$ is homeomorphic to $\Spec A$, and closed subsets of $\nPSpec A$ correspond to open subsets of $\ASpec A$, yielding a network of lattice isomorphisms among Ziegler, atom, Serre, null, coherent, thick, and localization lattices. These results unify several known spectra (Spec, Zg, ASpec) with new spectral data from premonoform objects, and provide practical tools for understanding subcategory lattices via topological invariants.

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 $A$.

Spectrum of an abelian category via premonoform objects

TL;DR

The paper introduces the pspectrum

of an abelian category and its noetherian subspace

to classify subcategories via closed/extension-closed subsets, establishing a bridge between torsion classes and spectral data. It develops prem monoform/premoform objects, connects them to extreme/prime ideals in noncommutative and commutative rings, and demonstrates how these objects yield structural classifications of Serre, localizing, and torsion subcategories. A new Serre-based topology on

(via Serre psupports) provides bijections between Serre subcategories of noetherian objects and Serre closed/extension-closed subsets of

, and, in the locally noetherian setting, a correspondence between localizing subcategories and Serre-closed, Serre-extension-closed subsets. In the commutative noetherian case,

is homeomorphic to

, and closed subsets of

correspond to open subsets of

, yielding a network of lattice isomorphisms among Ziegler, atom, Serre, null, coherent, thick, and localization lattices. These results unify several known spectra (Spec, Zg, ASpec) with new spectral data from premonoform objects, and provide practical tools for understanding subcategory lattices via topological invariants.

Abstract

Let

be an abelian category. In this paper, we study

, a topological space formed by equivalence classes derived from an equivalence relation on (noetherian) premonoform objects. We classify torsion classes of

via closed subclasses of

. We introduce a new topology on

and we classify Serre subcategories of

and localizing subcategories of

using this topology. If

is a commutative noetherian ring, we show that

is homeomorphic to

. Moreover, there is a one-to-one correspondence between the closed subsets of

and the open subsets of

, the atom spectrum of

. Finally, we explore the relationships between the new subctegories of

and subsets of

introduced in this paper, and the known subcategories of

and subsets of other spectra of

Paper Structure (5 sections, 51 theorems, 22 equations)

This paper contains 5 sections, 51 theorems, 22 equations.

Introduction
Premonoform objects in an abelian category
Premonoform modules over a noncommutative ring
A new topological space for an abelian category
Premonoform modules over a commutative ring

Key Result

Theorem 1

There exists an injective map $\varPhi\mapsto \mathop{\mathrm{nPSupp}}\nolimits^{-1}\varPhi$ between closed and extension-closed subclasses of $\mathop{\mathrm{nPSpec}}\nolimits\mathcal{A}$ and torsion classes of $\mathcal{A}$. This map has a left inverse given by $\mathcal{X}\mapsto\mathop{\mathrm{

Theorems & Definitions (103)

Theorem 1
Theorem 2
Theorem 3
Theorem 4
Theorem 5
Theorem 6
Theorem 7
Definition 1
Lemma 1
proof
...and 93 more

Spectrum of an abelian category via premonoform objects

TL;DR

Abstract

Spectrum of an abelian category via premonoform objects

Authors

TL;DR

Abstract

Table of Contents

Key Result

Theorems & Definitions (103)