Atom spectra of symmetric monoidal abelian categories and classification of subcategories

TL;DR

This paper generalizes classification results for subcategories in module categories to symmetric monoidal abelian categories using the orbit atom spectrum, unifying and extending classical theories.

Contribution

It introduces the orbit atom spectrum and applies it to classify subcategories in symmetric monoidal abelian categories, extending classical results.

Findings

Classifies torsion-free classes via subsets of the orbit atom spectrum.

Recovers classical classifications for commutative noetherian rings.

Provides new classifications for graded modules, coherent sheaves, and dg modules.

Abstract

We extend the classification results for torsion classes and torsion-free classes in the category of finitely generated modules over a commutative noetherian ring to suitable symmetric monoidal closed noetherian abelian categories. Our main tool is the orbit atom spectrum, defined as the quotient of Kanda's atom spectrum by the action induced by tensoring with invertible objects. We prove that, under natural tensor-theoretic assumptions, several classes of subcategories collapse to Serre subcategories or torsion-free classes. Moreover, torsion-free classes compatible with the tensor structure are classified by arbitrary subsets of the orbit atom spectrum. As applications, we recover the classical classifications for commutative noetherian rings and obtain analogues for graded modules, coherent sheaves, and dg modules.