Classifying subcategories of modules over a commutative noetherian ring

TL;DR

This paper extends Hovey's bijection and Serre subcategory results from specific quotient rings to all commutative noetherian rings, and relates localizing subcategories to subsets of the spectrum of R.

Contribution

It generalizes the classification of subcategories of modules over a broad class of rings and connects derived category substructures to prime spectra.

Findings

Every coherent subcategory of finitely presented modules over a noetherian ring is a Serre subcategory.

Established a module version of Neeman's bijection between localizing subcategories and subsets of Spec R.

Extended Hovey's isomorphism to all commutative noetherian rings.

Abstract

Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-modules is a Serre subcategory. In this paper, it is proved that this holds whenever R is a commutative noetherian ring. This paper also yields a module version of the bijection between the set of localizing subcategories of the derived category of R-modules and the set of subsets of Spec R which was given by Neeman.