Filtrations in module categories, derived categories and prime spectra

TL;DR

This paper introduces new categorical and spectral filtrations in module and derived categories over noetherian rings, linking them to prime spectra and extending classical classification results.

Contribution

It defines n-coherent subsets and n-uniform subcategories, establishing their relationships with n-wide subcategories and providing a unified filtration framework.

Findings

Established filtrations of subcategories in Mod R and D(Mod R)

Connected subcategory classifications to prime spectrum subsets

Extended classical classification theorems using new filtrations

Abstract

Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria.