Filtrations in module categories, derived categories and prime spectra

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.

arXiv comment17 pages, to appear in IMRN

References in corpus (6)

• Stratifying triangulated categories
• Classifying Serre subcategories via atom spectrum
• Tensor triangular geometry and KK-theory
• Flat ring epimorphisms and universal localisations of commutative rings
• Localizing subcategories in the Bootstrap category of separable C*-algebras
• Cohomological dimensions of specialization-closed subsets and subcategories of modules

Cited by in corpus (1)

• Relative tilting theory in abelian categories I: Auslander-Buchweitz-Reiten approximations theory in subcategories and cotorsion pairs