Filtrations in Module Categories, Derived Categories, and Prime Spectra

Abstract

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