Cohomological dimensions of specialization-closed subsets and subcategories of modules

TL;DR

This paper investigates the relationship between specialization-closed subsets of Spec R and subcategories of modules, introducing n-wide subcategories to analyze cohomological dimensions in a commutative noetherian ring context.

Contribution

It characterizes specialization-closed subsets via subcategory closure properties and introduces n-wide subcategories to study cohomological dimensions.

Findings

Characterization of specialization-closed subsets in terms of module subcategories

Introduction of n-wide subcategories for cohomological dimension analysis

Criteria for when a subset has cohomological dimension at most n

Abstract

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n.