Supports of intersections of thick subcategories and proxy smallness

TL;DR

This paper investigates the conditions under which the support of the intersection of thick subcategories equals the intersection of their supports in the derived category of a noetherian ring, leading to classifications and characterizations.

Contribution

It provides new criteria for support equalities in thick subcategories and characterizes proxy small subcategories and their classifications.

Findings

Established conditions for support equality in thick subcategories.

Characterized proxy small subcategories.

Classified certain thick subcategories.

Abstract

Let R be a commutative noetherian ring. Let D^b(R) be the bounded derived category of finitely generated R-modules. Let X and Y be thick subcategories of D^b(R). In this paper, we consider the question asking when the equality Supp(X\cap Y)=Supp X\cap Supp Y holds, and give several answers. As applications, we obtain a characterization of the proxy small subcategories, and classifications of certain thick subcategories.