Classifying preaisles of derived categories of complete intersections

TL;DR

This paper classifies certain subcategories within the derived category of complete intersection rings, extending previous classifications of thick and resolving subcategories, thus deepening understanding of their structure.

Contribution

It provides a complete classification of preaisles in the bounded derived category of complete intersection rings, unifying and extending prior classification results.

Findings

Classifies preaisles containing R and closed under summands for complete intersections

Includes classification of thick subcategories of the singularity category

Includes classification of resolving subcategories of finitely generated modules

Abstract

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod}R$ the category of finitely generated $R$-modules, by $\operatorname{D^b}(R)$ the bounded derived category of $\operatorname{mod}R$, and by $\operatorname{D_{sg}}(R)$ the singularity category of $R$. The main result of this paper provides, when $R$ is a complete intersection, a complete classification of the preaisles of $\operatorname{D^b}(R)$ containing $R$ and closed under direct summands, which includes as restrictions the classification of thick subcategories of $\operatorname{D_{sg}}(R)$ due to Stevenson, and the classification of resolving subcategories of $\operatorname{mod}R$ due to Dao and Takahashi.