Classifying preaisles of derived categories of complete intersections

Ryo Takahashi

Vol. 339 (2025), No. 2, 345–409

DOI: 10.2140/pjm.2025.339.345

Abstract

Let $R$ be a commutative noetherian ring. Denote by $modR$ the category of finitely generated $R$-modules, by $D^b(R)$ the bounded derived category of $modR$, and by $D_{sg}(R)$ 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 $D^b(R)$ containing $R$ and closed under direct summands, which includes as restrictions the classification of thick subcategories of $D_{sg}(R)$ due to Stevenson, and the classification of resolving subcategories of $modR$ due to Dao and Takahashi.

Keywords

derived category, complete intersection, (pre)(co)aisle, module category, resolving subcategory, thick subcategory, singularity category, hypersurface, perfect complex, maximal Cohen–Macaulay module/complex, $t$-structure, Koszul complex, projective dimension, G-dimension, filtration by supports (sp-filtration)

Mathematical Subject Classification

Primary: 13D09

Secondary: 13C60

Milestones

Received: 9 August 2024

Revised: 4 September 2025

Accepted: 8 September 2025

Published: 21 September 2025