Localizing and colocalizing subcategories on schemes

TL;DR

This paper classifies certain subcategories of derived categories of schemes based on their localization and colocalization properties, revealing a correspondence with subsets of the scheme and linking colocalizing subcategories to orthogonal complements of localizing ones.

Contribution

It provides a classification of $ ensor$-ideal localizing and $ ext{H}$-coideal colocalizing subcategories of derived categories of schemes, connecting them to subsets and orthogonal complements.

Findings

$ ensor$-ideal localizing subcategories correspond to subsets of the scheme

$ ext{H}$-coideal colocalizing subcategories are classified similarly

Every colocalizing subcategory of this form is an orthogonal complement of a localizing subcategory

Abstract

A full triangulated subcategory $\mathsf{L} \subset \mathsf{T}$ of triangulated category $\mathsf{T}$ is \emph{localizing} if it is stable for coproducts. If, further, $\mathsf{T}$ is $\otimes$-triangulated, we say that $\mathsf{L}$ is $\otimes$-ideal if $F \otimes G \in \mathsf{L}$ for all $G \in \mathsf{L}$ and all $F \in \mathsf{T}$. Analogously, a full triangulated subcategory $\mathsf{C} \subset \mathsf{T}$ is \emph{colocalizing} if it is stable for products. If, further, $\mathsf{T}$ is \emph{closed}, \textit{i.e.} $\otimes$-triangulated with internal homs (denoted $[-,-]$), we say that $\mathsf{C}$ is $\mathcal{H}$-coideal if $[F, G] \in \mathsf{C}$ for all $G \in \mathsf{C}$ and all $F \in \mathsf{T}$. For a point generated concentrated scheme $X$, we prove that all $\otimes$-ideal localizing subcategories of $\mathbf{D}_{qc}(X)$ are classified by the subsets of $X$. As a consequence, we prove that for $\mathcal{H}$-coideal colocalizing subcategories of $\mathbf{D}_{qc}(X)$ the same holds. Moreover, every such colocalizing subcategory $\mathsf{C}$ is of the form $\mathsf{C} = \mathsf{L}^\perp$, where $\mathsf{L}$ is a $\otimes$-ideal localizing subcategory.