Localizing and colocalizing subcategories on schemes

Abstract

A full triangulated subcategory $\mathsf{L}\subset \mathsf{T}$ of triangulated category $\mathsf{T}$ is localizing if it is stable for coproducts. If, further, $\mathsf{T}$ is $\otimes$-triangulated, we say that $\mathsf{H}$ 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 colocalizing if it is stable for products. If, further, $\mathsf{T}$ is closed, 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 of ${\mathbf{D}}_{qc}(X)$.

21 pages. V3: Improved section 6 and further minor changes. V2: Changed title, revamped introduction, added new section discussing the improvement of the present set up with respect to the usual Noetherian hypothesis