Localizing and colocalizing subcategories on schemes
Leovigildo Alonso, Ana Jerem\'ias, Eduardo Loureiro

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 of triangulated category is \emph{localizing} if it is stable for coproducts. If, further, is -triangulated, we say that is -ideal if for all and all . Analogously, a full triangulated subcategory is \emph{colocalizing} if it is stable for products. If, further, is \emph{closed}, \textit{i.e.} -triangulated with internal homs (denoted ), we say that is -coideal if for all and all . For a point generated concentrated scheme , we prove that all -ideal localizing subcategories of are classified by the subsets of . As a…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Rings, Modules, and Algebras
