The derived category of the projective line

TL;DR

This paper classifies all localizing subcategories of the derived category of quasi-coherent sheaves on the projective line over a field, focusing on those that are kernels of cohomological functors to Grothendieck categories.

Contribution

It provides a complete classification of certain localizing subcategories in the derived category of the projective line, a previously unresolved problem.

Findings

Complete classification of localizing subcategories as kernels of cohomological functors

Identification of the structure of these subcategories on the projective line

Advancement in understanding the derived category of algebraic curves

Abstract

We examine the localizing subcategories of the derived category of quasi-coherent sheaves on the projective line over a field. We provide a complete classification of all such subcategories which arise as the kernel of a cohomological functor to a Grothendieck category.