A note on thick subcategories of stable derived categories

TL;DR

This paper establishes a bijection between certain thick subcategories in exact categories and their stable derived categories, enabling classification and generation of subcategories in algebraic and geometric contexts.

Contribution

It introduces a new bijection linking thick subcategories containing projectives to those in stable derived categories, aiding classification and generation tasks.

Findings

Classifies thick subcategories of finitely generated modules over local complete intersections.

Provides generators for the category of coherent sheaves on schemes.

Establishes a bijection between thick subcategories in exact and stable derived categories.

Abstract

For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection we classify thick subcategories of finitely generated modules over local complete intersections and produce generators for the category of coherent sheaves on a separated noetherian scheme with an ample family.