A descent principle for compactly supported extensions of functors

TL;DR

This paper investigates conditions under which a cohomology theory for algebraic varieties admits a compactly supported version satisfying a long exact sequence, linking descent properties with the existence of such extensions.

Contribution

It establishes an equivalence between categories of hypersheaves and characterizes when compactly supported cohomology extensions exist based on descent properties.

Findings

Proves the equivalence between hypersheaf categories and compactly supported extensions.

Shows that descent for abstract blowups guarantees the existence of compactly supported cohomology.

Derives classical results like the weight filtration from the main theorem.

Abstract

A characteristic property of cohomology with compact support is the long exact sequence that connects the compactly supported cohomology groups of a space, an open subspace and its complement. Given an arbitrary cohomology theory of algebraic varieties, one can ask whether a compactly supported version exists, satisfying such a long exact sequence. This is the case whenever the cohomology theory satisfies descent for abstract blowups (also known as proper cdh descent). We make this precise by proving an equivalence between certain categories of hypersheaves. We show how several classical and non-trivial results, such as the existence of a unique weight filtration on cohomology with compact support, can be derived from this theorem.