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A descent principle for compactly supported extensions of functors

Josefien Kuijper
Vol. 8 (2023), No. 3, 489–529
DOI: 10.2140/akt.2023.8.489
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 nontrivial results, such as the existence of a unique weight filtration on cohomology with compact support, can be derived from this theorem.

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Keywords
proper cdh descent, cohomology with compact support, varieties
Mathematical Subject Classification
Primary: 14F99
Secondary: 14F42, 18F10, 55U99
Milestones
Received: 12 December 2022
Revised: 11 May 2023
Accepted: 2 June 2023
Published: 27 August 2023
Authors
Josefien Kuijper
Department of Mathematics
Stockholm University
Stockholm
Sweden
© 2023 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).