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.
Keywords
proper cdh descent, cohomology with compact support,
varieties
