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A characteristic property of cohomology with compact support is the longexact sequence that connects the compactly supported cohomology groups of aspace, an open subspace and its complement. Given an arbitrary cohomologytheory of algebraic varieties, one can ask whether a compactly supportedversion exists, satisfying such a long exact sequence. This is the casewhenever the cohomology theory satisfies descent for abstract blowups (alsoknown as proper cdh descent). We make this precise by proving an equivalencebetween certain categories of hypersheaves. We show how several classical andnon-trivial results, such as the existence of a unique weight filtration oncohomology with compact support, can be derived from this theorem.

A descent principle for compactly supported extensions of functors