  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.
