group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In compactly supported cohomology cocycles and coboundaries on some space are required to have compact support: to be non-trivial only over a compact subspace/compact subobject of the base.
Discussion of compactly supported de Rham cohomology includes for instance the following.
A standard textbook account is in
A brief account is also in section 11 of
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