Holmstrom D25 The standard types of homology and cohomology

More about cohomology, homology, cohomology with compact support, and Borel-Moore homology. Maybe also relative groups and groups with support in a subscheme. This obviously connects with representable theories and the six functors, and should maybe be more integrated into it.

Somewhere I must explain relative cohomology and cohomology with supports, maybe together with the four types of theories, or under F20 Relative theories. See both these entries when writing up.

Cohomology with compact supports Locally finite homology

It would be nice to see how the six functors formalism implies the Bloch-Ogus axioms, and where exactly we use that the spectrum is a Bloch-Ogus spectrum and not just any spectrum.

There is something by Lipman about duality and the four theories I think.

Maybe this is also a good place to explain things like relative cohomology and cohomology with supports in a closed subscheme, and what functorialities these satisfy, and how to define them from a sheaf theory pov and from a motivic stable homotopy pov.

I think Thomason states somewhere that one cannot in general define homology of a topos, only cohomology (???). What is (cf. Flat cohomology) Flat homology then in general, or Etale homology?

Ref: Kahn - Calculations in etale cohomology

An example: For “singular homology of varieties”, one constructs a complex $C(X)$. Taking the homology of $C(X) \otimes^L A$ gives singular homology with coeffs in $A$, and taking homology (?) of $RHom(C(X), A)$ gives singular cohomology with coeffs in $A$. This indicates a way of understanding the unification of geometric objects and coefficients.

nLab page on D25 The standard types of homology and cohomology