Holmstrom Cohomological descent

Weibel in Thomason’s obituary

Weibel’s Thomason obituary sketches the following things: For a diagram of spectra indexed by a small category J, the levelwise holims turn out to assemble to a spectrum, called the hypercohomology. Special cases: Hypercohomology of a complex of sheaves, homotopy fixed point spectrum. Explanation of Thomason’s homotopy limit problem (special cases being the Segal conjecture and the Quillen-Lichtenbaum conjecture). For a presheaf $F$, with values in spectra, we get the hypercohomology spectrum together with an augmentation map $F(X) \to \mathbb{H}^{\bullet}(X ; F)$. Cohomological descent means that for every open $U$, the augmentation is a weak equivalence. Cohomological descent implies Mayer-Vietoris (the M-V square is homotopy cartesian), and for a Noetherian scheme of finite Krull dimension, cohomological descent for the Zariski topology is equivalent to M-V. There is also the hypercohomology spectral sequence, and special cases of this are explained to some extent.

At least for a scheme which is quasiprojective over a finite-dimensional Noetherian ring, we have the presheaf of Quillen K-theory spectra, and hence we can construct the corresponding Zariski and etale hypercohomology spectra. For $X$ regular, the Quillen K-theory spectrum has CD for the Zariski topology. For nonregular X, there is a fringing effect related to Bass K-groups which means that the Quillen K-theory spectrum cannot have CD, but it is the (-1)-connected cover of another K-theory spectrum (Bass?) which does have cohomological descent for the Zariski topology/ (still for any X q-proj over a fin-dim Noeth ring).

Can also consider K-theory with finite coefficients; it the homotopy groups of a certain spectrum, which can be described in many ways, for example as the smash product with the mod m Moore spectrum. Such a smash prod preserves cd, so the mod m spectrum also has Zariski cd as above.

More about the Quillen-Lichtenbaum conjecture: $(K/m)[\beta^{-1}]$ satisfies etale cohomological descent under some hypotheses. Comparison between K-theory and etale K-theory, see article for details.

nLab page on Cohomological descent