Holmstrom C20 Sheaf cohomology and Cech cohomology

Examples of hypercohomology should include the logarithmic de Rham complex in Hodge III. Recall that there is a canonical iso $H^*(U, \mathbf{C} ) = \mathbf{H}^*(X, \Omega^{\bullet}_X(\log Y ) )$ where $U$ (smooth) is embedded into $X$ (smooth) and $Y$ is the complement, a NCD. This iso also gives a Hodge filtration on cohomology of $U$. Check source to get details right.

Generalized sheaf cohomology, mention here and treat in detail under simplicial techniques.

Verdier’s hypercovering theorem: “Sheaf cohomology of a site can computed as the colimit of the Cech cohomology of all hypercoverings”. See end of Barnea-Schlank for references and a nice proof.

Barnea-Schlank (v6) has a discussion on page 59, with references, on how abelian sheaf cohomology is a Hom group (using Jardine’s model structure on simplicial presheaves). The formula is

$H^n(C, \tilde{F}) \cong [*, K(F, n)]$

See B-S for more discussion!!! They also discuss the simplicial sheaf/hypercovering understanding of what Cech cohomology really is.

Cech cohomology Hypercohomology

Sheaf cohomology theories

These can take as input any complex of sheaves, for example versions of motivic complexes: Zariski cohomology, Etale cohomology, Cohomology for the h-topology and the qfh-topology, Nisnevich cohomology, cdh-cohomology, Flat cohomology and maybe Flat homology, Infinitesimal cohomology. See also Sheaf cohomology and Cohomology with compact supports, Coherent cohomology

Discuss maybe generalized sheaf cohomology and the nLab perspective

Generalized sheaf cohomology,

Discussion of why sheaf cohomology is more important in algebraic geometry than in differential geometry: http://mathoverflow.net/questions/17545/sheaves-and-bundles-in-differential-geometry

nLab page on C20 Sheaf cohomology and Cech cohomology