Hypercohomology

General Introduction
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Online References
For hypercohomology of complexes not bounded from below, see Voevodsky and Suslin: BlochKato conj and motivic cohomology with finite coeffs, file in Voevodsky folder, pp 4.
For hypercohomology of pointed simplicial sheaves, see Appendix 1 of Voevodsky: Motivic cohomology with Z/2 coeffs, published version.
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Paper References
Lubkin has a section on relative hypercohomology.
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Definition
Ref: Discussion with Scholl, Oct 2007.
To compute ordinary Sheaf cohomology of a sheaf on some space/scheme (in general a site with as the final object, I guess), one takes an injective resolution of the sheaf, , applies the global section functor to get a complex of abelian groups, and then takes the cohomology of that complex. One shows that a homotopy equivalence between injective resolutions gives same cohomology.
Let be a complex of sheaves. Then we can find a quasiisomorphism of complexes to a complex of injectives. (And also, I think, for two such quasiIMs, there is a homotopy equivalence between the two complexes of injectives, making the whole thing commute.
Now the hypercohomology of the complex is defined by taking the cohomology of the global sections complex of this complex of injectives. Note that we get back usual sheaf cohomology as a special case: And injective resolution is nothing but a quasiisomorphism from the complex .
Memo ("apply blindly"): ("anyone who uses another convention should be shot"). Similarly, .
For any complex of abelian groups , the cohomology is the homotopy classes of maps from to .
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Verdier: Topologie sur les espaces de cohomologie d’un complexe de faisceaux analytiques a cohomologie coherente. Computes hypercohomology (also with compact supports) by constructing a complex of Frechet nuclear spaces with the right topology. This is done in a way that is functorial in both and .
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Other Information
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