Hypercohomology

• General Introduction

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• Search results

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• Online References

  For hypercohomology of complexes not bounded from below, see Voevodsky and Suslin: Bloch-Kato 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 F on some space/scheme X (in general a site with X as the final object, I guess), one takes an injective resolution of the sheaf, F \mapsto I^{\bullet}, 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 F^0 \to F^1 \to \ldots be a complex of sheaves. Then we can find a quasi-isomorphism of complexes F^{\bullet} \to I^{\bullet} to a complex of injectives. (And also, I think, for two such quasi-IMs, there is a homotopy equivalence between the two complexes of injectives, making the whole thing commute.

  Now the hypercohomology of the complex F^{\bullet} 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 quasi-isomorphism from the complex F \to 0 \to 0 \to \ldots.

  Memo ("apply blindly"): H^{i+n}(F^{\bullet}) = H^i(F^{\bullet}[n]) ("anyone who uses another convention should be shot"). Similarly, L(M(n), s) = L(M, s+n).

  For any complex of abelian groups A^{\bullet}, the cohomology H^n(A) is the homotopy classes of maps from \mathbb{Z}[-n] to A^{\bullet}.

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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 X and F.

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• Other Information

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• General Introduction
• Search results
• Online References
• Paper References
• Definition
• Properties
• Standard theorems
• Open Problems
• Connections to Number Theory
• Computations and Examples
• History and Applications
• Some Research Articles
• Other Information
• Comments Posted
• Comments