Holmstrom Hypercohomology

Hypercohomology

Example: Beilinson has given conjectures for versions of universal cohomology which woould arise as hypercohomology in the Zariski topology of certain complexes of sheaves.

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Hypercohomology

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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Hypercohomology

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Hypercohomology

CT (Category theory), AG (Algebraic geometry)

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Hypercohomology

Cat

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Hypercohomology

Lubkin has a section on relative hypercohomology.

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Hypercohomology

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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Hypercohomology

For hypercohomology of complexes not bounded from below, see Voevodsky and Suslin: Bloch-Kato conj and motivic cohom 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.

nLab page on Hypercohomology