nLab
hypercohomology

Context

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Contents

Idea

Ordinary abelian sheaf cohomology is often considered exclusively with coefficients being Eilenberg-MacLane objects B nA\mathbf{B}^n A given by chain complexes concentrated in degree nn. If more generally any chain complex is allowed as a coefficient object, one speaks for emphasis of hyper-cohomology.

If abelian sheaf cohomology is thought of given by the derived functor of the global sections functor, then hypercohomology is given by the corresponding hyper-derived functor.

In the literature, hypercohomology is typically denoted by blackboard bold.

In terms of the general nPOV on cohomology (as described there) this just means the following:

in a given (∞,1)-topos H\mathbf{H} (which if we are thinking of describing abelian sheaf cohomology over some site is the (∞,1)-category of (∞,1)-sheaves on that site) “ordinary” cohomology in degree nn with coefficients in a group object A is just

H n(X,A):=π 0H(X,K(A,n)), H^n(X,A) := \pi_0 \mathbf{H}(X, K(A,n)) \,,

where K(A,n)=B nAK(A,n) = \mathbf{B}^n A is the Eilenberg-MacLane object with AA in degree nn. Typically this is the complex of sheaves [00A000][\cdots \to 0 \to 0 \to A \to 0 \to 0 \to \cdots \to 0] turned into a simplicial sheaf using the Dold-Kan correspondence

Ξ:Ch KanCplx \Xi : Ch_\bullet \to KanCplx

and then interpreted as an ∞-stack using the model structure on simplicial presheaves.

As the discussion at cohomology amplifies, this definition of cohomology in terms of the derived hom-space H(,)\mathbf{H}(-,-) depends in no way on the coefficient object being an Eilenberg-MacLane object. It could be any object. Only some familiar properties of cohomology (related to the notion of degree) do depend the coefficients being Eilenberg-MacLane objects.

We could take a completely arbitrary coefficient object KK and consider

H(X,K):=π 0H(X,K). H(X,K) := \pi_0 \mathbf{H}(X,K) \,.

This is then called nonabelian cohomology. The notion of hypercohomology lies in between Eilenberg-MacLane-type cohomology and fully general nonabelian cohomology. For hypercohomology we allow the coefficient object to be a general sheaf of chain complexes A =[A 2A 1A 0]A_\bullet = [\cdots \to A_2 \to A_1 \to A_0], or rather the simplicial presheaf ΞA \Xi A_\bullet represented by that. Then hypercohomology is

H(X,A ):=π 0H(X,ΞA ). H(X,A_\bullet) := \pi_0\mathbf{H}(X,\Xi A_\bullet) \,.

For a bit more on this see also the discussion at abelian sheaf cohomology.

Examples

  • Deligne cohomology is hypercohomology for complexes of sheaves of differential forms of the form

    [0C (,U(1))d dRlogΩ 1()d dRΩ 2()d dRΩ n1()d dRΩ n()]. [ \cdots \to 0 \to C^\infty(-,U(1)) \stackrel{d_{dR} log}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \Omega^{n-1}(-) \stackrel{d_{dR}}{\to} \Omega^n(-) ] \,.
  • In the context of algebraic geometry, “de Rham cohomology” refers to the hypercohomology of the de Rham complex Ω \Omega^\bullet. At least for nice varieties, say smooth? and proper, this “de Rham cohomology” agrees with the classical de Rham cohomology of the analytification. See this discussion on MathOverflow. Note that, in the algebraic category, the cohomology of the constant sheaf ̲\underline{\mathbb{C}} is not the right thing to consider, for constant sheaves on irreducible spaces are flasque?, hence acyclic?.

Revised on August 26, 2012 19:35:16 by Urs Schreiber (89.204.137.239)