nLab
hypercohomology

Context

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