De Rham–Weil theorem

In algebraic topology, the De Rham–Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question.

Let [math]\displaystyle{ \mathcal F }[/math] be a sheaf on a topological space [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \mathcal F^\bullet }[/math] a resolution of [math]\displaystyle{ \mathcal F }[/math] by acyclic sheaves. Then

[math]\displaystyle{ H^q(X,\mathcal F) \cong H^q(\mathcal F^\bullet(X)), }[/math]

where [math]\displaystyle{ H^q(X,\mathcal F) }[/math] denotes the [math]\displaystyle{ q }[/math]-th sheaf cohomology group of [math]\displaystyle{ X }[/math] with coefficients in [math]\displaystyle{ \mathcal F. }[/math]

The De Rham–Weil theorem follows from the more general fact that derived functors may be computed using acyclic resolutions instead of simply injective resolutions.