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 $$\mathcal F$$ be a sheaf on a topological space $$X$$ and  $$\mathcal F^\bullet$$ a resolution of  $$\mathcal F$$ by acyclic sheaves. Then


 * $$ H^q(X,\mathcal F) \cong H^q(\mathcal F^\bullet(X)), $$

where $$H^q(X,\mathcal F)$$ denotes the $$q$$-th sheaf cohomology group of $$X$$ with coefficients in  $$\mathcal F.$$

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.