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

$H^q(X,{ℱ})\cong H^q({{ℱ}}^{\bullet }(X)),$

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

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.