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

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

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

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.