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 ${\displaystyle {\mathcal {F}}^{\bullet }}$ a resolution of ${\mathcal {F}}$ 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 $q$-th sheaf cohomology group of $X$ 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.

This article incorporates material from De Rham–Weil theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.