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.

See also

• de Rham theorem

References

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