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.

References

• De Rham, Georges (1931). Sur l'analysis situs des variétés à n dimensions. Thèses de l'entre-deux-guerres. Vol. 129.
• Samelson, Hans (1967). "On de Rham's theorem". Topology. 6 (4): 427–432. doi:10.1016/0040-9383(67)90002-X.
• Weil, André (1952). "Sur les théorèmes de de Rham". Commentarii Mathematici Helvetici. 26: 119–145. doi:10.1007/BF02564296. S2CID 124799328.

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