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.

See also

• de Rham theorem

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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