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

{{cite book |url=http://www.numdam.org/item/THESE_1931__129__1_0/|title=Sur l'analysis situs des variétés à n dimensions|series=Thèses de l'entre-deux-guerres |year=1931|volume= 129 |last1=De Rham |first1=Georges|author1-link=Georges de Rham }}
{{cite journal |doi=10.1016/0040-9383(67)90002-X|title=On de Rham's theorem |year=1967 |last1=Samelson |first1=Hans|author1-link=Hans Samelson |journal=Topology |volume=6 |issue=4 |pages=427–432 |doi-access=free }}
{{cite journal |doi=10.1007/BF02564296|title=Sur les théorèmes de de Rham |year=1952 |last1=Weil |first1=André |author-link = André Weil|journal=Commentarii Mathematici Helvetici |volume=26 |pages=119–145 |s2cid=124799328 }}

Category:Homological algebra

Category:Sheaf theory

Category:Theorems in algebraic geometry

De Rham–Weil theorem

See also

References