De Rham–Weil theorem explained

In algebraic topology, the De Rham–Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question.

Let

l F be a sheaf on a topological space X and l F ^{\bullet} a resolution of l F by acyclic sheaves. Then H ^{q(X,lF)} \cong H ^{q(lF} ^{\bullet(X)),}

where

H ^{q(X,lF)} denotes the q -th sheaf cohomology group of X with coefficients in l 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

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