Examples
There are four so-called classical Weil cohomology theories:
- singular (=Betti) cohomology, regarding varieties over C as topological spaces using their analytic topology (see GAGA)
- de Rham cohomology over a base field of characteristic zero: over C defined by differential forms and in general by means of the complex of Kähler differentials (see algebraic de Rham cohomology)
- l-adic cohomology for varieties over fields of characteristic different from l
- crystalline cohomology
The proofs of the axioms in the case of Betti and de Rham cohomology are comparatively easy and classical, whereas for l-adic cohomology, for example, most of the above properties are deep theorems.
The vanishing of Betti cohomology groups exceeding twice the dimension is clear from the fact that a (complex) manifold of complex dimension n has real dimension 2n, so these higher cohomology groups vanish (for example by comparing them to simplicial (co)homology). The cycle map also has a down-to-earth explanation: given any (complex-)i-dimensional sub-variety of (the compact manifold) X of complex dimension n, one can integrate a differential (2n−i)-form along this sub-variety. The classical statement of Poincaré duality is, that this gives a non-degenerate pairing:
-
- ,
thus (via the comparison of de Rham cohomology and Betti cohomology) an isomorphism:
Read more about this topic: Weil Cohomology Theory
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