- Weil cohomology theory
In

algebraic geometry , a subfield ofmathematics , a**Weil cohomology**or**Weil cohomology theory**is a cohomology satisfying certain axioms concerning the interplay ofalgebraic cycles and cohomology groups. The name is in honour ofAndré Weil . Weil cohomology theories play an important role in the theory of motives, insofar as the category of Chow motives is a universal Weil cohomology theory.**Definition**A "Weil cohomology theory" is a contravariant

functor "H

^{*}": {"smooth projective varieties" over a field "k"} → {"graded K-algebras"}subject to the following axioms:

*"H^{n}(X)" vanishes for "n" negative or "n > 2 · dim X". Every "H^{n}(X)" is a finite-dimensional "K"-vector space .

*"H^{2r}(X)" is isomorphic to "K" (so-called orientation map)

*There is aPoincaré duality , i.e. a non-degenerate pairing:"H^{i}(X)" × "H^{2r-i}(X) → H^{2r}(X) ≅ K"

*There is a canonical Künneth isomorphism :"H^{*}(X)" ⊗ "H^{*}(Y)" → "H^{*}(X × Y) ≅ K"

*There is a "cycle-map":γ_{"X"}: "Z^{i}(X)" → "H^{2i}(X)", where the former group means algebraic cycles of codimension "i", satisfying certain compatibility conditions with respect to functoriality of "H", the Künneth isomorphism and such that for "X" a point, the cycle map is the inclusion ℤ ⊂ "K".

*"Weak Lefschetz axiom": For any smooth hyperplane section "i: W ⊂ X" (i.e. "W = X ∩ H", "H" some hyperplane in the ambient projective space), the maps "i^{*}: H^{n}(X)" → "H^{n}(W)" are isomorphisms for "n ≤ dim X-2" and a monomorphism for "n ≤ dim X-1".

*"Hard Lefschetz axiom": Again let "W" be a hyperplane section and "w = γ_{"X"}(W) ∈ H^{2}(X)" its image under the cycle class map. The "Lefschetz operator" "L : H^{n}(X)" → "H^{n+2}(X)" maps "x" to "x·w" (the dot denotes the product in the algebra "H^{*}(X)". The axiom is that:L^{dim X-n}: H^{n}(X) → H^{2 dim X-n}(X)is an isomorphism ("n ≤ dim X").The field "K" is not to be confused with "k"; the former is a field of characteristic zero, called the "coefficient field", whereas the base field "k" can be arbitrary.

**Examples**There are four so-called classical Weil cohomology theories:

*singular (=Betti) cohomology, regarding varieties over ℂ as topological spaces using their analytic topology (seeGAGA )

*de Rham cohomology over a base field of characteristic zero: over ℂ defined bydifferential forms and in general by means of the complex of Kähler differentials (seealgebraic 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 "r" has real dimension "2r", 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-)"n"-dimensional subvariety of (the compact manifold) "X" of complex dimension "r", one can integrate a differential ("2r-n")-form along this subvariety. The classical statement of Poincaré duality is, that this gives a non-degenerate pairing:"H

_{n}(X)" ⊗ "H_{dR}^{2r-n}(X)" → ℂ,thus (via the comparison of de Rham cohomology and Betti cohomology) an isomorphism, "H_{n}(X) ≅ H_{dR}^{2r-n}(X)^{∨}≅ H^{n}(X)".**References*** | year=1994 (contains proofs of all of the axioms for Betti and de-Rham cohomology)

* (idem for "l"-adic cohomology)

* | year=1968 | chapter=Algebraic cycles and the Weil conjectures | pages=359–386

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