Weil cohomology theory

Fix a base field k of arbitrary characteristic and a "coefficient field" K of characteristic zero. A Weil cohomology theory is a contravariant functor H^*: \{\text{smooth projective varieties over } k \} \longrightarrow \{\text{graded } K\text{-algebras}\} satisfying the axioms below. For each smooth projective algebraic variety X of dimension n over k, then the graded K-algebra H^*(X) = \bigoplus\nolimits_i H^i(X) is required to satisfy the following: • H^i(X) is a finite-dimensional K-vector space for each integer i. • H^i(X) = 0 for each i 2n. • H^{2n}(X) is isomorphic to K (the so-called orientation map). • Poincaré duality: there is a perfect pairing H^i(X) \times H^{2n-i}(X) \to H^{2n}(X) \cong K. • There is a canonical Künneth isomorphism H^*(X) \otimes H^*(Y) \to H^*(X\times Y). • For each integer r, there is a cycle map defined on the group Z^r(X) of algebraic cycles of codimension r on X, \gamma_X : Z^r(X) \to H^{2r}(X), satisfying certain compatibility conditions with respect to functoriality of H and the Künneth isomorphism. If X is a point, the cycle map is required to be the inclusion Z ⊂ K. • Weak Lefschetz axiom: For any smooth hyperplane section j: W ⊂ X (i.e. W = X ∩ H, H some hyperplane in the ambient projective space), the maps j^*: H^i(X) \to H^i(W) are isomorphisms for i \leqslant n-2 and injections for i \leqslant n-1. • Hard Lefschetz axiom: Let W be a hyperplane section and w =\gamma_X(W) \in H^2(X) be its image under the cycle class map. The Lefschetz operator is defined as \begin{cases} L: H^i(X) \to H^{i+2}(X) \\ x \mapsto x \cdot w, \end{cases} where the dot denotes the product in the algebra H^*(X). Then L^i : H^{n-i}(X) \to H^{n+i}(X) is an isomorphism for i = 1, ..., n. ==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), • \ell-adic cohomology for varieties over fields of characteristic different from \ell, • crystalline cohomology. The proofs of the axioms for Betti cohomology and de Rham cohomology are comparatively easy and classical. For \ell-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 de Rham cycle map also has a down-to-earth explanation: Given a subvariety Y of complex codimension r in a complete variety X of complex dimension n, the real dimension of Y is 2n−2r, so one can integrate any differential (2n−2r)-form along Y to produce a complex number. This induces a linear functional \textstyle\int_Y \colon \; H^{2n-2r}_{\text{dR}}(X) \to \mathbf{C}. By Poincaré duality, to give such a functional is equivalent to giving an element of H^{2r}_{\text{dR}}(X); that element is the image of Y under the cycle map. == See also ==