Weil Cohomology Theory - Definition

Definition

A Weil cohomology theory is a contravariant functor:

H*: {smooth projective varieties over a field k} → {graded K-algebras}

subject to the axioms below. Note that 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. Suppose X is a smooth projective algebraic variety of dimension n, then the graded K-algebra H*(X) = ⊕Hi(X) is subject to the following:

1. Hi(X) are finite-dimensional K-vector spaces.
2. Hi(X) vanish for i < 0 or i > 2n.
3. H2n(X) is isomorphic to K (so-called orientation map).
4. There is a Poincaré duality, i.e. a non-degenerate pairing: Hi(X) × H2n−i(X) → H2n(X) ≅ K.
5. There is a canonical Künneth isomorphism: H*(X) ⊗ H*(Y) → H*(X × Y).
6. There is a cycle-map: γ_X: Zi(X) → H2i(X), where the former group means algebraic cycles of codimension i, satisfying certain compatibility conditions with respect to functionality of H, the Künneth isomorphism and such that for X a point, the cycle map is the inclusion Z ⊂ K.
7. 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*: Hi(X) → Hi(W) are isomorphisms for i ≤ n-2 and a monomorphism for i ≤ n-1.
8. Hard Lefschetz axiom: Again let W be a hyperplane section and w = γ_X(W) ∈ H2(X)be its image under the cycle class map. The Lefschetz operator L: Hi(X) → Hi+2(X) maps x to x·w (the dot denotes the product in the algebra H*(X)). The axiom states that Li: Hn−i(X) → Hn+i(X) is an isomorphism for i=1, ..., n.