Weil cohomology theory

A Weil cohomology theory is a contravariant functor:

${\displaystyle H^{*}:\{{\text{smooth projective varieties over a field }}k\}\longrightarrow \{{\text{graded }}K{\text{-algebras}}\},}$

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

${\displaystyle H^{*}(X)=\bigoplus \nolimits _{i}H^{i}(X)}$

is subject to the following:

• ${\displaystyle H^{i}(X)}$ are finite-dimensional K-vector spaces.

• ${\displaystyle H^{i}(X)}$ vanish for i < 0 or i > 2n.

• ${\displaystyle H^{2n}(X)}$ is isomorphic to K (so-called orientation map).

• There is Poincaré duality, i.e. a non-degenerate pairing:

${\displaystyle H^{i}(X)\times H^{2n-i}(X)\to H^{2n}(X)\cong K.}$

• There is a canonical Künneth isomorphism:

${\displaystyle H^{*}(X)\otimes H^{*}(Y)\to H^{*}(X\times Y).}$

• There is a cycle-map:

${\displaystyle \gamma _{X}:Z^{i}(X)\to H^{2i}(X),}$

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 j: W ⊂ X (i.e. W = X ∩ H, H some hyperplane in the ambient projective space), the maps:

${\displaystyle j^{*}:H^{i}(X)\to H^{i}(W),}$

are isomorphisms for ${\displaystyle i\leqslant n-2}$ and a monomorphism for ${\displaystyle i\leqslant n-1.}$

• Hard Lefschetz axiom: Let W be a hyperplane section and ${\displaystyle w=\gamma _{X}(W)\in H^{2}(X)}$ be its image under the cycle class map. The Lefschetz operator is defined as

${\displaystyle {\begin{cases}L:H^{i}(X)\to H^{i+2}(X)\\x\mapsto x\cdot w\end{cases}}}$

where the dot denotes the product in the algebra ${\displaystyle H^{*}(X).}$ Then

${\displaystyle L^{i}:H^{n-i}(X)\to H^{n+i}(X)}$

is an isomorphism for i = 1, ..., n.

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:

${\displaystyle H_{i}(X)\otimes H_{\text{dR}}^{2n-i}(X)\to \mathbf {C} ,}$

thus (via the comparison of de Rham cohomology and Betti cohomology) an isomorphism:

${\displaystyle H_{i}(X)\cong H_{\text{dR}}^{2n-i}(X)^{\vee }\cong H^{i}(X).}$

• Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: Wiley, doi:10.1002/9781118032527, ISBN 978-0-471-05059-9, MR 1288523 (contains proofs of all of the axioms for Betti and de-Rham cohomology)
• Milne, James S. (1980), Étale cohomology, Princeton, NJ: Princeton University Press, ISBN 978-0-691-08238-7 (idem for l-adic cohomology)
• Kleiman, S. L. (1968), "Algebraic cycles and the Weil conjectures", Dix exposés sur la cohomologie des schémas, Amsterdam: North-Holland, pp. 359–386, MR 0292838