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

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

satisfying the axioms below. For each smooth projective algebraic variety X of dimension n over k, then the graded K-algebra

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

is required to satisfy the following:

• ${\displaystyle H^{i}(X)}$ is a finite-dimensional K-vector space for each integer i.

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

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

• Poincaré duality: there is a perfect pairing

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

• There is a canonical Künneth isomorphism

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

• For each integer r, there is a cycle map defined on the group ${\displaystyle Z^{r}(X)}$ of algebraic cycles of codimension r on X,

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

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

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

are isomorphisms for ${\displaystyle i\leqslant n-2}$ and injections 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),

• ${\displaystyle \ell }$-adic cohomology for varieties over fields of characteristic different from ${\displaystyle \ell }$,

• crystalline cohomology.

The proofs of the axioms for Betti cohomology and de Rham cohomology are comparatively easy and classical. For ${\displaystyle \ell }$-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 ${\displaystyle \textstyle \int _{Y}\colon \;H_{\text{dR}}^{2n-2r}(X)\to \mathbf {C} }$. By Poincaré duality, to give such a functional is equivalent to giving an element of ${\displaystyle H_{\text{dR}}^{2r}(X)}$; that element is the image of Y under the cycle map.

• 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