Weil cohomology theory

In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil. Weil cohomology theories play an important role in the theory of motives, insofar as the category of Chow motives is universal for Weil cohomology theories in the sense that any Weil cohomology theory factors through Chow motives. Note that, however, the category of Chow motives does not give a Weil cohomology theory since it is not abelian.

Definition

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.

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)

• 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:

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

References

• 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