Weil cohomology

Cohomology of algebraic varieties with coefficients in a field of characteristic zero, with formal properties required to obtain the Lefschetz formula for the number of fixed points. The necessity for such a theory was pointed out by A. Weil [1], who showed that the rationality of the zeta-function and $L$- function of a variety over a finite field follow from the Lefschetz formula, whereas the remaining hypotheses about the zeta-function can naturally be formulated in cohomological terms. Let the variety $X$ be a projective smooth connected scheme over a fixed algebraically closed field $k$ and let $K$ be a field of characteristic zero. Then Weil cohomology with coefficient field $K$ is a contravariant functor $X \rightarrow H ^ {*} ( X)$ from the category of varieties into the category of finite-dimensional graded anti-commutative $K$- algebras, which satisfies the following conditions:

1) If $n= { \mathop{\rm dim} } ( X)$, then $H ^ {2n} ( X)$ is isomorphic to $K$, and the mapping

$$H ^ {i} ( X) \times H ^ {2n-i} ( X) \rightarrow H ^ {2n} ( X) ,$$

defined by the multiplication in $H ^ {*} ( X)$, is non-degenerate for all $i$;

2) $H ^ {*} ( X) \otimes _ {K} H ^ {*} ( Y) \widetilde \rightarrow H ^ {*} ( X \times Y)$( Künneth formula);

3) Mapping of cycles. There exists a functorial homomorphism $\gamma _ {X}$ from the group $C ^ {p} ( X)$ of algebraic cycles in $X$ of codimension $p$ into $H ^ {2p} ( X)$ which maps the direct product of cycles to the tensor product and is non-trivial in the sense that, for a point $P$, $\gamma _ {P}$ becomes the canonical imbedding of $\mathbf Z$ into $K$. The number

$$b _ {i} ( X) = \mathop{\rm dim} _ {K} H ^ {i} ( X)$$

is known as the $i$- th Betti number of the variety $X$.

Examples. If $k = \mathbf C$, classical cohomology of complex manifolds with coefficients in $\mathbf C$ is a Weil cohomology. If $l$ is a prime number distinct from the characteristic of the field $k$, then étale $l$- adic cohomology

$$X \mapsto \left [ \lim\limits _ {\\vec{nu} } H _ {et} ^ {*} ( X, \mathbf Z / l ^ \nu \mathbf Z ) \right ] \otimes _ {\mathbf Z _ {l} } \mathbf Q _ {l}$$

is a Weil cohomology with coefficients in the field $\mathbf Q _ {l}$.

The Lefschetz formula

$$\langle u \cdot \Delta \rangle = \sum_{i=0}^ { 2n } (- 1) ^ {i} \mathop{\rm Tr} ( u _ {i} )$$

is valid for Weil cohomology. In the above formula, $\langle u \cdot \Delta \rangle$ is the intersection index in $X \times X$ of the graph $\Gamma$ of the morphism $u : X \rightarrow X$ with the diagonal $\Delta \subset X \times X$, which may also be interpreted as the number of fixed points of the endomorphism $u$, while ${ \mathop{\rm Tr} } ( u _ {i} )$ is the trace of the endomorphism $u _ {i}$ which is induced by $u$ in $H ^ {i} ( X)$. Moreover, this formula is also valid for correspondences, i.e. elements $u \in H ^ {2n} ( X \times X)$.

References

[1]	A. Weil, "Numbers of solutions of equations in finite fields" Bull. Amer. Math. Soc. , 55 (1949) pp. 497–508 MR0029393 Zbl 0032.39402
[2]	S.L. Kleiman, "Algebraic cycles and the Weil conjectures" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.), Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 359–386 MR0292838 Zbl 0198.25902
[a1]	A. Grothendieck, "The cohomology theory of abstract algebraic varieties" J.A. Todd (ed.) , Proc. Internat. Congress Mathematicians (Edinburgh, 1958) , Cambridge Univ. Press (1960) pp. 103–118 MR0130879 Zbl 0119.36902
[a2]	A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie $\ell$-adique et fonctions $L$. SGA 5" , Lect. notes in math. , 589 , Springer (1977) MR491704
[a3]	J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) MR0559531 Zbl 0433.14012
[a4]	E. Freitag, R. Kiehl, "Étale cohomology and the Weil conjecture" , Springer (1988) MR0926276 Zbl 0643.14012
[a5]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 272 MR0463157 Zbl 0367.14001