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
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[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