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 , who showed that the rationality of the zeta-function and -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 be a projective smooth connected scheme over a fixed algebraically closed field and let be a field of characteristic zero. Then Weil cohomology with coefficient field is a contravariant functor from the category of varieties into the category of finite-dimensional graded anti-commutative -algebras, which satisfies the following conditions:
1) If , then is isomorphic to , and the mapping
defined by the multiplication in , is non-degenerate for all ;
2) (Künneth formula);
3) Mapping of cycles. There exists a functorial homomorphism from the group of algebraic cycles in of codimension into which maps the direct product of cycles to the tensor product and is non-trivial in the sense that, for a point , becomes the canonical imbedding of into . The number
is known as the -th Betti number of the variety .
Examples. If , classical cohomology of complex manifolds with coefficients in is a Weil cohomology. If is a prime number distinct from the characteristic of the field , then étale -adic cohomology
is a Weil cohomology with coefficients in the field .
The Lefschetz formula
is valid for Weil cohomology. In the above formula, is the intersection index in of the graph of the morphism with the diagonal , which may also be interpreted as the number of fixed points of the endomorphism , while is the trace of the endomorphism which is induced by in . Moreover, this formula is also valid for correspondences, i.e. elements .
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Weil cohomology. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Weil_cohomology&oldid=24012