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
-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
.