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. Any Weil cohomology theory factors uniquely through the category of Chow motives, but the category of Chow motives itself is not a Weil cohomology theory, since it is not an abelian category.
Definition.
Fix a base field "k" of arbitrary characteristic and a "coefficient field" "K" of characteristic zero. A "Weil cohomology theory" is a contravariant functor 
satsfying the axioms below. For each smooth projective algebraic variety "X" of dimension "n" over "k", then the graded "K"-algebra 
is required to satisfy the following:
Examples.
There are four so-called classical Weil cohomology theories:
The proofs of the axioms for Betti cohomology and de Rham cohomology are comparatively easy and classical. For formula_17-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 2"n", so these higher cohomology groups vanish (for example by comparing them to simplicial (co)homology). 
The de Rham cycle map also has a down-to-earth explanation: Given a subvariety "Y" of complex codimension "r" in a complete variety "X" of complex dimension "n", the real dimension of "Y" is 2"n"−2"r", so one can integrate any differential (2"n"−2"r")-form along "Y" to produce a complex number. This induces a linear functional formula_20. By Poincaré duality, to give such a functional is equivalent to giving an element of formula_21; that element is the image of "Y" under the cycle map.