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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. Weil cohomology theories play an important role in the theory of motives, insofar as the category of Chow motives is universal for Weil cohomology theories in the sense that any Weil cohomology theory factors through Chow motives. Note that, however, the category of Chow motives does not give a Weil cohomology theory since it is not abelian.

Definition


A Weil cohomology theory is a contravariant functor:

subject to the axioms below. Note that the field K is not to be confused with k; the former is a field of characteristic zero, called the coefficient field, whereas the base field k can be arbitrary. Suppose X is a smooth projective algebraic variety of dimension n, then the graded K-algebra

is subject to the following:

  • vanish for i < 0 or i > 2n.
  • is isomorphic to K (so-called orientation map).
  • There is a canonical Künneth isomorphism:
  • There is a cycle-map:
  • Weak Lefschetz axiom: For any smooth hyperplane section j: WX (i.e. W = XH, H some hyperplane in the ambient projective space), the maps:
  • Hard Lefschetz axiom: Let W be a hyperplane section and be its image under the cycle class map. The Lefschetz operator is defined as

Examples


There are four so-called classical Weil cohomology theories:

The proofs of the axioms in the case of Betti and de Rham cohomology are comparatively easy and classical, whereas for l-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 2n, so these higher cohomology groups vanish (for example by comparing them to simplicial (co)homology). The cycle map also has a down-to-earth explanation: given any (complex-)i-dimensional sub-variety of (the compact manifold) X of complex dimension n, one can integrate a differential (2n−i)-form along this sub-variety. The classical statement of Poincaré duality is, that this gives a non-degenerate pairing:

thus (via the comparison of de Rham cohomology and Betti cohomology) an isomorphism:

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