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
H ∗ : { smooth projective varieties over k } ⟶ { graded K -algebras } {\displaystyle H^{*}:\{{\text{smooth projective varieties over }}k\}\longrightarrow \{{\text{graded }}K{\text{-algebras}}\}}
satisfying the axioms below. For each smooth projective algebraic variety X of dimension n over k, the graded K-algebra
H ∗ ( X ) = ⨁ i H i ( X ) {\displaystyle H^{*}(X)=\bigoplus \nolimits _{i}H^{i}(X)}
is required to satisfy the following:
H i ( X ) {\displaystyle H^{i}(X)} is a finite-dimensional K-vector space for each integer i.
H i ( X ) = 0 {\displaystyle H^{i}(X)=0} for each i < 0 or i > 2n.
H 2 n ( X ) {\displaystyle H^{2n}(X)} is isomorphic to K (the so-called orientation map). Poincaré duality: there is a perfect pairing H i ( X ) × H 2 n − i ( X ) → H 2 n ( X ) ≅ K . {\displaystyle H^{i}(X)\times H^{2n-i}(X)\to H^{2n}(X)\cong K.}
There is a canonical Künneth isomorphism H ∗ ( X ) ⊗ H ∗ ( Y ) → H ∗ ( X × Y ) . {\displaystyle H^{*}(X)\otimes H^{*}(Y)\to H^{*}(X\times Y).}
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