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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: ::::''H *'': → 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'' ''H *(X)'' = ⊕''Hi(X)'' is subject to the following: #''Hi(X)'' are finite-dimensional ''K''-vector spaces. #''Hi(X)'' vanish for ''i < 0'' or ''i > 2n''. #''H2n(X)'' is isomorphic to ''K'' (so-called orientation map). #There is a Poincaré duality, i.e. a non-degenerate pairing: ''Hi(X)'' × ''H2n−i(X) → H2n(X) ≅ K''. #There is a canonical Künneth isomorphism: ''H *(X)'' ⊗ ''H *(Y)'' → ''H *(X × Y)''. #There is a ''cycle-map'': γ''X'': ''Zi(X)'' → ''H2i(X)'', where the former group means algebraic cycles of codimension ''i'', satisfying certain compatibility conditions with respect to functionality of ''H'', the Künneth isomorphism and such that for ''X'' a point, the cycle map is the inclusion Z ⊂ ''K''. #''Weak Lefschetz axiom'': For any smooth hyperplane section ''j: W ⊂ X'' (i.e. ''W = X ∩ H'', ''H'' some hyperplane in the ambient projective space), the maps ''j *: Hi(X)'' → ''Hi(W)'' are isomorphisms for ''i ≤ n-2'' and a monomorphism for ''i ≤ n-1''. #''Hard Lefschetz axiom'': Again let ''W'' be a hyperplane section and ''w'' = γ''X''(''W'') ∈ ''H''2''(X)''be its image under the cycle class map. The ''Lefschetz operator'' ''L: Hi(X)'' → ''Hi+2(X)'' maps ''x'' to ''x·w'' (the dot denotes the product in the algebra ''H *(X)''). The axiom states that ''Li: Hn−i(X) → Hn+i(X)'' is an isomorphism for ''i=1, ..., n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Weil cohomology theory」の詳細全文を読む スポンサード リンク
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