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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 [[motive (algebraic geometry)|motives]], insofar as the [[category (mathematics)|category]] of [[Chow motive]]s is a universal Weil cohomology theory in the sense that any Weil cohomology function factors through Chow motives. Note that, however, the category of Chow motives does not give a Weil cohomology theory since it is not [[abelian category|abelian]].
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| ==Definition==
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| A ''Weil cohomology theory'' is a [[functor#Covariance and contravariance|contravariant functor]]:
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| ::::''H<sup>*</sup>'': {smooth projective [[algebraic variety|varieties]] over a field ''k''} → {graded ''K''-algebras}
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| 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 variety|projective algebraic variety]] of dimension ''n'', then the [[graded algebra|graded]] ''[[K-algebra]]'' ''H<sup>*</sup>(X)'' = ⊕''H<sup>i</sup>(X)'' is subject to the following:
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| #''H<sup>i</sup>(X)'' are finite-dimensional ''K''-[[vector space]]s.
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| #''H<sup>i</sup>(X)'' vanish for ''i < 0'' or ''i > 2n''.
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| #''H<sup>2n</sup>(X)'' is isomorphic to ''K'' (so-called orientation map).
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| #There is a [[Poincaré duality]], i.e. a non-degenerate pairing: ''H<sup>i</sup>(X)'' × ''H<sup>2n−i</sup>(X) → H<sup>2n</sup>(X) ≅ K''.
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| #There is a canonical [[Künneth theorem|Künneth]] isomorphism: ''H<sup>*</sup>(X)'' ⊗ ''H<sup>*</sup>(Y)'' → ''H<sup>*</sup>(X × Y)''.
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| #There is a ''cycle-map'': γ<sub>''X''</sub>: ''Z<sup>i</sup>(X)'' → ''H<sup>2i</sup>(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''.
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| #''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<sup>*</sup>: H<sup>i</sup>(X)'' → ''H<sup>i</sup>(W)'' are isomorphisms for ''i ≤ n-2'' and a monomorphism for ''i ≤ n-1''.
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| #''Hard Lefschetz axiom'': Again let ''W'' be a hyperplane section and ''w'' = γ<sub>''X''</sub>(''W'') ∈ ''H''<sup>2</sup>''(X)''be its image under the cycle class map. The ''Lefschetz operator'' ''L: H<sup>i</sup>(X)'' → ''H<sup>i+2</sup>(X)'' maps ''x'' to ''x·w'' (the dot denotes the product in the algebra ''H<sup>*</sup>(X)''). The axiom states that ''L<sup>i</sup>: H<sup>n−i</sup>(X) → H<sup>n+i</sup>(X)'' is an isomorphism for ''i=1, ..., n''.
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| ==Examples==
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| There are four so-called classical Weil cohomology theories:
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| *[[Betti cohomology|singular (=Betti) cohomology]], regarding varieties over '''C''' as topological spaces using their [[analytic topology]] (see [[GAGA]])
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| *[[de Rham cohomology]] over a base field of [[characteristic (algebra)|characteristic]] zero: over '''C''' defined by [[differential forms]] and in general by means of the complex of Kähler differentials (see [[algebraic de Rham cohomology]])
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| *[[etale cohomology|l-adic cohomology]] for varieties over fields of characteristic different from ''l''
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| *[[crystalline cohomology]]
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| 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.
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| 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 homology|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:
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| ::<math>H_i (X) \otimes H_{\text{dR}}^{2n-i}(X) \rightarrow \mathbf{C}</math>,
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| thus (via the comparison of de Rham cohomology and Betti cohomology) an isomorphism:
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| ::<math>H_i (X) \cong H_{\text{dR}}^{2n-i}(X)^{\vee} \cong H^i(X).</math>
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| ==References==
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| * {{Citation | last1=Griffiths | first1=Phillip | last2=Harris | first2=Joseph | title=Principles of algebraic geometry | publisher=Wiley | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | id={{MathSciNet | id = 1288523}} | year=1994}} (contains proofs of all of the axioms for Betti and de-Rham cohomology)
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| * {{Citation | last1=Milne | first1=James S. | title=Étale cohomology | publisher=[[Princeton University Press]] | location=Princeton, NJ | isbn=978-0-691-08238-7 | year=1980}} (idem for ''l''-adic cohomology)
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| * {{Citation | last1=Kleiman | first1=S. L. | title=Dix exposés sur la cohomologie des schémas | publisher=North-Holland | location=Amsterdam | id={{MathSciNet | id = 0292838}} | year=1968 | chapter=Algebraic cycles and the Weil conjectures | pages=359–386}}
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| [[Category:Topological methods of algebraic geometry]]
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| [[Category:Cohomology theories]]
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