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单词 elladicetaleCohomology
释义

-adic étale cohomology


Let X be a scheme over a field k having algebraic closureMathworldPlanetmath k¯.Let (Xkk¯)ét be the small étale site on Xkk¯,and let /ln denote the sheaf on (Xkk¯)ét associated to the group scheme /ln for some fixed prime l.Finally, let Γ be the global sections functor on the category of étale sheaves on (Xkk¯)ét.

The l-adic étale cohomology of X is

Héti(X,l)=lllimn(RiΓ)(/ln),.

where Ri denotes taking the i-th right-derived functor.

This apparently appalling definition is necessary to ensure that (for l not equal to the characteristicPlanetmathPlanetmath of k) étale cohomology is the appropriate generalization of de Rham cohomologyMathworldPlanetmath on a complex manifold.For example, on a scheme of dimension n, the cohomology groupsPlanetmathPlanetmath Hi vanish for i>2n and we have a version of Poincaré duality.Grothendieck introduced étale cohomology as a tool to prove the Weil conjectures, and indeed it is what Deligne used to prove them.

These references are approximately in order of difficulty and of generality and precision.

References

  • 1 J. S. Milne, Lectures on Étale Cohomology, 1998, available on the web at http://www.jmilne.org/math/http://www.jmilne.org/math/
  • 2 James S. Milne, Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton N.J., 1980
  • 3 Deligne et al., Séminaires en Gèometrie Algèbrique 412, available on the web athttp://www.math.mcgill.ca/ archibal/SGA/SGA.htmlhttp://www.math.mcgill.ca/ archibal/SGA/SGA.html
  • 4 Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3, available on the web athttp://www.math.mcgill.ca/ archibal/SGA/SGA.htmlhttp://www.math.mcgill.ca/ archibal/SGA/SGA.html
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