-adic étale cohomology
Let be a scheme over a field having algebraic closure .Let be the small étale site on ,and let denote the sheaf on associated to the group scheme for some fixed prime .Finally, let be the global sections functor on the category of étale sheaves on .
The -adic étale cohomology of is
where denotes taking the -th right-derived functor.
This apparently appalling definition is necessary to ensure that (for not equal to the characteristic of ) étale cohomology is the appropriate generalization of de Rham cohomology
on a complex manifold.For example, on a scheme of dimension , the cohomology groups
vanish for 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 4, 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