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I am looking for a reference (if it exists) that makes the link between cohomology of sheaves for sites and Galois cohomology :

quickly said, I would like to see Galois cohomology (at least in the commutative case) as the cohomology of a sheaf over the étale site of extensions of k.

By the way, what is a reference for cohomology of sites ?


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Milne's notes on étale cohomology. Tamme's book on étale cohomology. – Mariano Suárez-Alvarez Mar 1 '10 at 15:24
Both of Mariano's references are good. Also, there's plenty of lectures notes and unofficial write-ups of this material all over the web. For example, try googling "mcgill seminar on cohomology" (no quotes). – Cam McLeman Mar 1 '10 at 15:37
up vote 6 down vote accepted

The two references from my comment above, now with links!

  • Milne, James S. Étale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, N.J., 1980. xiii+323 pp. MR0559531 You can get another set of notes on étale cohomology from his web page: «in comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes».
  • Tamme, Günter. Introduction to étale cohomology. Translated from the German by Manfred Kolster. Universitext. Springer-Verlag, Berlin, 1994. x+186 pp. MR1317816
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I will also add:

E. Freitag, R. Kiehl: "Etale cohomology and the Weil conjectures"

There are also some notes for a course by de Jong given at Columbia University which can be found here:

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The notes aren't there, but I love the picture. – Tyler Lawson Mar 1 '10 at 18:10
The notes are there, actually. – José Figueroa-O'Farrill Mar 1 '10 at 18:48
You are correct. I am getting too old to notice links on a webpage. – Tyler Lawson Mar 1 '10 at 18:55
I didn't add a direct link to the notes so as a) people could see the photo b) I could also add SGA4 1/2 to the references :) – Frank Mar 1 '10 at 21:30

Barry Mazur has also written an article about this: Notes on étale cohomology of number fields. I hope this helps.

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A more permanent copy at numdam: – Regenbogen Mar 2 '10 at 0:00

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