What is the best textbook (or book) for studying Etale cohomology?

7$\begingroup$ FrietagKiehl, "Etale Cohomology and the Weil Conjectures" is pretty nice. As is Tamme, "Introduction to Etale Cohomology". Both are out of print. Milne's "Etale Cohomology" is in print, but I prefer his notes: jmilne.org/math/CourseNotes/lec.html $\endgroup$– B RNov 10, 2011 at 21:04

3$\begingroup$ Also, SGA 4.5, which is once again available. $\endgroup$– B RNov 10, 2011 at 21:09

2$\begingroup$ There's a very nice bird's eye view in V.I. Danilov, "Cohomology of algebraic varieties" MR1392958. $\endgroup$– Dan PetersenNov 10, 2011 at 21:28

2$\begingroup$ Ali, I don't think there is a "Royal Road" to etale cohomology. If you have easy access to SGA 4.5, try it out. Maybe use it in conjunction with Milne's notes (and/or book) for things you don't understand. If you can get access to one of the other books, even better! Why choose just one? $\endgroup$– B RNov 10, 2011 at 21:40

3$\begingroup$ Artin, Michael (1962), Grothendieck topologies, Harvard University, Dept. of Mathematics $\endgroup$– NielsNov 11, 2011 at 9:20
7 Answers
Not a textbook, but a free PDF by J.S. Milne, http://www.jmilne.org/math/CourseNotes/LEC.pdf, pretty good IMHO.

$\begingroup$ Thank you. I think I have it. Do you think its one of the best? $\endgroup$– user16974Nov 10, 2011 at 21:03

$\begingroup$ I've not read dozens, but this is a good one, for sure. $\endgroup$– PortlandNov 10, 2011 at 21:25

$\begingroup$ It might be worth pointing out that Milne's notes are done using varieties, while his book is done using schemes. Of course, a lot of the general stuff about sites, sheaves, etc., is the same either way. $\endgroup$ Nov 10, 2011 at 21:34

4$\begingroup$ Both his course notes and his text book are good, but they are quite different. The book is more about the technical details whereas the notes give a very good overview with many examples, culminating in a proof of the Weil conjectures. Generally speaking, I strongly recommend Milne's course notes on any topic. Not that I have read all of them, but it happened more than once to me (which implies I'm a bad learner) that I looked into some standard textbook in order to learn some topic and only realized later that Milne's notes were way better. $\endgroup$ Nov 10, 2011 at 22:09
I'll complement the list of well known books on the subject by some freely available documents, which I find userfriendly.
Here are great lecture notes , from a course that de Jong (of Stacks Project fame) gave in 2009.
Edgar José Martins Dias Costa's short dissertation on the subject .
Evan Jenkins's notes of a seminar on étale cohomology (click on the pdf icons).
The arXiv notes of a minicourse given by a fine expositor, Antoine Ducros, which also cover analytical aspects of étale cohomology (used for Berkovich spaces).
And finally a historic survey (in French unfortunately) on the genesis and successes of étale cohomology.
It was written by Illusie, one of Grothendieck's most brilliant students, who acknowledges the help he received in his reminiscences from luminaries such as Serre and Deligne.

6$\begingroup$ Dear Georges, Thanks especially for the link to the Illusie notes. (And I don't think there's any need to apologize for notes in French.) Best wishes, Matthew $\endgroup$– EmertonNov 11, 2011 at 14:06

$\begingroup$ There are also another Luc Illusie's great lecture notes: Old and new in étale cohomology $\endgroup$– tttbaseJan 7, 2017 at 15:45
Lei Fu, Étale Cohomology Theory is also nice and has not been mentioned yet.
And the lecture notes of Alexander Schmidt: http://theorics.yichuanshen.de/etalekohomologie/ (unfortunately in German)
My first exposure to étale cohomology was through Bjorn Poonen's notes Rational Points on Varieties, Ch. 6. Not all of the big theorems are mentioned there, but it provides a great introduction to those who have had no previous dealings with the subject.

1$\begingroup$ +1: very nice, unexpected reference. $\endgroup$ Jan 4, 2014 at 21:19
I would highly recommend these notes by Donu Arapura for a good overview of etale cohomology, as well as this short paper by Tom Sutherland for an even quicker overview.

2$\begingroup$ Thanks, but these are quite rough, so caveat lector. $\endgroup$ Sep 26, 2012 at 10:58
In the web page of Uwe Jannsen there are great lecture notes of (étale cohomology) courses. In particular:
Sommersemester 2015: Étale Kohomologie (Eng).

1$\begingroup$ Also his notes about Weil I and II: mathematik.uniregensburg.de/Jannsen/home/Weilgesamteng.pdf and mathematik.uniregensburg.de/Jannsen/home/… $\endgroup$– user19475Jan 7, 2017 at 4:59
Here are some extra references:
 Amazeen, Étale and ProÉtale Fundamental Groups;
 Belmans, Grothendieck Topologies and Étale Cohomology;
 BergströmRydh, Étale Cohomology Spring 2016;
 Conrad (?), Étale Cohomology;
 Hajj Chehade, Sheaf Cohomology on Sites and the Leray Spectral Sequence, Chapter 3;
 Klingler, Étale Cohomology and the Weil Conjectures;
 Kunkel, Étale Fundamental Group: An Exposition;
 Laskar, Étale Cohomology;
 Puttick, Galois Groups and the Étale Fundamental Group;
 Robinson, Étale Cohomology;
 Sarlin, The Étale Fundamental Group, Étale Homotopy and Anabelian Geometry;
 Szamuely, Galois Groups and Fundamental Groups, Chapter 5;
 The Stacks Project Authors, Étale Cohomology;
 Yang, Fundamental Groups of Schemes;
 Zarabara, Étale Cohomology over $\mathrm{Spec}(k)$.