# Textbook for Etale Cohomology

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

• Frietag-Kiehl, "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 – B R Nov 10 '11 at 21:04
• Also, SGA 4.5, which is once again available. – B R Nov 10 '11 at 21:09
• There's a very nice bird's eye view in V.I. Danilov, "Cohomology of algebraic varieties" MR1392958. – Dan Petersen Nov 10 '11 at 21:28
• 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? – B R Nov 10 '11 at 21:40
• Artin, Michael (1962), Grothendieck topologies, Harvard University, Dept. of Mathematics – Niels Nov 11 '11 at 9:20

Not a textbook, but a free PDF by J.S. Milne, http://www.jmilne.org/math/CourseNotes/LEC.pdf, pretty good IMHO.

• Thank you. I think I have it. Do you think its one of the best? – user16974 Nov 10 '11 at 21:03
• I've not read dozens, but this is a good one, for sure. – Portland Nov 10 '11 at 21:25
• 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. – Keenan Kidwell Nov 10 '11 at 21:34
• 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. – Robert Kucharczyk Nov 10 '11 at 22:09

I'll complement the list of well known books on the subject by some freely available documents, which I find user-friendly.

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 mini-course 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.

• 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 – Emerton Nov 11 '11 at 14:06
• There are also another Luc Illusie's great lecture notes: Old and new in étale cohomology – tttbase Jan 7 '17 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/etale-kohomologie/ (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: very nice, unexpected reference. – Georges Elencwajg Jan 4 '14 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.

• Thanks, but these are quite rough, so caveat lector. – Donu Arapura Sep 26 '12 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).