# étale fundamental group as unification of galois theory and covering theory

what is a good reference if one wants to learn the basic theory of étale cohomology, étale fundamental group and in particular relationships between galois theory and covering theory (unified via the étale fundamental group). My background actually lies in algebraic topology, but I am familiar with schemes, sheaves and sheaf cohomology. I plan on doing a reading seminar with friends, and we want to get ideas which material to cover. We planned to do as much as one usually would do in a 2-hour a week course going for one semester, i.e., we would need material for about 12 sessions to meet.

Any information is appreciated.

-
Try Szamuely's Galois Groups and Fundamental Groups. It used to be freely available online, but alas, those happy days are over. –  Qiaochu Yuan May 23 '12 at 13:57
Szamuely's Galois groups and fundamental groups gives a good introduction to all the topics you mention except étale cohomology. Conversely, I think most references for étale cohomology omit detailed study of the étale fundamental group. –  Zhen Lin May 23 '12 at 13:57
SGA 1 is the basic reference for etale fundamental groups, and it remains a good one. There is a freely available LaTeXed version on the Arxiv. Regards, –  Matt E May 23 '12 at 15:05
There's also Milne's "Lecture notes on etale cohomology". He doesn't spend tons of time with the etale fundamental group either, but it's a nice read and I think it's worth at least taking a look at his exposition: jmilne.org/math/CourseNotes/LEC.pdf –  Aaron Mazel-Gee May 26 '12 at 10:40
Also, as an algebraic topologist you might appreciate Sullivan's "MIT notes", which discuss etale homotopy theory (among other things): maths.ed.ac.uk/~aar/books/gtop.pdf –  Aaron Mazel-Gee May 26 '12 at 10:44

Here are some texts which might be of interest for understanding the geometry of finite étale covers:

1. Tamas Szamuely, Galois groups and fundamental groups.
2. SGA 1 (available on arxiv).
3. Michel Raynaud, Anneaux locaux Henséliens
4. Deligne, P., Le groupe fondamental de la droite projective moins trois points, in: Galois groups over Q, editors Y. Ihara, K. Ribet, J.-P. Serre, MSRI Publications 16, Springer, 1989, 79-297.

The analogue of 'covering space' of a base $S$ will be finite étale morphisms $X\to S$, in the following sense: if you have a point $s\in S$, there is a 'geometric point' $\overline{s}$ (spectrum of a separably closed field) and a "tiny ball for the étale topology" which is the spectrum of a strict henselization, called the strict localization together with morphisms $\overline{s}\to S_{(\overline{s})} \to S$. Being finite étale is preserved by base change, so you'll have the base change of $X/S$ to $X_{S_{(\overline{s})}}:=X\times_S S_{(\overline{s})}$ over $S_{(\overline{s})}$ and $X_{\overline{s}}:=X\times_{S} \overline{s}$. Basic commutative algebra allows one to classify finite étale algebras over a field or a strict henselization, and you get a disjoint union of the bases in these base changes. The scheme $X_{\overline{s}}$ is the fiber of your base point, each component of which sits inside a copy of $S_{(\overline{s})}$ in $X_{S_{(\overline{s})}}$:

$$\begin{matrix} X_{\overline{s}}&{\rightarrow}&X_{S_{(\overline{s})}}&{\rightarrow}&X\\ {\downarrow}& & \downarrow & & \downarrow \\ \overline{s} & \rightarrow & S_{(\overline{s})} & \rightarrow & S \end{matrix}$$

so that a finite étale morphism is 'locally for the étale topology' a trivial covering.

The functor sending $X$ to $X_{\overline{s}}$ is called the fiber functor at $\overline{s}$ and the automorphism group of this functor is defined to be $\pi_1(S,\overline{s})$ - the fundamental group of $S$ at the geometric point $\overline{s}$. Each $X_{\overline{s}}$ is endowed with an action of this group, and the first step is to realize that the geometry of a cover $X/S$ is somewhat encoded by how $\pi_1(S,\overline{s})$ acts on the fibers.

Edit: Coming back to Galois theory, one can in particular show that if $K$ is a field then a finite étale $K$-algebra is just any finite product of finite separable extensions of $L$. Taking $S=Spec(K)$, a separable closure $K^{sep}/K$ gives an associated geometric point $Spec(K^{sep})\to Spec(K)$ and one finds that that $\pi_1(S,\overline{s})$ is the absolute Galois group $Gal(K^{sep}/K)$. (For example this is worked out in Szamuely's text under the name "Grothendieck's version of Galois theory" ).

-
What a nice, easy to remember handle you have chosen, dear vgty6h7uij :-) –  Georges Elencwajg Dec 16 '12 at 11:15
@GeorgesElencwajg After a long enough time, none of us will be remembered. :) –  vgty6h7uij Dec 24 '12 at 11:20