The pro-étale fundamental group of a scheme was introduced by Bhatt and Scholze. It generalizes the fundamental groups of schemes introduced by Grothendieck in SGA1 and SGA3. The corresponding category of covers of a scheme X consists of schemes Y -> X that are étale and satisfy the valuative criterion of properness. They are called "geometric covers". As we do not assume Y -> X to be of finite type, we get more than just finite étale covers. The basic example is given by a nodal curve and a cover by an infnite chain of P^1's glued in a suitable way. This cover was already "detected" by the SGA3 fundamental group, but for more complicated schemes (e.g. an elliptic curve with two points glued) one gets more. The prominent feature of the pro-étale fundamental group is that its finite-dimensional Q_\ell-representations are able to detect all Q_\ell-local systems on X. The greater generality comes at the price of working with a more complicated class of topological groups - Noohi groups. This class includes profinite and prodiscrete groups. An important feature is that it also includes groups like GL_n(Q_\ell).
In this thesis, I prove some fundamental results for the pro-étale fundamental group, that generalize the results of Grothendieck on the usual étale fundamental group to this more general context. The main results concern the homotopy exact sequence of the pro-étale fundamental groups arising from a proper morphism of connected (Nagata) noetherian schemes with geometrically connected and reduced fibers. There are two separate cases: over a general base scheme S and over a spectrum of a field k. Over the general base, one does not have exactness on the left of the sequence and the main difficulty is the exactness in the middle. Moreover, one has to use a suitable notion of exactness, involving some kind of topological closures. Over Spec(k), one can drop the properness assumption and the sequence is exact even on the level of abstract groups. In this case, we show additionally the exactness on the left (i.e. that the "geometric" fundamental group embeds into the "arithmetic" one) and this is the most difficult part. As in the classical case, the statements on exactness of the pro-étale fundamental groups translate to a statement in terms of geometric covers. We provide a detailed dictionary between the two languages. In terms of covers, the main theorems we prove are as follows. For a geometrically connected scheme X of finite type over a field k, we show that a connected geometric cover of X_\bar{k} can be dominated by a geometric cover defined over a finite extension l of the base field k. Unlike in the case of finite étale covers, this is non-trivial. Over a general base scheme S (and with assumptions about X -> S as above), we prove existence of an "infinite Stein factorization" of a geometric cover Y -> X. The resulting scheme is a geometric cover of S. If Y -> X is finite, it matches the usual Stein factorization applied to Y -> X -> S. For infinite covers, the standard definition of Stein factorization fails and we proceed differently: using a descent argument along a large pro-étale cover. Over a field, we refine an abstract version of the van Kampen theorem to provide a presentation of the pro-étale fundamental group in terms of a free product of profinite groups, discrete groups and relations (up to a certain form of completion). We use it to study the Galois actions. We also prove the Künneth formula for the pro-étale fundamental groups, generalizing the classical result of Grothendieck. We use the abstract van Kampen in the proof, again.
