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The pro-étale fundamental group. Bhatt & Scholze (2015, §7) have introduced a variant of the étale fundamental group called the pro-étale fundamental group. It is constructed by considering, instead of finite étale covers, maps which are both étale and satisfy the valuative criterion of properness.
Oct 12, 2020 · The goal of this talk is to give an introduction to the etale fundamental group. We begin with some motivation from topology, and then proceed to study the appropriate algebraic analogue
We define (finally) the pro-étale fundamental group. Definition 3.3. The pro-étale fundamental group of (X, x¯) is ppro-et(X, x¯) = Aut Fx¯. Theorem 3.4. The pair (Cov X, Fx¯) is an infinite Galois category. Proof. We omit the verifications that Cov X has colimits and finite limits, that F commutes with
There is a profinite group p, unique up to isomorphism, such that FEt X ˇp-FSet. Given a geometric point x of X, we can define p and the equivalence functorially in (X, x). It is the étale fundamental group pet(X, x). Often pet(X, x) is the desired analogue of the topological fundamental group. This can be
Zˆ is an example of a pro nite group, i.e., an inverse limit of finite groups. By definition, every etale fundamental group is a profinite group. We endow a profinite group the topology induced from the product topology. So every profinite group is compact by Tychonoff’s theorem. The following fact matches our intuition about ...
compact-open topology is called the pro-´etale fundamental group ˇproet 1 (X;x). Then (1) (Loc X;ev x) is equivalent to the category of continuous representations of ˇ pro et 1 (X;x) on discrete sets; (2) the pro-finite completion of ˇproet 1 (X;x) is the etale fundamental group´ ˇet 1 (X;x); (3) the pro-discrete completion of ˇproet 1 ...
October 15, 2014. In this lecture our only goal is to give lots of examples of etale fun-damental groups so that the reader gets some feel for them. Some of the examples will involve scheme-theoretic concepts that we have not covered such as normality, smoothness, dimension etc.
