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Analogue in algebraic geometry
- The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
en.wikipedia.org/wiki/Étale_fundamental_group
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The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Oct 12, 2020 · ETALE FUNDAMENTAL GROUPS 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 of the fundamental group. 1. The classical fundamental group
- Dexter Chua
- 3 Etale Covers
- FE
- Corollary. If we have a composition
- FE
- 5 Galois Theory
- FE
- A Faithfully flat morphisms
- A B : A-Mod ! B-Mod
- A C is
- M C = B A M ;
Introduction Etale Morphisms Etale Covers The Etale Fundamental Group Galois Theory Appendix A Faithfully at morphisms
De nition (Etale cover). An ( nite) etale cover is a morphism that is nite and etale. We write tX
A trivial cover of X for the category of etale covers of X. is one that is a nite disjoint union of copies of X. Lemma. Etale covers are closed under pullback and composition, and satisfy fpqc descent. Lemma. A ne, and in particular etale morphisms are separated. It turns out being an etale cover imposes strong conditions on the map. Lemma. A nite ...
q p Z Y X where p and p q are etale covers, then so is q. Proof. By fpqc descent, we may assume that Z and Y are both trivial covers, in which case the proposition is clear.
By de nition, jF(Y )j = deg Y . De nition (Etale fundamental group). Let X be a scheme and x a base point. The etale fundamental group 1(X; x) is de ned to be Aut F, the group of all automorphisms of the functor F : tX ! Sets. FE In the case of covering theory or Galois theory, this functor F is \representable". For Galois groups, this is represent...
The usual proof of the usual Fundamental Theorem of Galois Theory can pretty much be carried over to the general case if we can say the word \Galois". Fortunately, the word is not too di cult to utter. De nition (Galois cover). A Galois cover of X is an element Y 2 tX such that
X X if there is a morphism X ! X . For each X , pick a point x 2 F(X ). Then if X X , we pick the morphism that sends x to x for use in the pro-system. Then this pro-system pro-represents F. Proof. This is indeed a pro-system since the pullback of two Galois covers is yet another Galois cover of X. There is a natural transformation colim Hom(X ; Y ...
In this appendix, we document some important facts about at and faithfully at morphisms. De nition. Let f : A ! B be a ring homomorphism. We say f is at if the functor
is exact. De nition. A morphism p : Y ! X is at if for all y 2 Y and x = f(y), the map OX;x ! OY;y is at. This in particular implies the pullback functor p : QCoh(Y ) ! QCoh(X) is exact. If X is quasi-compact and quasi-separated, then p being exact implies p being at. Lemma. Compositions and pullbacks of at maps are at. Proof. We only have to show ...
at. But if M is a exact sequence of chain complexes, then B A C
where we think of M as an A-module via g. So this is exact.
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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.
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Also, the étale fundamental group of a field is its Galois group. On the other hand, for smooth varieties X over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the profinite completion of the latter.
In general, for a field F, the etale fundamental group of SpecF is the absolute Galois group Gal( F s /F ), where F s is the separable closure of F (by the same argument). The absolute Galois group of Q contains monstrous information about all number fields and
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.
