Étale fundamental group

The étale fundamental group is studied in algebraic geometry . It is an analogue of the fundamental group of topological spaces for schemas . It generalizes the concept of the Galois group and was introduced by Alexander Grothendieck and Claude Chevalley .

The étale fundamental group of a schema denotes the automorphisms of the fiber functor belonging to the category of Galois superpositions (i.e. finite étale superpositions) of , which assigns the fiber above it to a base point. ${\displaystyle X}$ ${\displaystyle X}$

In the case of the spectrum of a body , choosing a base point is equivalent to choosing a separable degree . In this way the algebraic fundamental group with base point can be canonically identified with the Galois group of the Galois extension . This interpretation is known as Grothendieck's Galois theory. ${\displaystyle k}$ ${\displaystyle k_{s}/k}$ ${\displaystyle k_{s}}$ ${\displaystyle k_{s}/k}$

The case of an actual schema of finite type over an algebraically closed field with the characteristic zero can be reduced to the case thanks to the Lefschetz principle . In this case, Serre's GAGA (or Riemann's existence theorem in the case of Riemann surfaces ) allows us to identify the étale fundamental group with the pro-finite completion of the topological fundamental group of . ${\displaystyle X}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle X(\mathbb {C} )}$

In particular, the étal fundamental groups of the affine line over an algebraic closed field with the characteristic zero is trivial. Contrary to intuition, the fundamental group of an affine straight line in a positive characteristic is not trivial, since Artin-Schreier extensions exist.

The Grothendieck conjecture of Anabelian geometry makes specific statements about the étale fundamental group .

Ultimately, Grothendieck's concept culminated in his introduction of motivic Galois groups. The motivic Galois group of the category of the zero-dimensional motifs of a number field is nothing else than the étale fundamental group of and can therefore be identified with the absolute Galois group of . ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$

literature

• Alexander Grothendieck , Michel Raynaud : Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3 ), Société Mathématique de France , 2003 (1971), Exposés V, IX, X .
• James Milne : Lectures on Etale Cohomology , lecture notes available online , especially Chapter 3.
• Jacob Murre : Lectures on an Introduction to Grothendieck's Theory of the Fundamental Group , Tata Institute, 1967.
• Jean-Pierre Serre : Géométrie algébrique et géométrie analytique , NUMDAM , Université de Grenoble. Annales de l'Institut Fourier 6, 1956 (GAGA).

Web links

• Holger Brenner, Fundamental Group and Vector Bundle, wikiversity