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 notion of Galois group and was introduced by Alexander Grothendieck and Claude Chevalley.

The étale fundamental group of a scheme called the automorphisms of Faserfunktors the category of Galoisüberlagerungen (ie finite étale overlays ) of which assigns a base point, the fiber over this.

In the case of the spectrum of a body corresponds to the choice of a base point of choosing a separable financial statements. In this way, the algebraic fundamental group can be identified canonically with base point with the Galois group of a Galois extension. This interpretation is referred to as Grothendiecksche Galois theory.

The case of an actual scheme of finite type over an algebraically closed field of characteristic zero can be reduced to the case thanks to the Lefschetz principle. In this case, now allows us Serres GAGA (or Riemann's existence theorem in the case of Riemann surfaces ), the étale fundamental group of the profinite completion of the topological fundamental group of identify.

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

About the étale fundamental group makes the general Grothendieck 's conjecture of anabelschen geometry specific statements.

Ultimately, Grothendieck's concept culminated in his introduction motivic Galois groups. The motivic Galois group of the category of zero-dimensional motives of a number field is nothing more than the étale fundamental group of and therefore can be identified with the absolute Galois group of.