The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
In algebraic topology, the fundamental group
is defined as the group of homotopy classes of loops based at
This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology.
In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space.
This is more promising: finite étale morphisms of algebraic varieties are the appropriate analogue of covering spaces of topological spaces.
often fails to have a "universal cover" that is finite over
, so one must consider the entire category of finite étale coverings of
be a connected and locally noetherian scheme, let
and abstractly it is the Yoneda functor represented by
[1] It also means that we have given an isomorphism of functors: In particular, we have a marked point
We then make the following definition: the étale fundamental group
can be shown to be isomorphic to the absolute Galois group
is equivalent to giving a separably closed extension field
is connected) there is an exact sequence of profinite groups: For a scheme
, the complex numbers, there is a close relation between the étale fundamental group of
The algebraic fundamental group, as it is typically called in this case, is the profinite completion of
This is a consequence of the Riemann existence theorem, which says that all finite étale coverings of
(i.e., open Riemann surfaces) is well understood; this determines the algebraic fundamental group.
More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.
of positive characteristic, the results are different, since Artin–Schreier coverings exist in this situation.
For example, the fundamental group of the affine line
which takes into account only covers that are tamely ramified along
[3][4] For example, the tame fundamental group of the affine line is zero.
is entirely determined by its etale homotopy group.
Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups.
[6] Friedlander (1982) studies higher étale homotopy groups by means of the étale homotopy type of a scheme.
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.
For geometrically unibranch schemes (e.g., normal schemes), the two approaches agree, but in general the pro-étale fundamental group is a finer invariant: its profinite completion is the étale fundamental group.