Étale fundamental group
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Topological analogue/informal discussion
In algebraic topology, the fundamental group  of a pointed topological space  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. Unfortunately, an algebraic variety  often fails to have a "universal cover" that is finite over , so one must consider the entire category of finite étale coverings of . One can then define the étale fundamental group as an inverse limit of finite automorphism groups. Formal definition
Let  be a connected and locally noetherian scheme, let  be a geometric point of  and let  be the category of pairs  such that  is a finite étale morphism from a scheme  Morphisms  in this category are morphisms  as schemes over   This category has a natural functor to the category of sets, namely the functor

geometrically this is the fiber of  over  and abstractly it is the Yoneda functor represented by  in the category of schemes over . The functor  is typically not representable in ; however, it is pro-representable in , in fact by Galois covers of . This means that we have a projective system  in , indexed by a directed set  where the  are Galois covers of , i.e., finite étale schemes over  such that . It also means that we have given an isomorphism of functors 
. In particular, we have a marked point  of the projective system. For two such  the map  induces a group homomorphism

which produces a projective system of automorphism groups from the projective system . We then make the following definition: the étale fundamental group  of  at  is the inverse limit

with the inverse limit topology. The functor  is now a functor from  to the category of finite and continuous -sets, and establishes an equivalence of categories between  and the category of finite and continuous -sets. Examples and theorems

The most basic example of is , the étale fundamental group of a field . Essentially by definition, the fundamental group of  can be shown to be isomorphic to the absolute Galois group . More precisely, the choice of a geometric point of  is equivalent to giving a separably closed extension field , and the étale fundamental group with respect to that base point identifies with the Galois group . This interpretation of the Galois group is known as Grothendieck's Galois theory. More generally, for any geometrically connected variety  over a field  (i.e.,  is such that  is connected) there is an exact sequence of profinite groups

Schemes over a field of characteristic zero
For a scheme  that is of finite type over , the complex numbers, there is a close relation between the étale fundamental group of  and the usual, topological, fundamental group of , the complex analytic space attached to . 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  stem from ones of . In particular, as the fundamental group of smooth curves over  (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. Schemes over a field of positive characteristic and the tame fundamental group
For an algebraically closed field  of positive characteristic, the results are different, since Artin–Schreier coverings exist in this situation. For example, the fundamental group of the affine line  is not topologically finitely generated. The tame fundamental group of some scheme U is a quotient of the usual fundamental group of  which takes into account only covers that are tamely ramified along , where  is some compactification and  is the complement of  in . For example, the tame fundamental group of the affine line is zero. Affine schemes over a field of characteristic p 
It turns out that every affine scheme  is a -space, in the sense that the etale homotopy type of  is entirely determined by its etale homotopy group. Note  where  is a geometric point. Further topics
From a category-theoretic point of view, the fundamental group is a functor
{Pointed algebraic varieties} → {Profinite groups}. The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions). Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups. studies higher étale homotopy groups by means of the étale homotopy type of a scheme. The pro-étale fundamental group
 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. See also
 étale morphism
 Fundamental group
 Fundamental group scheme

Notes

References 
 	
 
 
 
 

Scheme theory
Topological methods of algebraic geometry