The algebraic fundamental group is the fundamental group of a scheme, as defined by Grothendieck in SGA1. It is essentially the fundamental group as seen by étale homotopy, the étale fundamental group.
In arithmetic geometry one also speaks of the arithmetic fundamental group.
Let be a connected scheme. Recall that a finite étale cover of is a finite flat surjection such that each fibre at a point is the spectrum of a finite étale algebra over the local ring at . Fix a geometric point .
For a finite étale cover, , we consider the geometric fibre, , over , and denote by its underlying set. This gives a set-valued functor on the category of finite étale covers of .
The algebraic fundamental group, is defined to be the automorphism group of this functor.
For more on this area, see at étale homotopy.
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or in a lengthier form:
An earlier version is to be found here.
A paper on a closely related subject is