algebraic fundamental group

Noun

 * 1)  A group which is an analogue for schemes of the fundamental group for topological spaces.
 * 2) * 1997,, The Algebraic Fundamental Group, Leila Schneps, Pierre Lochak (editors), Geometric Galois Actions 1: Around Grothendieck's Esquisse D'un Programme, , , page 78,
 * Serre constructed an example of an algebraic variety $$X$$ over a number field $$K$$ plus two embeddings of $$K$$ into $$\mathbb C$$ such that the geometric fundamental groups of $$X_1(\mathbb C)$$ and $$X_2(\mathbb C)$$ are not isomorphic, while the algebraic fundamental groups (i.e. their profinite completions) clearly are isomorphic, see (Serre).

Usage notes

 * In algebraic topology, the fundamental group $$\pi_1(X, x)$$ of a pointed topological space $$(X, x)$$ is defined as the group of homotopy classes of loops based at $$x$$. This definition works well for spaces such as real and complex manifolds, but is unsatisfactory for an algebraic variety equipped with the Zariski topology.
 * In the classification of covering spaces, the fundamental group turns out to be exactly the group of deck transformations (cover transformations) of the universal covering space. This is more useful: finite (i.e., finitely generated) étale morphisms are the appropriate analogue of covering maps. However, an algebraic variety $$X$$ does not in general have a universal cover that is finite over $$X$$, so the entire category of finite étale coverings of $$X$$ must be considered. The algebraic fundamental group or étale fundamental group is then defined as an inverse limit of finite automorphism groups.