Algebraic fundamental group

English

Template:Lb A group which is an analogue for schemes of the fundamental group for topological spaces.
- 1997, Template:W, The Algebraic Fundamental Group, Leila Schneps, Pierre Lochak (editors), Geometric Galois Actions 1: Around Grothendieck's Esquisse D'un Programme, Template:W, Template:W, page 78,
  Serre constructed an example of an algebraic variety $X$ over a number field $K$ plus two embeddings of $K$ into $ℂ$ such that the geometric fundamental groups of $X_{1} (ℂ)$ and $X_{2} (ℂ)$ are not isomorphic, while the algebraic fundamental groups (i.e. their profinite completions) clearly are isomorphic, see (Serre).
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In algebraic topology, the fundamental group π1(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.