algebraic fundamental group

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Étale fundamental group

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Noun

algebraic fundamental group (plural algebraic fundamental groups)

1. (algebraic geometry) A group which is an analogue for schemes of the fundamental group for topological spaces.
   • 1997, Frans Oort, The Algebraic Fundamental Group, Leila Schneps, Pierre Lochak (editors), Geometric Galois Actions 1: Around Grothendieck's Esquisse D'un Programme, Cambridge University Press, London Mathematical Society, page 78,

     Serre constructed an example of an algebraic variety ${\displaystyle X}$ over a number field ${\displaystyle K}$ plus two embeddings of ${\displaystyle K}$ into ${\displaystyle \mathbb {C} }$ such that the geometric fundamental groups of ${\displaystyle X_{1}(\mathbb {C} )}$ and ${\displaystyle X_{2}(\mathbb {C} )}$ are not isomorphic, while the algebraic fundamental groups (i.e. their profinite completions) clearly are isomorphic, see (Serre).

   • 2011, Ingrid Bauer, Fabrizio Catanese, Roberto Pignatelli, “Surfaces of general type with geometric genus zero: a survey”, in Wolfgang Ebeling, Klaus Hulek, Knut Smoczyk, editors, Complex and Differential Geometry, Springer, page 11:

     Neves and Papadakis ([NP09]) constructed a numerical Campedelli surface with algebraic fundamental group ${\displaystyle \mathbb {Z} _{6}}$, while Lee and Park ([LP09]) constructed one with algebraic fundamental group ${\displaystyle \mathbb {Z} _{2}}$, and one with algebraic fundamental group ${\displaystyle \mathbb {Z} _{3}}$ was added in the second version of the same paper.

   • 2013, János Kollár, “Rationally Connected Varieties and Connected Groups”, in Károly Böröczky, Jr., János Kollár, Szamuely Tamas, editors, Higher Dimensional Varieties and Rational Points, Springer, page 70:

     In order to work in arbitrary characteristic, we should use the algebraic fundamental group, which we denote by ${\displaystyle \pi _{1}}$. (A very good introduction to algebraic fundamental groups is [24]. […] )

Usage notes

• In algebraic topology, the fundamental group ${\displaystyle \pi _{1}(X,x)}$ of a pointed topological space ${\displaystyle (X,x)}$ is defined as the group of homotopy classes of loops based at ${\displaystyle x}$. 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 ${\displaystyle X}$ does not in general have a universal cover that is finite over ${\displaystyle X}$, so the entire category of finite étale coverings of ${\displaystyle X}$ must be considered. The algebraic fundamental group or étale fundamental group is then defined as an inverse limit of finite automorphism groups.

Synonyms

• (algebraic analogue of fundamental group): étale fundamental group

Related terms

• fundamental group
• geometric fundamental group
• orbifold fundamental group
• topological fundamental group

Translations

Further reading

• Étale morphism on Wikipedia.Wikipedia
• Fundamental group on Wikipedia.Wikipedia
• Fundamental group scheme on Wikipedia.Wikipedia