algebraic fundamental group

algebraic fundamental group

English

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, 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, 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

algebraic analogue of fundamental group — See also translations at étale fundamental group

Further reading

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