étale fundamental group

Recall that in topologyMathworldPlanetmath, the fundamental groupMathworldPlanetmathPlanetmath π1(T) of a connected topological space T is defined to be the group of loops based at a point modulo homotopyMathworldPlanetmathPlanetmath. When one wants to obtain something similar in the algebraic categoryPlanetmathPlanetmathPlanetmathPlanetmath, this definition encounters problems. One cannot simply attempt to use the same definition, since the result will be wrong if one is working in positive characteristic. More to the point, the topology on a scheme fails to capture much of the stucture of the scheme. Simply choosing the “loop” to be an algebraic curve is not appropriate either, since in the most familiar case (over the complex numbers) such a “loop” has two real dimensions rather than one.

In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformationsMathworldPlanetmath of the universal covering space. This is more promising: surjectivePlanetmathPlanetmath étale morphismsMathworldPlanetmathPlanetmath are the appropriate generalizationPlanetmathPlanetmath of covering spaces. Unfortunately, the universal covering space is often an infiniteMathworldPlanetmath covering of the orignal space, which is unlikely to yield anything manageable in the algebraic category. Finite coverings, on the other hand are tractable, so one can define the algebraic fundamental group as an inverse limitMathworldPlanetmathPlanetmath of automorphism groupsMathworldPlanetmath. Note also that finite étale morphisms are closed maps as well as open maps.

Definition 1

Let X be a scheme, and let x be a geometric point of X. Then let C be the category of pairs (Y,π) such that π:YX is a finite étale morphism. Morphisms (Y,π)(Y,π) in this category are morphisms YY as schemes over X. If Y factors through Y as YYX then we obtain a morphism from AutC(Y)AutC(Y). This allows us to construct the étale fundamental group


This explanation follows [1].


  • 1 James Milne, Lectures on Étale CohomologyPlanetmathPlanetmath, http://www.jmilne.org/math/CourseNotes/math732.htmlonline course notes.
Title étale fundamental group
Canonical name etaleFundamentalGroup
Date of creation 2013-03-22 14:11:46
Last modified on 2013-03-22 14:11:46
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 5
Author archibal (4430)
Entry type Definition
Classification msc 14F35
Related topic EtaleMorphism
Related topic TopologicalSpace
Related topic FundamentalGroup
Related topic ClassificationOfCoveringSpaces