Editing Fundamental group (section)

===Étale fundamental group===

In [[algebraic geometry]], the so-called [[étale fundamental group]] is used as a replacement for the fundamental group.<ref>{{harvtxt|Grothendieck|Raynaud|2003}}.</ref> Since the [[Zariski topology]] on an [[algebraic variety]] or [[scheme (mathematics)|scheme]] ''X'' is much [[comparison of topologies|coarser]] than, say, the [[topological space|topology]] of open subsets in <math>\R^n,</math> it is no longer meaningful to consider continuous maps from an [[interval (mathematics)|interval]] to ''X''. Instead, the approach developed by [[Grothendieck]] consists in constructing <math>\pi_1^\text{et}</math> by considering all [[finite morphism|finite]] [[étale morphism|étale covers]] of ''X''. These serve as an algebro-geometric analogue of coverings with finite fibers.

This yields a theory applicable in situations where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a [[finite field]]. Also, the étale fundamental group of a [[field (mathematics)|field]] is its ([[absolute Galois group|absolute]]) [[Galois group]]. On the other hand, for smooth varieties ''X'' over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the [[profinite completion]] of the latter.<ref>{{harvtxt|Grothendieck|Raynaud|2003|loc=Exposé XII, Cor. 5.2}}.</ref>