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| The '''étale''' or '''algebraic fundamental group''' is an analogue in [[algebraic geometry]], for [[Scheme (mathematics)|schemes]], of the usual [[fundamental group]] of topological spaces.
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| ==Topological analogue/informal discussion==
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| In [[algebraic topology]], the fundamental group ''π''<sub>1</sub>(''X'',''x'') of a pointed topological space (''X'',''x'') is defined as the [[group_(Mathematics)|group]] of homotopy classes of loops based at ''x''. This definition works well for spaces such as real and complex [[manifold]]s, but gives undesirable results for an [[algebraic variety]] with the [[Zariski topology]].
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| In the classification of covering spaces, it is shown that the fundamental group is exactly the group of [[deck transformation]]s of the [[universal covering space]]. This is more promising: [[finite morphism|finite]] [[étale morphism]]s are the appropriate analogue of [[covering map|covering spaces]]. Unfortunately, an algebraic variety ''X'' often fails to have a "universal cover" that is finite over ''X'', so one must consider the entire category of finite étale coverings of ''X''. One can then define the étale fundamental group as an [[inverse limit]] of finite [[automorphism]] groups.
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| ==Formal definition==
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| Let <math>X</math> be a connected and locally [[noetherian scheme]], let <math>x</math> be a [[geometric point]] of <math>X,</math> and let <math>C</math> be the category of pairs <math>(Y,f)</math> such that <math>f \colon Y \to X</math> is a [[Étale morphism|finite étale morphism]] from a scheme <math>Y.</math> [[Morphism|Morphisms]] <math>(Y,f)\to (Y',f')</math> in this category are morphisms <math>Y\to Y'</math> as [[Scheme_(mathematics)|schemes]] over <math>X.</math> This category has a [[functor|natural functor]] to the category of sets, namely the functor
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| :<math>F(Y) = \operatorname{Hom}_X(x, Y);</math>
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| geometrically this is the fiber of <math>Y \to X</math> over <math>x,</math> and abstractly it is the [[Yoneda functor]] [[Representable functor|corepresented]] by <math>x.</math> The functor <math>F</math> is not representable, however, it is pro-representable, in fact by Galois covers of <math>X</math>. This means that we have a [[projective system]] <math>\{X_j \to X_i \mid i < j \in I\}</math> in <math>C</math>, indexed by a [[directed set]] <math>I,</math> where the <math>X_i</math> are Galois covers of <math>X</math>, i.e., finite étale schemes over <math>X,</math> such that <math>\#\operatorname{Aut}_X(X_i) = \operatorname{deg}(X_i/X)</math>. It also means that we have given an isomorphism of functors
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| :<math>F(Y) = \varinjlim_{i \in I} \operatorname{Hom}_C(X_i, Y)</math>.
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| In particular, we have a marked point <math>P\in \varprojlim_{i \in I} F(X_i)</math> of the projective system.
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| For two such <math>X_i, X_j</math> the map <math>X_j \to X_i</math> induces a group homomorphism
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| <math>\operatorname{Aut}_X(X_j) \to \operatorname{Aut}_X(X_i)</math>
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| which produces a projective system of automorphism groups from the projective system <math>\{X_i\}</math>. We then make the following definition: the ''étale fundamental group'' <math>\pi_1(X,x)</math> of <math>X</math> at <math>x</math> is the inverse limit
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| :<math> \pi_1(X,x) = \varprojlim_{i \in I} {\operatorname{Aut}}_X(X_i),</math>
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| with the inverse limit topology.
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| The functor <math>F</math> is now a functor from <math>C</math> to the category of finite and continuous <math>\pi_1(X,x)</math>-sets, and establishes an ''equivalence of categories'' between <math>C</math> and the category of finite and continuous <math>\pi_1(X,x)</math>-sets.<ref>
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| {{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Raynaud | first2=Michèle | title=Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques '''3''') | origyear=1971 | arxiv=math.AG/0206203 | publisher=[[Société Mathématique de France]] | location=Paris | isbn=978-2-85629-141-2 | year=2003 | pages=xviii+327, see Exp. V, IX, X.}}</ref>
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| ==Examples and theorems==
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| The most basic example of a fundamental group is π<sub>1</sub>(Spec ''k''), the fundamental group of a [[field (mathematics)|field]] ''k''. Essentially by definition, the fundamental group of ''k'' can be shown to be isomorphic to the absolute [[Galois group]] Gal (''k''<sub>sep</sub> / ''k''). More precisely, the choice of a geometric point of Spec (''k'') is equivalent to giving a [[separably closed field|separably closed]] extension field ''K'', and the fundamental group with respect to that base point identifies with the Galois group Gal (''K'' / ''k''). This interpretation of the Galois group is known as [[Grothendieck's Galois theory]].
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| More generally, for any geometrically connected variety ''X'' over a field ''k'' (i.e., ''X'' is such that ''X''<sub>sep</sub> := ''X'' ×<sub>''k''</sub> ''k''<sub>sep</sub> is connected) there is an [[exact sequence]] of profinite groups
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| :1 → π<sub>1</sub>(''X''<sub>sep</sub>, {{overline|x}}) → π<sub>1</sub>(''X'', {{overline|x}}) → Gal(''k''<sub>sep</sub> / ''k'') → 1. | |
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| ===Schemes over a field of characteristic zero===
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| For a scheme ''X'' that is of finite type over '''C''', the complex numbers, there is a close relation between the étale fundamental group of ''X'' and the usual, topological, fundamental group of ''X''('''C'''), the [[complex analytic space]] attached to ''X''. The algebraic fundamental group, as it is typically called in this case, is the [[profinite completion]] of π<sub>1</sub>(''X''). This is a consequence of the [[Riemann existence theorem]], which says that all finite étale coverings of ''X''('''C''') stem from ones of ''X''. In particular, as the fundamental group of smooth curves over '''C''' (i.e., open Riemann surfaces) is well-understood, this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.
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| ===Schemes over a field of positive characteristic and the tame fundamental group===
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| For an algebraically closed field ''k'' of positive characteristic, the results are different, since Artin-Schreier coverings exist in this situation. For example, the fundamental group of the [[affine line]] <math>\mathbf A^1_k</math> is not topologically [[finitely generated group|finitely generated]]. The ''tame fundamental group'' of some scheme ''U'' is a quotient of the usual fundamental group of ''U'' which takes into account only covers that are tamely ramified along ''D'', where ''X'' is some compactification and ''D'' is the complement of ''U'' in ''X''.<ref>{{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | last2=Murre | first2=Jacob P. | title=The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 208 | year=1971}}</ref><ref>{{Citation | last1=Schmidt | first1=Alexander | title=Tame coverings of arithmetic schemes | doi=10.1007/s002080100262 | year=2002 | journal=[[Mathematische Annalen]] | volume=322 | issue=1 | pages=1–18}}</ref> For example, the tame fundamental group of the affine line is zero.
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| ===Further topics===
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| From a [[category theory|categoric]] point of view, the fundamental group is a functor
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| :{''Algebraic Varieties''} → {''Profinite groups''}. | |
| The [[inverse Galois problem]] asks what groups can arise as fundamental groups (or Galois groups of field extensions). [[Anabelian geometry]], for example [[Grothendieck]]'s [[section conjecture]], seeks to identify classes of varieties which are determined by their fundamental groups.<ref>{{Harvard citations|last=Tamagawa|year=1997}}</ref>
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| The étale fundamental group π<sub>1</sub> admit a generalization to a kind of higher homotopy groups by means of the étale homotopy type.<ref>{{Citation | last1=Friedlander | first1=Eric M. | title=Étale homotopy of simplicial schemes | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-08288-2 | year=1982 | volume=104}}</ref>
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| ==See also==
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| * [[étale morphism]]
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| * [[Topological space]]
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| * [[Fundamental group]]
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| * [[Fundamental group scheme]]
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| == References ==
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| <references/>
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| * {{Citation | last1=Murre | first1=J. P. | title=Lectures on an introduction to Grothendieck's theory of the fundamental group | publisher=Tata Institute of Fundamental Research | location=Bombay | mr=0302650 | year=1967}}
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| * {{Citation | last1=Tamagawa | first1=Akio | title=The Grothendieck conjecture for affine curves | doi=10.1023/A:1000114400142 | mr=1478817 | year=1997 | journal=Compositio Mathematica | volume=109 | issue=2 | pages=135–194}}
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| {{PlanetMath attribution|id=5627|title=étale fundamental group}}
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| {{DEFAULTSORT:Etale Fundamental Group}}
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| [[Category:Scheme theory]]
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| [[Category:Topological methods of algebraic geometry]]
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Emilia Shryock is my name but you can contact me something you like. For years he's been residing in North Dakota and his family loves it. My working day job is a librarian. Body building is what my family and I appreciate.
Look at my homepage; http://bogangsa.com/board_avHE75/389128