Étale fundamental group

The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. In algebraic topology, the fundamental group π1(X,x) of a pointed topological space (X,x) is defined as the group of homotopy classes of loops based at x. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology. In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms are the appropriate analogue of 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. Let X {displaystyle X} be a connected and locally noetherian scheme, let x {displaystyle x} be a geometric point of X , {displaystyle X,} and let C {displaystyle C} be the category of pairs ( Y , f ) {displaystyle (Y,f)} such that f : Y → X {displaystyle fcolon Y o X} is a finite étale morphism from a scheme Y . {displaystyle Y.} Morphisms ( Y , f ) → ( Y ′ , f ′ ) {displaystyle (Y,f) o (Y',f')} in this category are morphisms Y → Y ′ {displaystyle Y o Y'} as schemes over X . {displaystyle X.} This category has a natural functor to the category of sets, namely the functor geometrically this is the fiber of Y → X {displaystyle Y o X} over x , {displaystyle x,} and abstractly it is the Yoneda functor represented by x {displaystyle x} in the category of schemes over X {displaystyle X} . The functor F {displaystyle F} is typically not representable in C {displaystyle C} ; however, it is pro-representable in C {displaystyle C} , in fact by Galois covers of X {displaystyle X} . This means that we have a projective system { X j → X i ∣ i < j ∈ I } {displaystyle {X_{j} o X_{i}mid i<jin I}} in C {displaystyle C} , indexed by a directed set I , {displaystyle I,} where the X i {displaystyle X_{i}} are Galois covers of X {displaystyle X} , i.e., finite étale schemes over X {displaystyle X} such that # Aut X ⁡ ( X i ) = deg ⁡ ( X i / X ) {displaystyle #operatorname {Aut} _{X}(X_{i})=operatorname {deg} (X_{i}/X)} . It also means that we have given an isomorphism of functors In particular, we have a marked point P ∈ lim ← i ∈ I ⁡ F ( X i ) {displaystyle Pin varprojlim _{iin I}F(X_{i})} of the projective system. For two such X i , X j {displaystyle X_{i},X_{j}} the map X j → X i {displaystyle X_{j} o X_{i}} induces a group homomorphism Aut X ⁡ ( X j ) → Aut X ⁡ ( X i ) {displaystyle operatorname {Aut} _{X}(X_{j}) o operatorname {Aut} _{X}(X_{i})} which produces a projective system of automorphism groups from the projective system { X i } {displaystyle {X_{i}}} . We then make the following definition: the étale fundamental group π 1 ( X , x ) {displaystyle pi _{1}(X,x)} of X {displaystyle X} at x {displaystyle x} is the inverse limit