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. In algebraic topology, the fundamental group of a pointed topological space is defined as the group of homotopy classes of loops based at . 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 of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety often fails to have a "universal cover" that is finite over , so one must consider the entire category of finite étale coverings of . One can then define the étale fundamental group as an inverse limit of finite automorphism groups. Let be a connected and locally noetherian scheme, let be a geometric point of and let be the category of pairs such that is a finite étale morphism from a scheme Morphisms in this category are morphisms as schemes over This category has a natural functor to the category of sets, namely the functor geometrically this is the fiber of over and abstractly it is the Yoneda functor represented by in the category of schemes over . The functor is typically not representable in ; however, it is pro-representable in , in fact by Galois covers of . This means that we have a projective system in , indexed by a directed set where the are Galois covers of , i.e., finite étale schemes over such that . It also means that we have given an isomorphism of functors In particular, we have a marked point of the projective system. For two such the map induces a group homomorphism which produces a projective system of automorphism groups from the projective system . We then make the following definition: the étale fundamental group of at is the inverse limit with the inverse limit topology.