Algebraic 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 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 ${\displaystyle X}$ be a connected and locally noetherian scheme, let ${\displaystyle x}$ be a geometric point of ${\displaystyle X,}$ and let ${\displaystyle C}$ be the category of pairs ${\displaystyle (Y,f)}$ such that ${\displaystyle f\colon Y\to X}$ is a finite étale morphism from a scheme ${\displaystyle Y.}$ Morphisms ${\displaystyle (Y,f)\to (Y',f')}$ in this category are morphisms ${\displaystyle Y\to Y'}$ as schemes over ${\displaystyle X.}$ This category has a natural functor to the category of sets, namely the functor

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