Étale fundamental group

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The étale fundamental group is an analogue for schemes of the usual fundamental group of topological spaces.

1 Topological analogue
2 Covering spaces
3 GAGA results
4 Notes
5 See also

[edit] Topological analogue

In algebraic topology, the fundamental group

π 1 (T)

of a connected topological space $T$ is defined to be the group of loops based at a point modulo homotopy. When one wants to obtain something similar in the algebraic category, this definition encounters problems.

One cannot simply attempt to use the same definition, since the result will be wrong if one is working in positive characteristic. More to the point, the topology on a scheme fails to capture much of the structure of the scheme. Simply choosing the "loop" to be an algebraic curve is not appropriate either, since in the most familiar case (over the complex numbers) such a "loop" has two real dimensions rather than one.

[edit] Covering spaces

This discussion follows Milne^[1].

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: surjective étale morphisms are the appropriate generalization of covering spaces. Unfortunately, the universal covering space is often an infinite covering of the original space, which is unlikely to yield anything manageable in the algebraic category. Finite coverings, on the other hand are tractable, so one can define the algebraic fundamental group as an inverse limit of automorphism groups.

Let $X$ be a scheme, let $x$ be a geometric point of $X,$ and let $C$ be the category of pairs $(Y, f)$ such that $f \colon Y \to X$ is a finite étale morphism ("finite étale schemes over $X$ "). Morphisms $(Y,f)\to (Y',f')$ in this category are morphisms $Y\to Y'$ as schemes over $X .$ This category has a natural functor given $x$ , namely the functor

$F(Y) = \operatorname{Hom}_X(x, Y);$

geometrically this is the fiber of $Y \to X$ over $x,$ and abstractly it is the covariant Yoneda functor "co-represented" by $x .$ The quotation marks are because $x \to X$ is not in fact a finite étale morphism, so that $F$ is not actually representable (in general). However, it is pro-representable, in fact by "Galois covers" of $X;$ this means that we have a projective system $\{X_j \to X_i \mid i < j \in I\}$ indexed by a directed set $I,$ where the $X i$ are of course finite étale schemes over $X,$

$\#\operatorname{Aut}_X(X_i) = \operatorname{deg}(X_i/X),$ and

$F(Y) = \varinjlim_{i \in I} \operatorname{Hom}_C(X_i, Y)$

(the subscript

C

is to emphasize that this Hom-set is in the category

C

Note that for two such $X i, X j$ the map $X_j \to X_i$ induces a group homomorphism

$\operatorname{Aut}_X(X_j) \to \operatorname{Aut}_X(X_i)$

which produces a projective system of automorphism groups from the projective system ${X i}$ . We then make the following definition: the étale fundamental group $π 1 (X, x)$ of $X$ at $x$ is the inverse limit

$\pi_1(X,x) = \varprojlim_{i \in I} {\operatorname{Aut}}_X(X_i).$

[edit] GAGA results

The general comparison machinery called GAGA gives the connection in the case of a compact Riemann surface, or more general complex non-singular complete variety V. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of π₁(V).