Étale fundamental group

The étale fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.

1 Topological analogue
2 Covering spaces
3 Examples and theorems
4 See also
5 References

Topological analogue

In algebraic topology, the fundamental group

$\pi_1(X)$

of a path connected topological space X 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 notion of a path does not make sense in general 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.

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: finite 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 $\pi_1(X,x)$ of $X$ at $x$ is the inverse limit

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

Examples and theorems

The most basic example of a fundamental group is π₁(Spec k), the fundamental group of a field k. Essentially by definition, the fundamental group of k can be shown to be isomorphic to the absolute Galois group Gal (k_sep / k). More precisely, the choice of a geometric point of Spec (k) is equivalent to giving a 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.

More generally, for any geometrically connected variety X over a field k (i.e., X is such that X_sep := X ×_k k_sep is connected) there is an exact sequence of profinite groups

1 → π₁(X_sep, x) → π₁(X, x) → Gal(k_sep / k) → 1.

Schemes over a field of characteristic zero

For a scheme X that is of finite type over C, the complex numbers, there is a close relation between the etale (also called algebraic) 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 π₁(X). This is a consequence of the Riemann existence theorem, which says that all finite etale 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.

Schemes over a field of positive characteristic and the tame fundamental group

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 $\mathbf A^1_k$ is not topologically 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.^[2]^[3] For example, the tame fundamental group of the affine line is zero.

Further topics

From a categoric point of view, the fundamental group is a functor

{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.^[4]

References

^ James Milne, Lectures on Étale Cohomology (online course notes)
^ Grothendieck, Alexander; Murre, Jacob P. (1971), The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, Vol. 208, Berlin, New York: Springer-Verlag
^ Schmidt, Alexander (2002), "Tame coverings of arithmetic schemes", Mathematische Annalen 322 (1): 1–18, doi:10.1007/s002080100262
^ (Tamagawa 1997)

Grothendieck, Alexandre; Raynaud, Michèle (2003) [1971], Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3), Paris: Société Mathématique de France, pp. xviii+327, arXiv:math.AG/0206203, ISBN 978-2-85629-141-2 , see Exp. V, IX, X.
Murre, J. P. (1967), Lectures on an introduction to Grothendieck's theory of the fundamental group, Bombay: Tata Institute of Fundamental Research, MR 0302650
Tamagawa, Akio (1997), "The Grothendieck conjecture for affine curves", Compositio Mathematica 109 (2): 135–194, doi:10.1023/A:1000114400142, MR 1478817

This article incorporates material from étale fundamental group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.