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.

Topological analogue/informal discussion

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.

Formal definition

Let $X$ be a connected and locally noetherian 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 from a scheme $Y.$ Morphisms $(Y,f)\to (Y',f')$ in this category are morphisms $Y\to Y'$ as schemes over $X.$ This category has a natural functor to the category of sets, 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 Yoneda functor represented by $x$ in the category of schemes over $X$ . The functor $F$ is typically not representable in $C$ ; however, it is pro-representable in $C$ , in fact by Galois covers of $X$ . This means that we have a projective system $\{X_{j}\to X_{i}\mid i<j\in I\}$ in $C$ , indexed by a directed set $I,$ where the $X_{i}$ are Galois covers of $X$ , i.e., finite étale schemes over $X$ such that $\#\operatorname {Aut}_{X}(X_{i})=\operatorname {deg}(X_{i}/X)$ .^[1] It also means that we have given an isomorphism of functors

F(Y)=\varinjlim _{{i\in I}}\operatorname {Hom}_{C}(X_{i},Y)

.

In particular, we have a marked point $P\in \varprojlim _{{i\in I}}F(X_{i})$ of the projective system.

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}),

with the inverse limit topology.

The functor $F$ is now a functor from $C$ to the category of finite and continuous $\pi_1(X,x)$ -sets, and establishes an equivalence of categories between $C$ and the category of finite and continuous $\pi_1(X,x)$ -sets.^[2]

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 étale 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 étale 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 ${\mathbf A}_{k}^{1}$ 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.^[3]^[4] For example, the tame fundamental group of the affine line is zero.

Further topics

From a category-theoretic point of view, the fundamental group is a functor

{Pointed 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.^[5]

The étale fundamental group π₁ admit a generalization to a kind of higher homotopy groups by means of the étale homotopy type.^[6]

Notes

↑ J. S. Milne, Lectures on Étale Cohomology, version 2.21: 26-27
↑ 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, see Exp. V, IX, X., arXiv:math.AG/0206203, ISBN 978-2-85629-141-2
↑ 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, arXiv:math/0005310, doi:10.1007/s002080100262
↑ (Tamagawa 1997)
↑ Friedlander, Eric M. (1982), Étale homotopy of simplicial schemes, Annals of Mathematics Studies, 104, Princeton University Press, ISBN 978-0-691-08288-2

References

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
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[1] J. S. Milne, Lectures on Étale Cohomology, version 2.21: 26-27

[2] 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, see Exp. V, IX, X., arXiv:math.AG/0206203, ISBN 978-2-85629-141-2

[3] 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

[4] Schmidt, Alexander (2002), "Tame coverings of arithmetic schemes", Mathematische Annalen, 322 (1): 1–18, arXiv:math/0005310, doi:10.1007/s002080100262

[5] (Tamagawa 1997)

[6] Friedlander, Eric M. (1982), Étale homotopy of simplicial schemes, Annals of Mathematics Studies, 104, Princeton University Press, ISBN 978-0-691-08288-2