Étale fundamental group

In algebraic topology, the fundamental group ${\displaystyle \pi _{1}(X,x)}$ of a pointed topological space ${\displaystyle (X,x)}$ is defined as the group of homotopy classes of loops based at ${\displaystyle x}$. 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 of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety ${\displaystyle X}$ often fails to have a "universal cover" that is finite over ${\displaystyle X}$, so one must consider the entire category of finite étale coverings of ${\displaystyle X}$. 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

${\displaystyle F(Y)=\operatorname {Hom} _{X}(x,Y);}$

geometrically this is the fiber of ${\displaystyle Y\to X}$ over ${\displaystyle x,}$ and abstractly it is the Yoneda functor represented by ${\displaystyle x}$ in the category of schemes over ${\displaystyle X}$. The functor ${\displaystyle F}$ is typically not representable in ${\displaystyle C}$; however, it is pro-representable in ${\displaystyle C}$, in fact by Galois covers of ${\displaystyle X}$. This means that we have a projective system ${\displaystyle \{X_{j}\to X_{i}\mid i<j\in I\}}$ in ${\displaystyle C}$, indexed by a directed set ${\displaystyle I,}$ where the ${\displaystyle X_{i}}$ are Galois covers of ${\displaystyle X}$, i.e., finite étale schemes over ${\displaystyle X}$ such that ${\displaystyle \#\operatorname {Aut} _{X}(X_{i})=\operatorname {deg} (X_{i}/X)}$.[1] It also means that we have given an isomorphism of functors

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

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

For two such ${\displaystyle X_{i},X_{j}}$ the map ${\displaystyle X_{j}\to X_{i}}$ induces a group homomorphism ${\displaystyle \operatorname {Aut} _{X}(X_{j})\to \operatorname {Aut} _{X}(X_{i})}$ which produces a projective system of automorphism groups from the projective system ${\displaystyle \{X_{i}\}}$. We then make the following definition: the étale fundamental group ${\displaystyle \pi _{1}(X,x)}$ of ${\displaystyle X}$ at ${\displaystyle x}$ is the inverse limit

${\displaystyle \pi _{1}(X,x)=\varprojlim _{i\in I}{\operatorname {Aut} }_{X}(X_{i}),}$

with the inverse limit topology.

The functor ${\displaystyle F}$ is now a functor from ${\displaystyle C}$ to the category of finite and continuous ${\displaystyle \pi _{1}(X,x)}$-sets, and establishes an equivalence of categories between ${\displaystyle C}$ and the category of finite and continuous ${\displaystyle \pi _{1}(X,x)}$-sets.[2]

The most basic example of is ${\displaystyle \pi _{1}(\operatorname {Spec} k)}$, the étale fundamental group of a field ${\displaystyle k}$. Essentially by definition, the fundamental group of ${\displaystyle k}$ can be shown to be isomorphic to the absolute Galois group ${\displaystyle \operatorname {Gal} (k^{sep}/k)}$. More precisely, the choice of a geometric point of ${\displaystyle \operatorname {Spec} (k)}$ is equivalent to giving a separably closed extension field ${\displaystyle K}$, and the étale fundamental group with respect to that base point identifies with the Galois group ${\displaystyle \operatorname {Gal} (K/k)}$. This interpretation of the Galois group is known as Grothendieck's Galois theory.

More generally, for any geometrically connected variety ${\displaystyle X}$ over a field ${\displaystyle k}$ (i.e., ${\displaystyle X}$ is such that ${\displaystyle X^{sep}:=X\times _{k}k^{sep}}$ is connected) there is an exact sequence of profinite groups

${\displaystyle 1\to \pi _{1}(X^{sep},{\overline {x}})\to \pi _{1}(X,{\overline {x}})\to \operatorname {Gal} (k^{sep}/k)\to 1.}$

Bhatt & Scholze (2015, §7) have introduced a variant of the étale fundamental group called the pro-étale fundamental group. It is constructed by considering, instead of finite étale covers, maps which are both étale and satisfy the valuative criterion of properness. For geometrically unibranch schemes (e.g., normal schemes), the two approaches agree, but in general the pro-étale fundamental group is a finer invariant: its profinite completion is the étale fundamental group.

• étale morphism
• Fundamental group
• Fundamental group scheme

• Bhatt, Bhargav; Scholze, Peter (2015), "The pro-étale topology for schemes", Astérisque: 99–201, arXiv:1309.1198, Bibcode:2013arXiv1309.1198B, MR 3379634
• Friedlander, Eric M. (1982), Étale homotopy of simplicial schemes, Annals of Mathematics Studies, vol. 104, Princeton University Press, ISBN 978-0-691-08288-2
• 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.