étale fundamental group

Topological analogue/informal discussion

In algebraic topology, the fundamental group

π

1

(

X

,

x

)

{\displaystyle \pi _{1}(X,x)}

of a pointed topological space

(

X

,

x

)

{\displaystyle (X,x)}

is defined as the group of homotopy classes of loops based at

x

{\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

X

{\displaystyle X}

often fails to have a "universal cover" that is finite over

X

{\displaystyle X}

, so one must consider the entire category of finite étale coverings of

X

{\displaystyle X}

. One can then define the étale fundamental group as an inverse limit of finite automorphism groups.

Formal definition

Let

X

{\displaystyle X}

be a connected and locally noetherian scheme, let

x

{\displaystyle x}

be a geometric point of

X

,

{\displaystyle X,}

and let

C

{\displaystyle C}

be the category of pairs

(

Y

,

f

)

{\displaystyle (Y,f)}

such that

f

:

Y

→

X

{\displaystyle f\colon Y\to X}

is a finite étale morphism from a scheme

Y

.

{\displaystyle Y.}

Morphisms

Examples and theorems

The most basic example of is

π

1

(

Spec

⁡

k

)

{\displaystyle \pi _{1}(\operatorname {Spec} k)}

, the étale fundamental group of a field

k

{\displaystyle k}

. Essentially by definition, the fundamental group of

k

{\displaystyle k}

can be shown to be isomorphic to the absolute Galois group

Gal

⁡

(

k

s

e

p

/

k

)

{\displaystyle \operatorname {Gal} (k^{sep}/k)}

. More precisely, the choice of a geometric point of

Spec

Schemes over a field of characteristic zero

For a scheme

X

{\displaystyle X}

that is of finite type over

C

{\displaystyle \mathbb {C} }

, the complex numbers, there is a close relation between the étale fundamental group of

X

{\displaystyle X}

and the usual, topological, fundamental group of

X

(

C

)

{\displaystyle X(\mathbb {C} )}

, the complex analytic space attached to

X

{\displaystyle X}

. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of

π

1

(

X

)

{\displaystyle \pi _{1}(X)}

. This is a consequence of the Riemann existence theorem, which says that all finite étale coverings of

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

For an algebraically closed field

k

{\displaystyle 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

A

k

1

{\displaystyle \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

{\displaystyle U}

which takes into account only covers that are tamely ramified along

D

{\displaystyle D}

, where

X

{\displaystyle X}

is some compactification and

D

{\displaystyle D}

is the complement of

U

{\displaystyle U}

in

Affine schemes over a field of characteristic p

It turns out that every affine scheme

X

⊂

A

k

n

{\displaystyle X\subset \mathbf {A} _{k}^{n}}

is a

K

(

π

,

1

)

{\displaystyle K(\pi ,1)}

-space, in the sense that the étale homotopy type of

X

{\displaystyle X}

is entirely determined by its étale homotopy group. Note

π

=

π

1

e

t

(

X

,

x

¯

)

{\displaystyle \pi =\pi _{1}^{et}(X,{\overline {x}})}

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.

Friedlander (1982) studies higher étale homotopy groups by means of the étale homotopy type of a scheme.

The pro-étale fundamental group

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.