Galois Extensions and Bundles — knowlified transcript

Core idea

User

Let's see, doesn't an argument of Grothendieck's give that a Galois extension of fields is equivalent data to a principal Gal(K/F)-bundle?

Assistant

Yes, with one important qualification: it is a principal bundle in the étale topology, not usually in the ordinary topology.

Let $K/F$ be a finite Galois extension and let

Then the corresponding principal $G$-bundle is

Here:

The group $G$ acts on $K$ by $F$-algebra automorphisms, hence contravariantly acts on $\operatorname{Spec}K$ over $\operatorname{Spec}F$. This makes

a $G$-torsor, i.e. a principal $G$-bundle, in the étale site.

The key algebraic condition is the Galois identity

given by

Geometrically, this says

That is the torsor condition: over $\operatorname{Spec}K$, the bundle trivializes into one copy of the fiber for each $g\in G$.

So the slogan is:

But more precisely:

determines a principal $G$-bundle

in the étale topology.

Conversely, a connected finite étale $G$-torsor over $\operatorname{Spec}F$ gives a finite Galois $F$-algebra. If the total space is connected, that algebra is a field extension $K/F$.

User

What is the etale topology? Also how do I type the accented e in macOS?

Assistant

The étale topology is a Grothendieck topology on schemes. It is not usually a topology on the point-set $|X|$. It is a rule for which morphisms into $X$ count as “open covers.”

For a scheme $X$, define the small étale site $X_{\mathrm{\acute{e}t}}$ as follows.

Objects:

where $U$ is a scheme and $U \to X$ is an étale morphism.

Morphisms:

are $X$-morphisms $V\to U$.

Covering families:

such that each $U_i\to U$ is étale and

So the slogan is:

An étale morphism is a morphism of schemes that is:

It is the algebraic-geometry analogue of a local diffeomorphism.

For fields, this becomes very concrete:

is finite étale iff $K/F$ is finite separable. Therefore a finite Galois extension $K/F$ gives an étale cover

This is why the principal bundle statement is made in the étale topology. In the ordinary Zariski topology, $\operatorname{Spec}F$ has only one point, so there are no nontrivial local covers that can trivialize $\operatorname{Spec}K\to\operatorname{Spec}F$. The étale topology adds algebraic covers like finite separable extensions.

For typing é on macOS:

So:

1. Press Option + e
2. Release
3. Press e

For uppercase:

giving É.

In LaTeX:

\'{e}tale

or directly:

étale

User

Sorry what is X and how is it related to K and F?

Assistant

In the general definition, $X$ is just an arbitrary scheme.

For your Galois-extension example, take

Then the étale cover/principal bundle is

So in this example:

The map

comes from the field inclusion

by contravariance of $\operatorname{Spec}$. In algebraic geometry, ring maps reverse direction:

induces

The reason I introduced a general $X$ was to define the étale topology on any scheme. For a general scheme $X$, objects of $X_{\acute{e}t}$ are étale maps

In the Galois case, the base scheme is specifically

and one distinguished étale object over it is

If $K/F$ is finite separable, this map is finite étale. If $K/F$ is finite Galois with group $G$, it is a principal $G$-bundle in the étale topology.

User

axiomatically define scheme and give examples.

Assistant

A scheme is a space locally modeled on commutative rings.

Fix $\mathbf{CRing}$ = category of commutative unital rings.

1. Locally ringed space

A locally ringed space is a pair

where:

Topological space: $X$ is a topological space.

Structure sheaf: $\mathcal O_X$ is a sheaf of commutative unital rings on $X$.

Local stalks: For every $x\in X$, the stalk

is a local ring, i.e. it has a unique maximal ideal.

The sheaf $\mathcal O_X$ is the “ring of functions” on $X$, though its elements may be algebraic rather than ordinary functions.

2. Affine scheme

Given $A\in \mathbf{CRing}$, define

as follows.

Points:

Closed sets: For each ideal $I\subseteq A$,

These define the Zariski topology.

Basic open sets: For $f\in A$,

Structure sheaf: Define

where $A_f$ is the localization of $A$ obtained by inverting $f$. These assignments glue to a sheaf $\mathcal O_{\operatorname{Spec}A}$.

Stalks: At $\mathfrak p\in \operatorname{Spec}A$,

Since $A_{\mathfrak p}$ is local, $\operatorname{Spec}A$ is a locally ringed space.

An affine scheme is a locally ringed space isomorphic to

for some $A\in\mathbf{CRing}$.

3. Scheme

A scheme is a locally ringed space

such that:

Affine local cover:

for open subsets $U_i\subseteq X$, and for each $i$, there exists a commutative unital ring $A_i$ such that

as locally ringed spaces.

So:

Examples

1. One-point scheme from a field

If $F$ is a field, then

It has one point, and its structure sheaf has stalk

This is the scheme appearing in the Galois-extension example:

2. The affine line over a field

For a field $k$,

Its points are prime ideals of $k[x]$. For example, if $k$ is algebraically closed, the closed points are

for $a\in k$, plus the generic point $(0)$.

3. Affine $n$-space

This is the scheme version of $n$-dimensional affine space over $k$.

4. The integer scheme

Its points are

where $(p)$ ranges over prime ideals generated by rational primes $p$.

The point $(0)$ is the generic point. The points $(p)$ are closed.

5. Projective space

Projective $n$-space over $k$ is

It is not usually affine, but it is a scheme because it has an affine open cover

with

So schemes include both affine objects like $\operatorname{Spec}A$ and objects glued from affine pieces, like $\mathbb P^n_k$.