Outline

The seminar Galois theory in algebra and topology had as a primary aim to recall the main results of Galois theory, introduce the theory of covering spaces and draw parallels between the two.

With that out of the way, we turned to Riemann surfaces. By definition, a complex variety combines both a ring theoretic and a topological component. Thus, working in this setting allows to bridge the two notions and combine the tools of these two fields in order to study appropriate objects. We used this bridge to compute the absolute Galois group of the field of meromorphic functions in one variable.

Finally, we kept exploring the notions close to that of covering spaces. We introduced local systems, their representations, and the Riemann-Hilbert correspondence.

The aim of these notes is not to give an exhaustive reference for the seminar: this duty is left to the main reference Galois Groups and Fundamental Groups and to the reports of the students, who did a very good job filling in the missing details. Instead, we will only state results and at most point out what plays a big role in the proofs. Moreover, we will take all of the necessary category theory constructions for granted and use this language freely. What we will try to do is to link these results together and motivate the direction the seminar took.

Conventions

We use a sans serif font to denote categories, for example \(\mathsf{Sets}\) denotes the category of sets. We use the standard notation \(k^s\) for the separable closure of a field.

Part 1: Galois and covering theories

We will start by discussing the Galois correspondences in field theory and covering theory. We will draw parallels between them and we will show the development of these results from the classical correspondences to more refined ones.

Galois theory

Galois theory deals with algebraic field extensions. Therefore, in the whole section all field extensions are algebraic, but not necessarily finite.

Finite (classical) results

Roughly speaking, Galois theory says the following two facts.

1. There exist certain finite field extensions for which the group of symmetries of the extension completely describes the set of intermediate extensions.
2. Any finite field can be embedded into one with the above property.

Essentially, this means that any problem that can be expressed in terms of extensions of fields, can be rephrased in group theoretic terms. I will not list them, but classically this led to a multitude of interesting results.

Let’s now recall the formal definitions of these informal ideas.

Definition 1: (finite) Galois extension

A finite extension of fields \(L|k\) is called Galois, if the following equality holds

\[ \# \operatorname{Aut}(L|k) = [L:k] ,\]

i.e. if the order of the group of automorphisms of \(L\) fixing \(k\) coincides with the degree of the extension. Analogously, this means that

\[ \# \operatorname{Hom}_{ k }(L, k^s) = [L:k] ,\]

where \(k^s\) denotes a separable closure of \(k\).

Galois extensions satisfy a maximality condition on the group of automorphisms. Let’s expand on this, since it is going to be very useful in generalizing this definition to other settings. Before doing so, recall that \(\operatorname{Aut}(L|k)\) acts on \(\operatorname{Hom}_k(L,k^s)\) on the right by precomposition.

Remark 2: characterization of Galois extensions

Consider a finite extension of fields \(L|k\). The following are equivalent:

• \(\# \operatorname{Aut}(L|k) = [L:k]\);
• for any \(\sigma \in \operatorname{Aut}(k^s|k)\), we have \(\sigma(L) \subset L\);
• the action of \(\operatorname{Aut}(L|k)\) on \(\operatorname{Hom}_k(L,k^s)\) is transitive;
• The only elements fixed by the whole \(\operatorname{Aut}(L|k)\) are exactly the elements of \(k\).

The maximality is then encoded in the following fact.

Remark 3: maximality of Galois groups

The group \(\operatorname{Aut}(k^s|k)\) acts on \(\operatorname{Hom}_k(L,k^s)\) on the left by postcomposition. The action of \(\operatorname{Aut}(L|k)^\text{op}\) on \(\operatorname{Hom}_k(L,k^s)\) recovers that of \(\operatorname{Aut}(k^s|k)\).

A more precise statement can be found in theorem 11 (+)(–). As a consequence, for any \(k\)-polynomial that admits a root in \(L\), it splits in linear factors.

Then, the Galois correspondence tells us the following.

Theorem 4: Galois correspondence

There is a one to one inclusion reversing correspondence between the set of subextensions of a Galois extension \(L|k\) and the set of subgroups of \(\operatorname{Gal}(L|k) \coloneqq \operatorname{Aut}(L|k)\). Moreover, normal subgroups correspond to Galois extension of \(k\).

Proof

The maps witnessing the correspondence are

\[ H \leq G \mapsto L^H := \{ x \in L \ |\ \sigma(x) = x \text{ for all } \sigma \in H \} \]

and

\[ (L|K|k) \mapsto \operatorname{Gal}(L|K) .\]

See theorem 1.2.5 of Galois Groups and Fundamental Groups for the actual proof. ∎

Before moving on to the theory of covering spaces, it is instructive to look at the theory of infinite extensions.

Infinite Galois theory

The first definition of Galois extensions does not make sense in the infinite case. We will replace it with the characterization given in remark 2 (+)(–).

Classically, it is important to distinguish the infinite case from the finite case, because of different behaviour at the level of the correspondence. Let’s investigate it.

First, we need to understand the group of automorphisms of an infinite Galois extension \(L|k\).

Lemma 5

There is an isomorphism of groups

\[ \operatorname{Aut}(L|k) \simeq \operatorname{lim} \operatorname{Gal}(K|k) ,\]

where \(\operatorname{lim}\) denotes the inverse limit of groups of the diagram of finite Galois subextensions \(L|K|k\). In particular, the Galois group of \(L|k\), denoted again by \(\operatorname{Gal}(L|k) : = \operatorname{Aut}(L|k)\), is a profinite group.

Proof

The proof can be read at proposition 1.3.5 of Galois Groups and Fundamental Groups. ∎

With this in our toolbox, we can see what differs from the classical case.

Warning 6

Assume \(L|k\) is an extension of countable order. In particular, the set of Galois subextensions of \(L|k\) is also going to be countable. One can prove that a profinite group is either finite or at least uncountable. Then, since any cyclic subgroup is at most countable and countable unions of countable sets are again countable, we know that any profinite group must have uncountably many subgroups! Therefore, for cardinality reasons, we cannot naïvely extend the finite Galois correspondence to the infinite case.

The above warning is a reinterpretation of an example due to Dedekind. It was then Krull who managed to adapt the Galois correspondence to the infinite case. The key was to add a topology to the Galois group.

Remark 7

An inverse limit of finite groups carries a natural topology: the inverse limit topology. In the case of the Galois group, this is also called the Krull topology.

Now we can finally state the infinite Galois correspondence.

Theorem 8: infinite Galois correspondence

There is a one to one inclusion reversing correspondence between the set of subextensions of a Galois extension \(L|k\) and the set of closed subgroups of \(\operatorname{Gal}(L|k)\).

The key observation is that \(\operatorname{Gal}(L|K) \leq \operatorname{Gal}(L|k)\) is a closed subgroup for any subextension \(L|K|k\).

Notice also that the above recovers the finite case of theorem 4 (+)(–). Indeed, one can see \(\operatorname{Gal}(L|k)\) as a discrete topological group. In particular all its subgroups are closed.

So far, we completely ignored one important part of the correspondence. Since it will play a crucial role later on, let’s recall it.

Theorem 9: infinite Galois correspondence - continuation

Along the Galois correspondence, normal subgroups of \(\operatorname{Gal}(L|k)\) correspond to Galois subextensions \(L|K|k\). Moreover, we have

\[ \operatorname{Gal}(K|k) \simeq \frac{ \operatorname{Gal}(L|k) }{ \operatorname{Gal}(L|K) } .\]

Proof

The proof can be read at proposition 1.3.11 of Galois Groups and Fundamental Groups. ∎

In particular, this part of the theorem tells us that there is a Galois group to rule them all. Indeed, all the Galois groups of Galois extensions of \(k\) are quotients of \(\operatorname{Gal}(k^s|k)\) along a closed subgroup.

A more global approach

We want to reinterpret the above results in a more “global” way. This work is mainly due to Grothendieck, and we highly suggest to read what follows in the language of schemes (especially when comparing with the results on the fundamental group). Practically speaking, this means that you should read all of the results by inverting the direction of the arrows.

Remark and notation 10

Theorem 9 (+)(–) leads us towards a change of perspective. Indeed, we started by studying a group that depends on the extension \(L\) of \(k\). After this theorem, we shift the focus to a group that only depends on \(k\): \(\operatorname{Gal}(k^s|k)\). To make this dependence more clear, we will start using the following notation

\[ \operatorname{Gal}(k) : = \operatorname{Gal}(k^s|k) \]

and we will start calling this object the absolute Galois group of \(k\).

Theorem 9: infinite Galois correspondence - continuation

Along the Galois correspondence, normal subgroups of \(\operatorname{Gal}(L|k)\) correspond to Galois subextensions \(L|K|k\). Moreover, we have

\[ \operatorname{Gal}(K|k) \simeq \frac{ \operatorname{Gal}(L|k) }{ \operatorname{Gal}(L|K) } .\]

It would be of interest to try to restate the Galois correspondence in terms of only this invariant of \(k\). Let’s see how to do this.

Let \(\mathsf{FinSep}_k\) denote the category of finite separable extensions of \(k\) and morphisms of \(k\)-algebras. Denote by \(\operatorname{Gal}(k)\)-\(\mathsf{Sets}_f^{ \text{tr} }\) the category of finite sets with a continuous and transitive action of \(\operatorname{Gal}(k)\) and \(\operatorname{Gal}(k)\)-equivariant morphisms.

Finally, pick a finite separable extension \(L|k\). Then, \(\operatorname{Gal}(k)\) acts on the left on \(\operatorname{Hom}_k(L,k^s)\) by postcomposition. As a consequence of theorem 8 (+)(–), the just defined action is continuous.

Theorem 11: Galois correspondence - Grothendieck version (v1)

The functor

\[ \operatorname{Hom}_k(-,k^s)\colon \mathsf{FinSep}_k^{\mathrm{op}} \to \operatorname{Gal}(k)\text{-}\mathsf{Sets}_f^{ \text{tr} } \]

is an equivalence of categories.

Moreover, if \(L|k\) is Galois, there exists an open normal subgroup \(U \triangleleft \operatorname{Gal}(k)\) and a \(\operatorname{Gal}(k)\)-equivariant bijection

\[ \operatorname{Hom}_k(L,k^s) \simeq \operatorname{Gal}(k)/U .\]

Proof

The proof can be read at proposition 1.5.2 of Galois Groups and Fundamental Groups. ∎

Notice that the last statement fully recovers the classical result, i.e. theorem 4 (+)(–). Nevertheless, the statement sounds a bit artificial: we need to impose some strong restrictions on the target category to obtain an equivalence. Let’s take a closer look at them.

Remark 12

The conditions we impose on the representations are the following: they have to be continuous, transitive and they act on finite sets. Here what we can say about them.

• Continuity: this cannot be avoided, as otherwise it would give access to any subset of \(\operatorname{Gal}(k)\). As discussed in warning 6 (+)(–), this would prevent us from getting essential surjectivity.

  Warning 6

  Assume \(L|k\) is an extension of countable order. In particular, the set of Galois subextensions of \(L|k\) is also going to be countable. One can prove that a profinite group is either finite or at least uncountable. Then, since any cyclic subgroup is at most countable and countable unions of countable sets are again countable, we know that any profinite group must have uncountably many subgroups! Therefore, for cardinality reasons, we cannot naïvely extend the finite Galois correspondence to the infinite case.

• Transitivity: there is no particular reason why this condition should be kept. As is going to be clear from theorem 24 (+)(–), this holds for representations arising from connected objects. From this point of view, field extensions are connected. (This is indeed true for the associated scheme.) In the section about Grothendieck’s reformulation we will introduce the appropriate algebraic notion corresponding to non-transitive representations.

  Theorem 24: Galois correspondence (v2)

  Let \(X\) be a connected and locally simply connected topological space and fix a point \(x \in X\). The functor

  \[ \operatorname{Fib}_x\colon \mathsf{Cov}_X \to \pi_1(X,x)\text{-}\mathsf{Sets} \]

  is an equivalence of categories. Moreover, connected covers correspond to sets with transitive action and Galois covers to quotients of \(\pi_1(X,x)\) by normal subgroups.

• Finiteness: also this condition is a bit artificial. It is needed because \(\operatorname{Hom}_k(L,k^s) \) is a discrete topological space. Similarly to Galois groups, one can endow this with the structure of a profinite set (hence with its topology), which would allow to extend the above result also to infinite extensions.

Let’s expand a bit on the last point. The point of the seminar Galois categories and étale fundamental groups is to generalize these constructions to more general base spaces. One could ask the same question about finiteness in that setting. Trying to answer that question would naturally lead to being interested in the pro-étale topology of Bhatt and Scholze.

Grothendieck’s reformulation

We suggest reading this section either after the section on the fundamental group. (If you have a solid base in scheme theory, read ahead but try to compare the results with the aforementioned section). Indeed, in this section we first want to highlight the geometric picture contained in the previous constructions and then generalize theorem 11 (+)(–) as explained in the transitivity point of remark 12 (+)(–).

Let’s start with the geometric intuition.

Remark 13

Recall that \(\operatorname{Spec}\colon \mathsf{Rings}^{\mathrm{op}} \to \mathsf{AffSch}\) is an equivalence of categories from the opposite category of the category of rings into affine schemes. From this and remark 3 (+)(–) we obtain that

\[ \operatorname{Hom}_{ \operatorname{Spec} k } (\operatorname{Spec} k^s, \operatorname{Spec} k^s) \simeq \operatorname{Hom}_k (k^s, k^s) \simeq \operatorname{Gal}(k) .\]

Remark 3: maximality of Galois groups

The group \(\operatorname{Aut}(k^s|k)\) acts on \(\operatorname{Hom}_k(L,k^s)\) on the left by postcomposition. The action of \(\operatorname{Aut}(L|k)^\text{op}\) on \(\operatorname{Hom}_k(L,k^s)\) recovers that of \(\operatorname{Aut}(k^s|k)\).

In particular, the absolute Galois group of \(k\) is defined as the group of deck transformations (see definition 18 (+)(–)) of \(\operatorname{Spec} k^s\). Moreover, since any algebraic field extension \(L|k\) embeds into \(k^s\), we can see \(\operatorname{Spec}k^s\) as a sort of universal cover of \(\operatorname{Spec}k\). (For this, compare with remark 29 (+)(–) and theorem 24 (+)(–) and recall that field extensions are connected over \(\operatorname{Spec}k\).)

Definition 18: deck transformations

Given a map of topological spaces \(p\colon Y \to X\), we define the group of deck transformations as the group \(\operatorname{Aut}(Y|X)\) of automorphisms of \(p\).

Remark 29

Let \(\widetilde{X}_x\) be the universal cover of \(X\) as of definition 27 (+)(–).

Definition 27: universal cover of a pointed space

Let \(X\) be a path-connected and locally simply connected topological space. Fix a point \(x \in X\). We call universal cover of \((X,x)\) the covering space \(\widetilde{X}_x \to X\) representing the functor \(\operatorname{Fib}_x\colon \mathsf{Cov}_X \to \mathsf{Sets}\).

• Any connected Galois cover \(Y\) is a quotient of the universal one. In particular it receives a map \(\widetilde{X}_x \to Y\). Moreover, if \(Y \simeq G \backslash \widetilde{X}_x\), we get an isomorphism \[ G \simeq \operatorname{Aut}(\widetilde{X}_x|Y) .\] Since \(\widetilde{X}_x \to Y\) is also the universal cover of \(Y\), we get \(G \simeq \pi_1(Y,y)\) for any \(y \in Y_x\).
• Finally, let’s recall how to compute the group of deck transformations of \(Y\): \[ \operatorname{Aut}(Y|X) \simeq \frac{ \operatorname{Aut}(\widetilde{X}_x|X) }{ \operatorname{Aut}(\widetilde{X}_x|Y) } .\]

Theorem 24: Galois correspondence (v2)

Let \(X\) be a connected and locally simply connected topological space and fix a point \(x \in X\). The functor

\[ \operatorname{Fib}_x\colon \mathsf{Cov}_X \to \pi_1(X,x)\text{-}\mathsf{Sets} \]

is an equivalence of categories. Moreover, connected covers correspond to sets with transitive action and Galois covers to quotients of \(\pi_1(X,x)\) by normal subgroups.

Finally, it turns out that the correct notion of fibre for Galois theory is that of geometric fibre. By this we mean that, for an algebraic extension \(L|k\), the fibre to look at is at the point \(\operatorname{Spec} k^s \to \operatorname{Spec} k\) corresponding to \(k \hookrightarrow k^s\). This is given by

\[ \operatorname{Hom}_{ \operatorname{Spec} k } (\operatorname{Spec} k^s, \operatorname{Spec} L) \simeq \operatorname{Hom}_k (L, k^s) .\]

As we pointed out in remark 12 (+)(–), we need to introduce the correct notion for a “non-connected covering” space of a field. Under the geometric fibre functor of the Galois correspondence of theorem 11 (+)(–), these objects should correspond to non-transitive actions of \(\operatorname{Gal}(k)\).

Definition 14: (finite) étale algebras

An étale algebra over a field \(k\) is a \(k\)-algebra \(A\) for which there is an isomorphism of \(k\)-algebras

\[ A \simeq \prod_{ i = 1 }^n L_i ,\]

where \(L_i|k\) is a finitely generated separable field extension. If \(L_i|k\) is a finite field extension for all \(i\), then \(A\) is called finite étale algebra.

The category of (finite) étale algebras, denoted by \(\mathsf{Ét}_k\) (resp. \(\mathsf{FinÉt}_k\)), is the full subcategory of \(\mathsf{Alg}_k\) spanned by (finite) étale algebras.

With a little algebraic geometry, it is easy to understand why these would be the correct notion for “non-connected coverings” of \(k\). Indeed, under the \(\operatorname{Spec}\) functor, finite products of rings are sent to the disjoint union of the spectra of the factors. In other words, we are essentially taking the disjoint unions of separable field extensions (which we consider as the connected covering spaces of a field).

Theorem 15: Galois correspondence - Grothendieck version (v2)

The functor

\[ \operatorname{Hom}_k(-,k^s)\colon \mathsf{FinÉt}_k^{\mathrm{op}} \to \operatorname{Gal}(k)\text{-}\mathsf{Sets}_f \]

is an equivalence of categories. The source category is that of definition 14 (+)(–).

Definition 14: (finite) étale algebras

An étale algebra over a field \(k\) is a \(k\)-algebra \(A\) for which there is an isomorphism of \(k\)-algebras

\[ A \simeq \prod_{ i = 1 }^n L_i ,\]

where \(L_i|k\) is a finitely generated separable field extension. If \(L_i|k\) is a finite field extension for all \(i\), then \(A\) is called finite étale algebra.

The category of (finite) étale algebras, denoted by \(\mathsf{Ét}_k\) (resp. \(\mathsf{FinÉt}_k\)), is the full subcategory of \(\mathsf{Alg}_k\) spanned by (finite) étale algebras.

This equivalence restricts to that of theorem 11 (+)(–) on finite separable field extensions.

Theorem 11: Galois correspondence - Grothendieck version (v1)

The functor

\[ \operatorname{Hom}_k(-,k^s)\colon \mathsf{FinSep}_k^{\mathrm{op}} \to \operatorname{Gal}(k)\text{-}\mathsf{Sets}_f^{ \text{tr} } \]

is an equivalence of categories.

Moreover, if \(L|k\) is Galois, there exists an open normal subgroup \(U \triangleleft \operatorname{Gal}(k)\) and a \(\operatorname{Gal}(k)\)-equivariant bijection

\[ \operatorname{Hom}_k(L,k^s) \simeq \operatorname{Gal}(k)/U .\]

Proof

The proof can be read at proposition 1.5.4 of Galois Groups and Fundamental Groups. ∎

Covering spaces

Let’s now shift our focus to covering spaces. We will first introduce the definition and their most important properties. Then we will turn to studying a classification result close in spirit to theorem 11 (+)(–).

A classical Galois correspondence

Definition 16: covering spaces

A covering space, or more simply cover, \(Y\) of \(X\) is a topological space with a continuous surjective morphism \(p\colon Y \to X\) satisfying the following condition. For any \(x \in X\) there is an open neighbourhood \(x \in U \subset X\) for which \(p^{-1}(U)\) is a disjoint union of open subsets \(V_i \subset Y\) and the restriction

\[ \left.p\right|_{ V_i }\colon V_i \to U \]

is a homeomorphism for all indices \(i\).

The category of covers of \(X\), denoted by \(\mathsf{Cov}_X\), is the full subcategory of \(\mathsf{Top}_X\) spanned by covers of \(X\). In particular, morphisms between covers make the diagram with the projections commute.

Let’s see a couple of examples that showcase a huge deal of the theory we are going to develop. First, recall that \(\mathbb{C}^\times\) is the multiplicative group of invertible elements of \(\mathbb{C}\). It carries the euclidean topology, and with it it is a deformation retract of the circle \(\mathbb{S}^1\).

Example 17: covering spaces

• Fix \(0 < n \in \mathbb{N}\). The map \(p_n\colon \mathbb{C}^\times \to \mathbb{C}^\times\) defined by \[ x \mapsto x^n \] is a covering space.
• The map \(\operatorname{exp}\colon \mathbb{C} \to \mathbb{C}^\times\) is a covering space.

The above induce covers of \(\mathbb{S}^1\). Moreover, let’s remark that the first family of examples is defined algebraically (hence we would expect it to make sense in the theory of schemes), whereas the last example is transcendental in nature. Indeed, one such morphism does not have an algebraic analogue. To understand why, let’s restrict the exponential map to the fibre of any point in the target. Intuitively, for an algebraic morphism the fibres would be the roots of a certain polynomial (depending on the point of \(\mathbb{C}^\times\)), hence they would only be finitely many and this is not the case.

Let’s turn now towards the Galois correspondence. Recall that, for a group \(G\) acting on a topological space \(X\), one can form the topological quotient \(G \backslash X\) being the space of orbits under the action. Moreover, if the action is even, the quotient map is itself a covering space.

Definition 18: deck transformations

Given a map of topological spaces \(p\colon Y \to X\), we define the group of deck transformations as the group \(\operatorname{Aut}(Y|X)\) of automorphisms of \(p\).

It is easy to show that the group of deck transformations acts evenly on \(Y\).

Definition 19: Galois covering

Consider a covering space \(p\colon Y \to X\) and denote by \(G : = \operatorname{Aut}(Y|X)\) the group of deck transformations of \(p\). The cover \(p\) is Galois if \(Y\) is connected, \(p\) factors as \(Y \to G \backslash Y \to X\) and the last map is a homeomorphism.

Equivalently, one can look at the induced action of \(\operatorname{Aut}(Y|X)\) on fibres. Recall the standard notation for fibres. Take a continuous map \(p\colon Y \to X\) and \(x \in X\). We denote the fibre of \(p\) at \(x\) by \(Y_x\).

Proposition 20

A covering space \(p\colon Y \to X\) is Galois iff \(Y\) is connected and \(\operatorname{Aut}(Y|X)\) acts transitively on one (and hence any) fibre \(Y_x\), for \(x \in X\).

Let’s remark that these notions strongly resemble the field theoretic definitions of remark 2 (+)(–). The definition resembles the last point on fixed points. Moreover, accepting that the set \(\operatorname{Hom}_k(L,k^s)\) represents the correct notion of fibre, the connection between proposition 20 (+)(–) and the third point of remark 2 (+)(–) is even more direct.

As promised, here is the Galois correspondence for covering spaces in the style of theorem 4 (+)(–).

Theorem 21: classical Galois correspondence for covering spaces

Fix a Galois covering space \(p\colon Y \to X\) and denote \(G : = \operatorname{Aut}(Y|X)\). There is a one to one inclusion reversing correspondence between the set of intermediate coverings \(Y \to Z \to X\) and the set of subgroups of \(G\). The covering \(Y \to Z\) is always Galois and normal subgroups correspond to Galois coverings \(Z \to X\).

Proof

The maps witnessing the correspondence are

\[ H \leq G \mapsto H \backslash Y \]

and

\[ (Y \to Z \to X ) \mapsto \operatorname{Aut}(Y|Z) .\]

The proof of the actual statements can be read at theorem 2.2.10 of Galois Groups and Fundamental Groups. ∎

The fundamental group

Remark and notation 10 (+)(–) provides a change of perspective for Galois theory. In this section we want to see how the same can be achieved in the theory of covering spaces. We will have to substitute the absolute Galois group with the fundamental group of our base \(X\).

We will assume you are already familiar with the notion of fundamental group of a topological space, but let us still quickly recall a possible definition.

Definition 22: fundamental group of a topological space

The fundamental group of a topological space \(X\) centered at \(x \in X\) is denoted by \(\pi_1(X,x)\) and is defined as follows. Its underlying set is given by the equivalence classes of continuous loops with endpoint \(x\) up to end point preserving homotopy. The group operation is induced by concatenation of loops.

To see how this group can replace the Galois group of a field extension, we need a way to lift paths from the base to its covering spaces. This is the content of the following fundamental result about covering spaces. As in topology, we denote by \(I\) the closed unit interval \([0,1] \subset \mathbb{R}\).

Theorem 23: path and homotopy lifting property

Fix a covering space \(p\colon Y \to X\) and a point \(y \in Y\). Call \(x : = p(y)\). Then

1. Given a path \(\gamma\colon I \to X\) with \(\gamma(0) = x\), there is a unique lift \(\widetilde{\gamma}\colon I \to Y\) such that \(p \circ \widetilde{\gamma} = \gamma\) and \(\widetilde{\gamma}(0) = y\).
2. Given a homotopy \(H\colon I^2 \to X\) between two paths \(\gamma\) and \(\sigma\), there is a unique lift to a homotopy \(\widetilde{H}\colon I^2 \to Y\) between \(\widetilde{\gamma}\) and \(\widetilde{\sigma}\) (the lifts of the previous point). Moreover, if \(H\) fixes endpoints, also \(\widetilde{H}\) does.

Proof

The proof can be read at lemma 2.3.2 of Galois Groups and Fundamental Groups. ∎

Theorem 23 allows us to define an action of \(\pi_1(X,x)\) on \(Y_x\) for all \(x \in X\). As a corollary, we can define a functor

\[ \operatorname{Fib}_x\colon \mathsf{Cov}_X \to \pi_1(X,x)\text{-}\mathsf{Sets} .\]

Theorem 24: Galois correspondence (v2)

Let \(X\) be a connected and locally simply connected topological space and fix a point \(x \in X\). The functor

\[ \operatorname{Fib}_x\colon \mathsf{Cov}_X \to \pi_1(X,x)\text{-}\mathsf{Sets} \]

is an equivalence of categories. Moreover, connected covers correspond to sets with transitive action and Galois covers to quotients of \(\pi_1(X,x)\) by normal subgroups.

Proof

The proof can be read at theorem 2.3.4 of Galois Groups and Fundamental Groups. ∎

Before commenting on the hypothesis, let’s compare theorem 24 with theorem 15 (+)(–).

Remark 25: Differences between algebra and topology

• On the face of it, the topological statement is a stronger result, as its image is the category of all \(\pi_1(X,x)\) sets, and not only the finite ones. This is not due to intrinsic differences between the two contexts and it is only due to the shortcomings of the seminar, as pointed out by remark 12 (+)(–).

  Remark 12

  The conditions we impose on the representations are the following: they have to be continuous, transitive and they act on finite sets. Here what we can say about them.

  • Continuity: this cannot be avoided, as otherwise it would give access to any subset of \(\operatorname{Gal}(k)\). As discussed in warning 6 (+)(–), this would prevent us from getting essential surjectivity.

    Warning 6

    Assume \(L|k\) is an extension of countable order. In particular, the set of Galois subextensions of \(L|k\) is also going to be countable. One can prove that a profinite group is either finite or at least uncountable. Then, since any cyclic subgroup is at most countable and countable unions of countable sets are again countable, we know that any profinite group must have uncountably many subgroups! Therefore, for cardinality reasons, we cannot naïvely extend the finite Galois correspondence to the infinite case.

  • Transitivity: there is no particular reason why this condition should be kept. As is going to be clear from theorem 24 (+)(–), this holds for representations arising from connected objects. From this point of view, field extensions are connected. (This is indeed true for the associated scheme.) In the section about Grothendieck’s reformulation we will introduce the appropriate algebraic notion corresponding to non-transitive representations.

    Theorem 24: Galois correspondence (v2)

    Let \(X\) be a connected and locally simply connected topological space and fix a point \(x \in X\). The functor

    \[ \operatorname{Fib}_x\colon \mathsf{Cov}_X \to \pi_1(X,x)\text{-}\mathsf{Sets} \]

    is an equivalence of categories. Moreover, connected covers correspond to sets with transitive action and Galois covers to quotients of \(\pi_1(X,x)\) by normal subgroups.

  • Finiteness: also this condition is a bit artificial. It is needed because \(\operatorname{Hom}_k(L,k^s) \) is a discrete topological space. Similarly to Galois groups, one can endow this with the structure of a profinite set (hence with its topology), which would allow to extend the above result also to infinite extensions.

• One could try to copy the statement of theorem 15 (+)(–). If we substituted \(\pi_1(X,x)\) with its profinite completion in theorem 24, we could recover a replica of the algebraic statement.

  Theorem 15: Galois correspondence - Grothendieck version (v2)

  The functor

  \[ \operatorname{Hom}_k(-,k^s)\colon \mathsf{FinÉt}_k^{\mathrm{op}} \to \operatorname{Gal}(k)\text{-}\mathsf{Sets}_f \]

  is an equivalence of categories. The source category is that of definition 14 (+)(–).

  Definition 14: (finite) étale algebras

  An étale algebra over a field \(k\) is a \(k\)-algebra \(A\) for which there is an isomorphism of \(k\)-algebras

  \[ A \simeq \prod_{ i = 1 }^n L_i ,\]

  where \(L_i|k\) is a finitely generated separable field extension. If \(L_i|k\) is a finite field extension for all \(i\), then \(A\) is called finite étale algebra.

  The category of (finite) étale algebras, denoted by \(\mathsf{Ét}_k\) (resp. \(\mathsf{FinÉt}_k\)), is the full subcategory of \(\mathsf{Alg}_k\) spanned by (finite) étale algebras.

  This equivalence restricts to that of theorem 11 (+)(–) on finite separable field extensions.

  Theorem 11: Galois correspondence - Grothendieck version (v1)

  The functor

  \[ \operatorname{Hom}_k(-,k^s)\colon \mathsf{FinSep}_k^{\mathrm{op}} \to \operatorname{Gal}(k)\text{-}\mathsf{Sets}_f^{ \text{tr} } \]

  is an equivalence of categories.

  Moreover, if \(L|k\) is Galois, there exists an open normal subgroup \(U \triangleleft \operatorname{Gal}(k)\) and a \(\operatorname{Gal}(k)\)-equivariant bijection

  \[ \operatorname{Hom}_k(L,k^s) \simeq \operatorname{Gal}(k)/U .\]

• What is actually different in the two situations is that in topology there naturally exist infinite covering spaces (i.e. spaces whose fibres are not finite) that are not limits of finite ones. A good example to keep in mind is, as usual, example 17 (+)(–). Notice that the infinite covering space is given by a transcendental function. Indeed, intuitively the fibres of an algebraic connected covering space are the roots of a given polynomial and clearly these cannot be infinite.

  Example 17: covering spaces

  • Fix \(0 < n \in \mathbb{N}\). The map \(p_n\colon \mathbb{C}^\times \to \mathbb{C}^\times\) defined by \[ x \mapsto x^n \] is a covering space.
  • The map \(\operatorname{exp}\colon \mathbb{C} \to \mathbb{C}^\times\) is a covering space.

Let’s now take a look at the hypothesis of theorem 24.

Remark 26: on the hypothesis of theorem 24 (+)(–)

Theorem 24: Galois correspondence (v2)

Let \(X\) be a connected and locally simply connected topological space and fix a point \(x \in X\). The functor

\[ \operatorname{Fib}_x\colon \mathsf{Cov}_X \to \pi_1(X,x)\text{-}\mathsf{Sets} \]

is an equivalence of categories. Moreover, connected covers correspond to sets with transitive action and Galois covers to quotients of \(\pi_1(X,x)\) by normal subgroups.

• Connected: This condition is only technical. The group \(\pi_1(X,x)\) cannot keep track of the various connected components. Swapping it for the fundamental groupoid and appropriately modifying the fibre functor allows one to generalize the above result to non-connected spaces.
• Locally simply connected: This condition is in some sense technical, in that it is used to construct a representative for the functor \(\operatorname{Fib}_x\), the universal cover of \(X\). It is also true that, if the space \(X\) is not locally path connected (which is implied by the condition we are considering), the fundamental group fails to have access to any interesting information about the space. (See for example this stackexchange question.) I don’t know if there are other more general results recovering a Galois correspondence in such settings, but one could argue that outside the realm of CW-complexes it is not geometry anymore.

The universal cover

So far, we obtained a result that looks strikingly close to the classical Galois correspondence. We are still to understand if there is any relation between \(\pi_1(X,x)\) and \(\operatorname{Gal}(k^s|k)\), other than playing a similar role in the Galois correspondence. Let’s see this by means of the so-called universal covering space of \(X\).

Definition 27: universal cover of a pointed space

Let \(X\) be a path-connected and locally simply connected topological space. Fix a point \(x \in X\). We call universal cover of \((X,x)\) the covering space \(\widetilde{X}_x \to X\) representing the functor \(\operatorname{Fib}_x\colon \mathsf{Cov}_X \to \mathsf{Sets}\).

This covering space exists and it is constructed in Construction 2.4.1 of Galois Groups and Fundamental Groups. For this section, we are mostly interested in the following result.

Proposition 28

There is an isomorphism \(\operatorname{Aut}(\widetilde{X}_x|X)^\mathrm{op} \simeq \pi_1(X,x)\), functorial in \(x \in X\).

Proof

The proof can be read at proposition 2.4.6 of Galois Groups and Fundamental Groups. ∎

This proposition gives us the direct comparison with the Galois group. Indeed the fundamental group, just as the absolute Galois group beforehand, is the group of automorphisms of the “universal” extension of the base.

Let’s finally state certain results, used in the proof of theorem 24 (+)(–), that highlight even more the similarities with the classical situation and justify remark 13 (+)(–).

Remark 29

Let \(\widetilde{X}_x\) be the universal cover of \(X\) as of definition 27 (+)(–).

Definition 27: universal cover of a pointed space

Let \(X\) be a path-connected and locally simply connected topological space. Fix a point \(x \in X\). We call universal cover of \((X,x)\) the covering space \(\widetilde{X}_x \to X\) representing the functor \(\operatorname{Fib}_x\colon \mathsf{Cov}_X \to \mathsf{Sets}\).

• Any connected Galois cover \(Y\) is a quotient of the universal one. In particular it receives a map \(\widetilde{X}_x \to Y\). Moreover, if \(Y \simeq G \backslash \widetilde{X}_x\), we get an isomorphism \[ G \simeq \operatorname{Aut}(\widetilde{X}_x|Y) .\] Since \(\widetilde{X}_x \to Y\) is also the universal cover of \(Y\), we get \(G \simeq \pi_1(Y,y)\) for any \(y \in Y_x\).
• Finally, let’s recall how to compute the group of deck transformations of \(Y\): \[ \operatorname{Aut}(Y|X) \simeq \frac{ \operatorname{Aut}(\widetilde{X}_x|X) }{ \operatorname{Aut}(\widetilde{X}_x|Y) } .\]

Let’s finish this section with a (sad) fact of life. Topological universal covers are often of infinite degree (i.e. their fibres are not finite). If this is the case, they are most likely not algebraic. Indeed, in algebraic geometry it is rare to have universal covers at one’s disposal. The case of fields is peculiar, since the algebraic closure of a field is the filtered colimit of finite field extensions.

Part 2: Riemann surfaces

References

Tamás Szamuely

Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2009. ISBN: 9780511627064.