In this post I want to talk a bit about how one can extend some of the ideas of Galois Theory to varieties. To begin, we remind ourselves how Galois Theory works over some field .
Galois Theory over a Field
We’ll be working in the category of finite separable extensions of . If we pick some algebraic closure then we can define a functor given by . (the group of natural isomorphisms of ) is a profinite group and the fundamental theorem is that gives an equivalence of categories between and transitive -sets.
Maybe the above description is a bit unfamiliar so lets break it down super carefully. Let be a finite separable extension. Then if we know that has precisely embeddings into and as defined above, is precisely this set of embeddings. is definitionally the set of natural isomorphisms of the functor which is the same thing as a collection of automorphisms of , one for each , which form a compatible system in the sense that if we have a map of -Algebras then
is a commutative diagram. In particular, for any , the automorphisms of an embedding are all induced by automorphisms of (exercise) and so a compatible system of such automorphisms is exactly an element of .
For certain special extensions we have that the elements of restricted to their actions on are also induced via pre-composition by automorphisms of . These are precisely the Galois extensions and of note is the fact that they are cofinal in the category in the sense that any finite separable extension of is contained within some Galois extension. If one sits down and traces through what this implies about , it is that can be computed as an inverse limit of the automorphism groups of all the Galois extensions of .
The reader who was excited by my first paragraph but immediately disgruntled by my second should have started to feel happier now.
Galois Theory over a Scheme
By being rather abstract in my constructions above, my aim was to hint that this theory might be generalized to other contexts. This can be very satisfactorily done and anyone interested in the details should read Lenstra’s online notes Galois Theory for Schemes.
The point is that one can define a special type of category called a Galois category equipped with a “fiber” functor . The fundamental theorem one proves is that gives an equivalence of categories between and -sets. The astute reader will notice that in the case of fields we got an equivalence between and transitive -sets. This came about because we didn’t consider the full Galois category associated to a field . This is the category of finite separable -algebras which we call . So for instance the full category also contains products of fields and this is how we get actions with multiple orbits. As in the case of fields we can consider Galois objects in a general Galois category and will be the limit of the automorphism groups of such objects.
Before I delve into schemes, I want to briefly give another example of a Galois category. Given a sufficiently nice topological space we can consider the category of covering spaces of . In this case the fiber functor is given by picking a base point and then mapping a cover of to its fiber over . Any undergrad has probably noticed the intense similarities between the theory of covering spaces and Galois theory. These similarities can be explained by the fact that both categories are Galois categories. Now onto schemes!
The field should be thought of geometrically as a single point corresponding to its unique prime ideal. Then the Galois category which we have constructed is what is called the finite etale site over . More generally, given some scheme we define the finite etale site over to be the category whose object are pairs where is finite etale and morphisms are maps satisfying . A finite etale morphism is a morphism which is both finite and etale. An etale morphism is defined to be flat and unramified where we remind the reader that an unramified morphism is defined to be locally of finite type and for each , we have is a finite separable field extension of where is the maximal ideal in the stalk . Maybe on a first pass, though, it’s best not to worry too much about the technical definition. The point is that an etale map is the geometric analog of a finite cover of topological spaces. In other words if is etale, then can’t look so different. In particular, at the very least they have to be the same dimension.
So lets check that this agrees with our understanding of . In particular, lets figure out for which schemes we have an etale map for a perfect field. is finite so for starters is some finite -algebra. Consider . It has one point and the local ring there is (since this ring is already local!). But the maximal ideal of is just and so the relevant ring in the definition of unramified is which is not even a field! A similar argument shows that will have to be reduced and so will be a product of fields.
So to recap, what we’ve been talking about is how, given a scheme , we can construct a Galois category . In the Galois category of covers of a topological space, the group was exactly the fundamental group of . So in the case of a scheme , we intuitively think of similarly as some sort of fundamental group and so we call the “etale fundamental group of ”.
In fact one can make a concrete connection between the two concepts. If is a complex variety, then we can give the analytic topology coming from the natural topology on . We call the resulting topological space . Then one can prove that the profinite completion of the fundamental group of is exactly the etale fundamental group of .
Galois Groups acting on Geometric Spaces
We are now going to temporarily switch gears. (Don’t worry everything will come together in the end). Take a field and pick an algebraic closure . Let be two schemes over and let be respectively. Given a map defined over we can uniquely lift to a map defined over . Explicitly, a map over is the same thing as maps which agree over . comes equipped with projections to and so composing the first of these with gives the desired maps.
The question we’re interested in is when we can do the converse. In other words, given a map over , when does it descend to a map over ? To answer this question, we consider the action of on . Take . Then acting with on on the right component gives us a map and so composing with we get a different map . Careful though! This new map is not defined over . To get this, we also need to have act on . So we define as
Then the maps that descend are precisely those that are invariant under this Galois action. After all, any map that descends will clearly have to be invariant under the Galois action, but we can argue that an invariant map will locally be defined by equations with coefficients in and so this will allow us to construct .
The Etale Fundamental Group of an Elliptic Curve
Lets work through a careful example of the above ideas. Let be an elliptic curve defined over some number field . If we then base change to , we get some elliptic curve which we can think of as a torus. Since tori have fundamental group , we expect to have a etale fundamental group that is . In fact, we recall that has a group structure and that the multiplication by map is etale. The kernel of has exactly elements and for any , we have that the translation by map is an automorphism of which commutes with . Thus we have found automorphisms of over itself via . It follows that the pairs are all Galois over and the inverse limit of their automorphism groups is . To confirm that this is in fact the full etale fundamental group of we need to show that the maps are cofinal in the etale site over . This can be done by showing that the etale maps are isogenies and noting that given an isogeny , we get a dual isogeny and their composition is .
Now, fix some and consider the automorphisms for . From our discussion in the previous section, we know that admits a action. But is defined over and therefore so are . Thus, is fixed by this Galois action and so the upshot is that acts on the set of automorphisms , . It’s clear that these actions are compatible over all and so we get a action on the etale fundamental group. The etale fundamental group we’ve constructed is precisely the product of the Tate module of over each prime, and the Galois action is indeed the usual one.
Lets conclude by sketching a slightly different way to see this action. Returning to defined over , we now try to compute its etale fundamental group. The map gives a functor of their etale sites and therefore etale fundamental groups. Similarly, we have the structure map . One shows that inside , one still has all the multiplication by maps and also a new class of etale morphisms given by base-changing by some finite extension of . Using these morphisms, one shows that the maps and (where denotes the etale fundamental group of ) are injective and surjective respectively. Thus we have an exact sequence
But in any short exact sequence of groups , we can take lifts of into and these lifts give an action of on by conjugation. This is again our desired action.