In this post we discuss the notion of Kummer theory in its general form, and how this leads to a proof of the (weak) Mordell-Weil theorem.
One first learns about Kummer theory in a first course in algebra. There, one learns that if is a field of characteristic , , then all the cyclic extensions of of order are of the form for some . The Galois group then being naturally identified with with corresponding to .
Things probably stay pretty calm on the ‘Kummer theory’ front of one’s learning for a while after this initial encounter. But, eventually one then learns, at least if you are an algebraic geometer, about the notion of a finite Galois cover of a scheme. It is a theory which, in a very literal sense, is the generalization of ‘Galois theory’ to the general theory of schemes. One then is confronted with an obvious question: in this ‘generalized Galois theory’ is there a ‘generalized Kummer theory’?
Namely, how does one describe connected cyclic Galois covers of a scheme ? Is there a relationship with the ‘roots of functions on ‘? Is it true that if is a finite cyclic Galois cover (of degree ) of , then naturally ? Of course, analogizing with the case above, one should assume that is invertible on (the replacement of ).
One can think about this from a more cohomological viewpoint. Namely, the connected cyclic Galois covers of (which is connected) with Galois group contained in correspond precisely the group . But, as we have discussed before we can describe this group equally well as . And thus, we see that we are trying to describe cohomology or, loosely, some sort of ‘abelianized’ Galois theory of .
Of course, something slightly more subtle must be going on here. Indeed, in the case of fields we needed to assume that contained an root of unity. What is the analogy here? What do we need to assume about ? Well, it turns out that the case of fields we learned in our youth was actually a bit misleading. The inclusion of the roots of unity in was a red herring which allowed us to ignore a technical detail. This detail, in terms of the étale cohomology perspective, was that Kummer theory is rightfully not about but instead of . Of course, if contains the roots of unity (in its global sections) we have an identification of and explaining the connection with classical Kummer theory.
Now then, once one has understood Kummer theory in this context, another natural question presents itself. Namely, what is but the -torsion of . One may then wonder what one can say about the -torsion of a general group scheme/abelian sheaf. This perspective, together with some elbow grease, will lead to a nice, conceptual, geometric proof of the weak Mordell-Weil theorem, that if is a number field, an abelian variety, then is finite for any . This is the hard step in the proof of the Mordell-Weil theorem that is finitely generated.
Remark: The other part of the Mordell-Weil theorem is the so-called ‘theory of heights’. One shows that if an abelian group possesses a height function (something roughly measuring size) then the Descent Theorem shows that being finite implies that is actually finitely generated (the obvious direction is obvious). One then uses the fact that has a natural height function associated to any ample line bundle on . In fact, there is a canonical height function called the Neron-Tate height function.
Hilbert’s theorem 90 in the abstract
Before we begin our pursuit of Kummer theory in earnest, we must begin by discussing what can be called the geometric version of Hilbert’s theorem 90. Recall firstly that Hilbert’s theorem 90 says that if is a cyclic Galois extension of order , then all are of the form for some . This follows from the general Galois computation that if is any Galois extension then , computed in the sense of group cohomology (for a reminder on group cohomology, look here).
Now, the vast generalization of Hilbert’s theorem 90 comes in the form of a theorem about the étale cohomology of the multiplicative group on the étale site of any scheme. Namely, one can show that for any scheme one has the following set of isomorphisms
Where, here denotes étale cohomology, the subscript denotes cohomology on the Zariski site of (i.e. ‘usual’ cohomology), denotes the ‘usual’ Picard group of , and denotes the line bundles on the étale site of .
To clarify the last point, we call a sheaf of abelian groups on a line bundle if there is an étale cover such that . Here, and for any scheme , the sheaf denotes the sheaf associating to any étale the group . In other words, it is the sheaf associated to the group scheme .
More generally, we call an abelian on quasicoherent if locally on it is the quotient of a power of . More specifically, there is an étale cover such that is a quotient of for some cardinal . One of the basic theorems in descent theory says that there is an equivalence of categories
given by associating to any its etalification which is denoted and associates to any the group .
Now, it’s clear that this map restricts to a map
and its clear (since the Zariski site contains the étale site) that this map is injective. Moreover, since every étale line bundle is clearly quasicoherent on the above equivalence tells us that there is some quasicoherent Zariski sheaf such that . That said, it’s not obvious that is actually trivializable on . This is the main content of the theorem.
Indeed, the other isomorphisms are entirely formal. Namely, the identification of (Zariski or étale) with (Zariski or étale) follows from the general yoga of torsors. Then, the isomorphisms of with holds true for sheaf cohomology on any site (see the Cech-to-derived spectral sequence, which holds on any site).
To prove that étalification is actually a surjective map one needs to perform a more in-depth analysis. I refer to Milne’s Lectures on Étale Cohomology. It was pointed out to me by my friend Minseon Shin that there is actually a relatively easy way to prove this. Namely, suppose that is a quasicoherent on which becomes a line bundle on . In particular, assume that there is an étale cover such that is trivial. By standard arguments we can assume that and are both affine. Thus, we have a module on such that (with ) free. We want to show that is a line bundle. Thus, we want to show that is finitely presented and flat. To see that it’s finitely presented merely choose a presentation
with a finitely generated -module. Note that we may assume that the generators of go to tensors of the form . Consider then the map defined by sending to and let be the kernel of this map. Note firstly that since
is surjective, it follows from faithful flatness that is surjective. So, we need only show that is also finitely generated. But, again, faithful flatness shows that . But, we didn’t use in the above to show that was finitely generated—we showed in general that if is finitely generated then is finitely generated. Thus, is finitely generated implies that is finitely generated, and so is finitely presented. So, it remains to show that is flat. But, for any -module we need to check that . But, again, by faithful flatness, it suffices to check that
the last equality since is flat.
Remark: Because of the above theorem, and since we’ll only talking about the étale or Zariski topology unless stated otherwise, we unambiguously denote the Picard group by .
Note that these isomorphisms do actually prove the classical version of Hilbert’s theorem 90. Namely, let be any field. Then, by the usual equivalence of (the latter being discrete -modules) which preserves cohomology we see that
But, by the above we know that the right hand side is equal to which is clearly (it’s a point!). Then, the usual injection proves the theorem in general.
Kummer theory in the abstract
Now that we have a general understand of Hilbert’s theorem 90 we are able to deduce the ‘true nature’ of Kummer theorem. Namely, the key will be the so-called Kummer sequence.
In particular, let us now suppose that is any scheme on which is invertible (i.e. is a scheme over ). Then, we claim that we have an exact sequence of sheaves on :
called the Kummer sequence on .
Note, that even though both and are actually affine group schemes on , that this is a sequence of étale sheaves on . In particular, the -power map is not necesarily surjective map of schemes necessarily—think about the map on -points if .
Remark: One can show that the map has trivial cokernel in the abelian category of, say, affine group schemes over . But, note that this has no more content since, definitionally, the cokernel is the sheaf representing the cokernel sheaf!
Now, it’s clear that the map is injective, and so we should say why the map is actually surjective (again, on the level of sheaves). This means that for any -scheme , and any section we can find some étale cover such that is an -power. But, this is simple. Namely, the scheme (i.e. locally adjoin an root of ) suffices. Note here that we see why it is necessary to assume that is invertible on —else the scheme just defined is not necessarily étale (i.e. is not a ‘separable polynomial’).
So, now with this sequence, we can give the fundamental theorem of Kummer theorem:
Theorem 1(Fundamental theorem of Kummer theory): Let be a scheme over . Then, there is a short exact sequence
Here, for an abelian group , denotes the -torsion of .
The proof of this theorem is utterly trivial given the setup we have. Namely, taking the long exact sequence associated to the Kummer sequence, and using Hilbert’s theorem 90 to identify with , immediately gives the result.
Remark: Using the monodromy perspective we can think about this as giving a short exact sequence describing the group cohomology .
A different interpretation: principal homogenous spaces
Principal homogenous spaces in general
Before we go on to give some applications of Kummer theory, let us give another interpretation of the above result. Namely, we should be able to geometrically describe the group , and this sequence in Theorem 1, without recourse to the Kummer sequence.
Why? Well, recall from the general yoga of torsors that classifies the group of -torsors. This does not sound very appealing, and certainly not very geometric. Namely, a -torsor is some sheaf of sets on with a transitive action of which is ‘locally trivial’ (i.e. locally has a section). That said, since is an affine group scheme over , we actually have a nice alternative way of thinking about -torsors.
Namely, let us fix a flat group scheme locally of finite type over . Then, call an -scheme equipped with an action
(meaning that for all -schemes we have an action of which is functorial in a principal homogenous space for if there is an étale cover such that is isomorphic to as a -space. Here we are thinking of as a -space with left multiplication. Equivalently, one can require that is a faithfully flat -scheme and the obvious map is an isomorphism. The phrase principal -bundle is also used in place of principal homogenous space for .
Let us give the key example of this construction. Namely, what are the principal homogenous spaces for the group ? Well, recall that a geometric line bundle on is an -scheme such that there exists a Zariski cover of such that is isomorphic, as a scheme, to .
We then define an -scheme associated to a geometric line bundle called the frame bundle of . We denote it by . Intuitively parameterizes ordered bases of the fibers of over . More explicitly, represents the functor on defined by
(where these are isomorphisms in category of geometric line bundles) and thus another reasonable name for would be the isomorphism sheaf .
Note that there is a natural action of on given by taking a and an isomorphism to the precomposition with the isomorphism . Moreover, note that this map is clearly a transitive action when is non-empty. Moreover, since is locally trivializable on the Zariski site, that locally has a section. Thus, is actually a -torsor
Recall also that there is an equivalence of categories between the category of geometric line bundles (where the morphisms are, on trivializing covers, linear) and the category of line bundles on . Namely, to a line bundle on we can consider the affine scheme where is the quasicoherent -algebra given by
with the usual multiplication. This construction yields the equivalence. In fact, the sheaf of sections of is precisely . Under this equivalence we can identify the torsor with the torsor which, again, to a associates the set of isomorphisms .
Remark: This is an important point which confused me for a long time. Namely, people will say that line bundles are -torsors. This is technically a lie. Namely, the most natural interpretation of this statement is that if is a line bundle then is a -torsor on . Of course, this makes no sense–there is no action. It is really the frame bundle naturally associated to which is the -torsor.
So, now, in general, to a principal homogenous space for we can associate -torsor on as, not shockingly, the isomorphism sheaf which associates to a the set of isomorphisms of -spaces . Now, in general there is no reason to believe that this should create a bijection between the set of -torsors and the set of principal homogenous spaces—in particular, there is no reason it should be surjective.
That said, we have the following:
Theorem 2: Let be a scheme and a smooth affine group scheme over . Then, there is an equivalence
given by .
Here the categories are both equipped with -equivariant morphisms.
Remark: For people that know about stacks, the above just says that if is smooth affine, then the stack is equal to the stack of principal homogenous spaces for . More generally, for any group scheme , one has that is equal to the stack of principal homogenous spaces if one expands the latter definition to include algebraic spaces with a locally trivial transitive -action.
The proof of this theorem follows fairly immediately from descent for affine morphisms. Namely, a -torsor is, definitionally, locally representable by a (relatively) affine scheme and thus, by descent, is globally representable. The smoothness condition is there for technical conditions.
The case for the roots of unity
So, now if is invertible on we know that is a smooth (étale!) group scheme over . Thus, the -torsors are all the isomorphism sheaf of some principal homogenous space for . Therefore, we’d like to have a nice description of these principal homogenous spaces.Just like the case of geometric line bundles, for which it is the line bundles (the sheaf of sections), which is simpler to understand, we will first understand the principal homogenous spaces for in terms of ‘coherent data’.
To this end, we consider pairs of the form where is a line bundle on and is an isomorphism of -modules:
We shall say that two pairs and are equivalent if there is an isomorphism such that the induced isomorphism carries to .
Note that this set is naturally a group with operation given by
where is shorthand for the natural isomorphism
where the inverse is given by where is shorthand for the natural isomorphism
which one can check really do define group operations. The identity of this group is obviously . Let us denote this group by .
Let us note that for the identity element we can compute its automorphism sheaf on given by
where the automorphisms are automorphisms of pairs as defined above. We claim then that there is a natural isomorphism between this sheaf and . Now, any automorphism of corresponds to a global section of . But, to respect the map we need precisely that . Thus, we see the result.
Thus, from the general yoga of torsors we know precisely that , the -torsors, correspond precisely to the twists of . More specifically, the torsors should correspond to pairs of a line bundle on and a morphism which is étale locally an isomorphism. But, from the section on the Picard group we know that this implies is really a line bundle on , and since is defined on and is locally an isomorphism it is an isomorphism. Thus, we see that is in bijective correspondence with .
So, now, what are the geometric structures associated to a pair ? Associated to such a pair an -scheme which we will denote . It will be a relatively affine -scheme, and thus, to specify it, we must only specific a quasicoherent -algebra. This is the algebra defined by
where the multiplication is defined in the usual way except we cycle powers past the -power back into the range using the isomorphism .
By construction, is a finite affine map. We claim, in fact, that it is étale. Indeed, to see that this morphism is étale consider an affine open in with . Let be an isomorphism between (for some -module ) with . Let be . Then, one can check that when defined correctly, and thus is étale over .
Moreover, note that the -points of (for a -scheme ) is just . Thus, we have a natural action of on . One can check that this action glues over the various , thus providing an action of on . Moreover, one can check that this actually makes a principal homogenous space for .
Then, one claims that defines a bijection between and the principal homogenous spaces over . This follows fairly formally from the observation we have already made that is in bijective correspondence with together with Theorem 2.
Finally, let us remark that under the identification of with we can naturally interpret the short exact sequence in Theorem 1. Namely, the map is merely . It’s clear that this map is surjective. Now, suppose that is in the kernel of this map. Then, . But, evidently all the maps on correspond to multiplication by an element of , and two such maps define isomorphic pairs if and only if they differ by an root of of . Thus we recover the sequence from Theorem 1 in this more geometric context:
where is the multiplication by map.
Let us now discuss some nice cases in which this theorem can be applied. In particular, let us examine the extreme cases where one of the outer terms in the sequence in Theorem 1 is zero. We shall make the blanket assumption that is invertible on , and that it is connected.
First, let’s assume that . The prototypical example we will consider is when is an integral proper variety over an algebraically closed field whose characteristic is coprime to . Then,
Moreover, since we may conclude from Theorem that
Moreover, since the map is multiplication by this corresponds on cohomology to the map , the multiplication by map. Thus, in fact, we see that
where the tranistion maps on the right are the multiplication by maps.
Let us now consider some specific examples:
- Let is an integral smooth projective curve of genus . Indeed, as is the usual proof, we know that if is the Jacobian of then this is an abelian variety over of dimension , and . Thus,
and thus we see that
- Similarly, if is an abelian variety, then we have the equality
and thus we see that
if . Thus, we can see that
Then, using the fact that , with the comultiplication coming from the multiplication on , is a connected cocommutative Hopf algebra it follows from the Hopf-Borel theorem that
- As a final example, we can make an interesting observation about Fano varieties. Namely, let us call a variety , where now we assume that is of characteristic , Fano if its canonical bundle is antiample. This means that if is the canonical bundle of over , then is ample. We claim that if is Fano, then is torsion-free.
Indeed, suppose that is non-zero for some . By standard group theory it suffices to assume that is for some prime Then, by the above we know that is non-zero. But, this implies (either by Hodge theory over or -adic Hodge theory) that is non-vanishing. That said, it follows from the ampleness of and the Kodaira-Nakano-Akizuki vanishing theorem that —contradiction.
Note also that our assumptions assure that all finite cyclic Galois covers are of the form . Indeed, this follows from our explicit description of the map from the last section (which is now an isomorphism!) and the fact that
Let us now consider the case when .
Theorem 1 then implies that we have an isomorphism
But, of course, we can’t make the connection between this result and -adic cohomology unless we assume further that is actually a scheme over where is a primitive root of unity. Then, we have the isomorphism
In particular, let’s think about the case when a field, then we know that
So, if happens to have the -roots of unity we see that
But, of course, correspond to the cyclic Galois extensions of cyclic of order . In fact, the association, following the general geometric picture of last time, is just maps to with Galois group .
More generally, assuming that is actually a scheme over we see again that all finite cyclic Galois covers of correspond to adjoining an -root of a global section of —generalizing ‘usual Kummer theory’.
The weak Mordell-Weil theorem
We now begin moving towards the proof of the weak Mordell-Weil theorem: that if is a number field and is an abelian variety, then is finite for any . But, before we actually get to the main meat of the proof, we’ll need to record some other facts of interest which will be useful in streamlining the exposition.
The Hochschild-Serre spectral sequence
It will be useful for us to discuss the Hochschild-Serre spectral sequence. Roughly, this is a spectral sequence which allows us to compare the étale cohomology between the domain and codomain of a finite Galois cover . Namely, we have the following:
Theorem 3(Hochschild-Serre spectral sequence): Let be a finite Galois cover with Galois group . Then, for any abelian sheaf on there is a spectral sequence
There are couple things which should be clarified here. First, the cohomology on the left is group cohomology. The action of on is the action coming from the action of on . Finally, on is just the restriction of to which, since this is an étale cover of , is nothing fancy.
This theorem will be useful in proving the weak Mordell-Weil theorem since, for example, the above shows that if is finite for all then is finite and, in fact,
which follows from the general machinery of spectral sequences (in fact, the definition of convergence).
This will allow us to essentially move to arbitrary finite Galois covers in our pursuit of showing that any particular étale cohomology group is finite.
Let us not prove this sequence exists, but give an indication about how one might about about proving it. Namely, note that since if finite Galois that for any sheaf for any sheaf on . Thus, we see that the global sections functor on is nothing more than the composition
Where is the usual pullback functor. But, note that since is exact, in the derived world we can ignore it.
Thus, we see that if we can show that for any injective sheaf on and then from the general sledgehammer of the Grothendieck-Serre spectral sequence we’ll have
which, upon rewriting using usual cohomological notation, is just the stated version of the Hochschild-Serre spectral sequence.
To prove that this condition holds, note that is just the Cech cohomology of for the étale cover . But, the Cech cohomology of an injective on any cover is trivial, giving the result.
Grothendieck’s lifting lemma
Another fact which we will need is that any abelian variety lifts to an abelian scheme for some dense open . One can proceed by the general theory of Neron models over Dedekind schemes, but this is too strong here. We mention here a method which bypasses this.
First, begin by noting that lifts to some smooth projective scheme , for some dense open . To see this, choose a projective embedding of into . Then, by examining the equations defining as a subscheme, we can clearly define in where is the open subsets obtained by removing the primes containing the coefficients (or just clear them out!). This might not actually be smooth over . But, by removing the finitely many primes containing the determinant of the Jacobian we can shrink even further to obtain a smooth projective variety over some dense open in .
Moreover, note that the identity section defines a point of which, by the valuative criterion for properness (since is a Dedekind scheme!), lifts to a map . We claim that this map is a section. But, this follows immediately from separatedness.
Thus, we see that for some dense open we have a smooth projective scheme together with a section such that this pair restricted to is just with the identity section.
Thus, to finish we appeal to the following amazing theorem of Grothendieck:
Theorem 4(Grothendieck): Let be a smooth projective morphism together with a section , such that is a connected locally Noetherian. Assume that for some geometric point of the pair and is an abelian variety, then has the unique structure of an abelian scheme with its identity section.
The proof of this is surprisingly easy, and essentially produces by bootstrapping first from the infinitesimal case (when is a nilpotent to a thickening of and the case of a DVR to the whole . A full proof can be found on page 124 of Mumford’s Geometric Invariant Theory.
Thus, we see that our projective scheme actually has the unique structure of an abelian scheme with as its identity section.
The proof of the weak Mordell-Weil theorem
So, let us now actually show that if is an abelian variety, where is a number field, then is finite.
We begin by noting that, as shown in the last section, we have a dense open and an abelian scheme lifting . Moreover, by passing to an even smaller open inside of , we may as well assume that is invertible on .
So, begin by considering the sequence
(which one can think about as a version of the Kummer sequence!) where the second map is the multiplication by map. This second map is clearly surjective since
So, take the long exact sequence in cohomology to obtain the piece
In particular, we obtain an injection . But, since is proper, and is a Dedekind scheme, we know that and thus we have an injection . Thus, we are reduced to showing that is finite.
Next, we claim that it suffices to prove this result under the assumption that is a scheme over which becomes trivial. Indeed, let us note that since is a finite étale group scheme, there exists some finite Galois cover , say with Galois group , such that is constant. Note then that by the Hochschild-Serre spectral sequence
But, for some finite extension which is a finite group, and thus the group cohomology is finite. Thus, it suffices to show that is finite as desired.
By similar reasoning, appealing again to the Hochschild-Serre spectral sequence, we may assume that is a scheme over where is a primitive -root of unity.
Now, is a product of constant group schemes of order dividing . Thus, it is a product of constant cyclic group schemes of the form for . Thus, it suffices to show that is finite. But, since is a scheme over Theorem 1 implies that we have a short exact sequence
But, all finite étale covers of are going to be the ring of -integers in some finite extension of . In particular, the general form of Dirichlet’s unit theorem and the finiteness of the class group implies that both of the outer terms of this sequence are finite (see here for a proof) and thus the middle term is finite as well.