I’ve been reading a lot recently about -divisible groups, which are algebro-geometric approximations to homotopy theory much like formal groups, though perhaps a bit better behaved. Recall that we can get a formal group by taking the completion of an elliptic curve at the identity; over a field of characteristic , this will be height 2 if the curve is supersingular, but only height 1 if it’s not. More generally, the sorts of base changes that invert things in the maximal ideal of your base ring tend to decrease the height of formal groups. The -divisible viewpoint says that formal groups in positive characteristic naturally sit inside objects whose height is constant under base change — in particular, every elliptic curve will give you a height 2 -divisible group, with some of this height being `non-formal’ if the curve is ordinary. (And more generally, you get a height -divisible group from an abelian variety of dimension in characteristic .)
In algebraic geometry and number theory, various results by Tate, Grothendieck, Messing and others argue the general point that the deformation theory of a grouplike object in characteristic is controlled by the deformation theory of its -divisible group. This idea has been key to a number of recent developments in homotopy theory, — in general, we hope that various moduli objects of -divisible groups are easier to study than the analogous moduli objects of formal groups, while offering the same level of access to the category of spectra. The key result in this field, due to Lurie, actually allows you to build spectra from 1-dimensional -divisible groups, in much the same way that the Goerss-Hopkins-Miller theorem lets you build spectra from deformations of formal groups. Reaching the very boundary of things I can claim to understand, Lurie’s theorem is a key component of Behrens and Lawson’s construction of the topological automorphic forms spectra.
A -divisible group is a certain sort of ind-scheme which can be canonically written as a colimit of finite flat commutative group schemes. So before we can say anything about -divisible groups, we have to know some things about these gadgets (which I’ll generally just call ‘group schemes’), and this post will fill in that background. (Presumably this is all well-known to the average algebraic geometer, but it was new to me, and hopefully it’ll help out other budding homotopy theorists curious about this subject.)
So: what’s a finite flat commutative group scheme? This is one of the places where the ‘functor of points’ perspective on schemes really comes in handy. This is where we think of a scheme over as a contravariant functor , with being the set of morphisms over . To make things even easier, we can think of this as a contravariant functor just on the category of affine schemes with maps to ; if is affine, this is equivalently a covariant functor on the category of -algebras, and we call the set of points of with values in . Since a morphism of schemes can be specified on a Zariski open cover of the domain, this functor satisfies a sheaf condition on with the topology where the open covers of an affine are its Zariski open covers. And as it turns out, the functors of points that arise from actual schemes are precisely those satisfying this sheaf condition.1
With this in mind, a commutative group scheme is a scheme whose functor of points lands in the category of abelian groups rather than the category of sets. This is extra structure, of course — you’re effectively giving a group structure on each set of points making the maps homomorphisms for each map of rings .
In particular, suppose that is affine and that is an affine group scheme (this will almost always be the case in what follows). Then the multiplication , identity , and inverse are all induced by maps , , and making various diagrams commute — in short, is given the structure of a Hopf algebra over . If is finite over , then is a finite -module; if is flat over , then is locally free over . In many cases, we’ll have local, so that is free. In this case, it makes sense to define the order of as the rank of over .
I’m going to be fixing the following notation: is a base scheme, is an arbitrary open affine of , is a finite flat commutative group over , and where is a finite free Hopf algebra. will generally be an -algebra, an -scheme, and a residue field of . Though I’ll do most of the below `globally’ just because I can, you don’t really lose any detail by taking or .
Example 1 For any finite abelian group , we can define as the group scheme that is copies of , with the group law induced by the group law of and the obvious isomorphism . Affine-locally, the Hopf algebra of this is the ring of functions with
where is the map that sends to 1 and all other elements to 0, and with . Such a group scheme is called constant.
The multiplicative group is the affine group scheme locally given by the Hopf algebra , where and . (Note that is the group of the units of under multiplication.) This isn’t finite — it’s actually 1-dimensional as a scheme — but the points of order are a finite closed subgroup scheme , locally given by the Hopf algebra and the same structure maps.
The additive group is the affine group scheme locally given by the Hopf algebra , where and . returns the additive group of . In characteristic , the th power map is a natural endomorphism of each and thus an endomorphism of , and its kernel is .
In particular, we can enumerate three order- group schemes over a field of characteristic : , , and . If is algebraically closed, these are all there will be.
Kernels and cokernels
This is only worth a paragraph or two, but: we can take kernels and cokernels. Clearly, the kernel of should be its fiber over the identity , which is . On points, .
Cokernels are a little more difficult. Of course, we should define as the sheaf associated to the functor , and then one has to check that this is again finite and flat (it’s clearly a group scheme), as well as the categorical quotient in the category of schemes.2 Moreover, the map is faithfully flat. This is specific to our specific case of finite flat group schemes, though. For example, acts on by multiplication, and the quotient via this definition is . But it also acts on in the same way, and there’s no good quotient in the category of schemes, or even in the category of sheaves on any of the sites we’d care to pick. The `right’ quotient is actually an object called a stack, which is a schemey sort of functor from the category of -schemes to the category of groupoids. We’ll get something that looks like the scheme with one extra point, the image of the origin, which has all of as its `automorphism group.’
(If this sounds scary, it’s just because it’s a one-sentence summary of a situation that’s irrelevant right now. Stacks are fairly important to what I’m working on, and we’ll talk more about them in due course.)
Over a field , finite flat commutative group schemes form an abelian category, which means that the kernels and cokernels described above behave as they do in the category of abelian groups. In particular, we can talk about monomorphisms and epimorphisms (closed subgroups and quotients) and the image of a morphism (the closed subgroup it factors through), and we have a good notion of `exact sequence.’
If we’re not over a field, this doesn’t have to be the case. For example, let and the finite flat group scheme described in the previous section. Multiplication by induces a map which is an isomorphism over the field of fractions of (the so-called generic fiber) but zero over the residue field (the special fiber). The group scheme kernel and cokernel of this are thus both zero over the generic fiber, which means that they’re zero in general, since they have to be finite and flat. But this map isn’t an isomorphism, since it evidently doesn’t have an inverse over the special fiber.
A few nice observations about the case of finite flat commutative group schemes over a field:
- We literally have .
- Using faithful flatness of quotients, one can observe that the orders of groups in an exact sequence are multiplicative: if is exact, then .
- Base change to a field extension is an exact functor.
One of the first unusual pieces of structure for finite flat commutative group schemes is a duality somewhat similar to Pontryagin duality, called Cartier duality. This is defined by
That is, is the functor . Though writing it this way makes it look like Pontryagin duality, it’s way easier to think of on the level of Hopf algebras. If and , then
Specifically, a map of Hopf algebras is equivalently an element with and (by the axioms of a Hopf algebra, this is automatically a unit). Since we want to represent the Cartier dual as an affine group scheme, we’d like this to be represented by maps to from some Hopf algebra. An obvious choice is the dual Hopf algebra , which inherits a multiplication from the comultiplication of , and a comultiplication from its multiplication. Of course, if is free and finite over , a is equivalently a map of -modules . If , then
if and only if . Likewise, the multiplicative unit of is , so the maps of unital algebras precisely correspond to the satisfying the conditions and . Finally, it’s clear that all of this is preserved by any base change whatsoever, giving us
the Cartier dual of a finite group scheme is Spec of its dual Hopf algebra.
In particular, a group and its dual have the same order, and we can conclude, as in the case of finite-dimensional vector spaces, that .
The connected-étale decomposition
The second interesting piece of structure is a natural exact sequence splitting a group into an ‘étale part’ and a ‘connected part’. The analogous thing for -divisible groups, which we’ll find very useful, is a decomposition of a -divisible group into a ‘formal part’ (which looks like the formal spectrum of a power series ring) and an ‘étale part’ (which will always look like the constant group ). First, a definition.
Definition 1 A map of schemes is étale if it is flat, locally of finite presentation, and for every and algebraic closure of the residue field , is a finite disjoint union of copies of .
What this is saying is that the fiber over any point of is Spec of a finite separable algebra over its residue field. This condition (together with the local finite presentation condition) is often described with the word unramified.
(By connected, all we mean here is Spec of a local ring.)
Theorem 2 Let be a finite flat commutative group scheme over , with complete, noetherian, and local. There is a unique, natural exact sequence
where is connected and is étale. If with a perfect field, then this sequence splits.
Proof: is obviously the connected component of the identity. This is a closed subscheme, and since is local, is still connected, so the restriction of the multiplication map to this subscheme factors through . Thus, is a closed subgroup. In Hopf algebra language, is a finite product of local extensions of , and factors through the projection to one of them, which will then be .
corresponds to the maximal étale subalgebra of . To get at this, base change to the residue field of , making a product of finite local -algebras , each of whose residue fields will be a finite extension of . The separable closure of in is of the form , by the primitive element theorem; using Hensel’s lemma, one can lift this to an element in , giving an embedding of a (maximal) finite separable extension of into each . By the uniqueness part of Hensel’s lemma, this subalgebra is unique, and it’s clearly étale over . Standard stuff about étale morphisms tells us that for some polynomials and such that is a unit in the localization. Now by Hensel’s lemma again, we can (uniquely) lift these polynomials to polynomials over satisfying the same condition, and get a subalgebra that is étale over and maximal among étale subalgebras. (If there were an étale subalgebra containing this, its reduction to would have to be the same, and it would have to be the same as by Nakayama’s lemma.)
Any map from a connected group to an étale group is trivial, so the composition is zero. On the other hand, reducing to and base changing to the algebraic closure, becomes the union of the closed geometric points, one of which is in each connected component, so the sequence becomes exact. Since these base changes are exact and faithful for finite flat group schemes, is the desired exact sequence.
For naturality, observe that if is a map of finite group schemes, then must map to by topology.
Now suppose that the base scheme is a perfect field , and let be the reduction of (the spectrum of mod its nilradical). Since is perfect, is again a reduced ring, and so is a reduced scheme — thus, the multiplication map sends this into , making a closed subgroup. Finally, is a map of group schemes that becomes an isomorphism after base changing to (where both group schemes are constant), and thus it’s an isomorphism over . The inclusion splits the exact sequence, and is evidently unique.
The last thing I want to cover in this post is a pair of special maps that will play an important role in the structure theory to follow. All this is over a field of characteristic . The first, Frobenius, is familiar to attentive students of positive characteristic. There’s a map induced by the map on , and for any -scheme , we define to be the fiber product . If is affine, this is , and so we should think of this as with th roots of elements of adjoined: if , , then in this ring, and will thus be a th root of . (If is perfect, this has a natural isomorphism with itself, but it’s worth it for the following to treat it as different.)
There’s an ‘absolute Frobenius’ map given by the identity on points and on structure sheaves, but this isn’t quite what we want. Rather, we should note that if is the structure map, then , and and thus induce a map to the pullback . This is what we’ll call the Frobenius morphism. The following diagram might help:
Again, let’s consider a ring . The Frobenius map should just raise everything to the th power, but the argument here is that it’s morally a map rather than , since , so that to multiply by on the target , one must multiply by its th root on the source.
If is a cocommutative Hopf algebra, we can in fact extend this to a map realizing the th power map. To do this, apply , landing by cocommutativity in the subspace of symmetric tensors, i.e. those invariant under the -action. This is generated as a -vector space by tensors of the form , so there’s a unique -linear map sending to . The composition is the Verschiebung morphism , and schemifying everything gives us the Verschiebung morphism . The composition (equivalently, ) is the th power map we wanted; likewise, is the th power map on .
To wrap this up and tie some things together, note that the definitions of and are dual! Indeed, was given by comultiplying times, noticing that you were in the -invariant subspace of the -fold tensor product, and mapping this to in the obvious way; but likewise, could be thought of as starting in , mapping to the -fold symmetric power of (which is the -coinvariants of the -fold tensor product), lifting to the -fold tensor product, and multiplying times to get an element of . Thus we see that is dual to .
That’s all for now. In the next post, we’ll define -divisible groups and extend these definitions to that area.
1 Ever since SGA 3, people have found it useful to think about functors of points on other categories or satisfying other sheaf conditions, considerably generalizing the theory of schemes. So right now we say that we’re working on the ‘Zariski site,’ and later on we might move to the ‘fppf site.’ If you haven’t seen this before, don’t worry about it — it’ll get its own post if and when it comes up.