Deformations of p-divisible Groups

I’ve made the official decision to not do a proof of anything with the deformation theory of {p}-divisible groups, but now that I’ve motivated it I’ll still state the results. The proof is incredibly long and tedious. It should be interesting to look at what this functor is, since it probably isn’t what you think it is. We’ll construct the moduli functor, but it isn’t just {p}-divisible groups up to isomorphism or isogeny, but involves the notion of a quasi-isogeny.

A few definitions are needed. We’ll work in great generality, so let {S} be a base scheme with your favorite properties. Our {p}-divisible groups will only be required to be fppf sheaves on {S} (with the {p}-divisible group property). An isogeny {f:G\rightarrow G'} of {p}-divisible groups is a surjection of sheaves with finite locally free kernel. An example is multiplication by {p} since the kernel is {G_1} which is by definition of a {p}-divisible group a finite locally free group scheme (of order {p^h}).

The group {\mathrm{Hom}_S(G,G')} is a torsion-free {\mathbb{Z}_p}-module. We can make the sheaf version by taking the Zariski sheaf of germs of functions {\mathcal{H}om_S(G,G')}. A quasi-isogeny {G\rightarrow G'} is a section {\rho\in\mathcal{H}om_S(G,G')\otimes_\mathbb{Z} \mathbb{Q}} with the property that {p^n\rho} is an isogeny for some integer {n}.

Now we have enough to write down the moduli functor that we want. We have everything over an algebraically closed (I think a descent argument allows us to do this all over a perfect field) field, {k}, of characteristic {p>0} and {W} its ring of Witt vectors. Consider the category {\mathrm{Nilp}_W} of locally Noetherian schemes, {S}, over {W} such that {p\mathcal{O}_S} is locally nilpotent.

Fix a {p}-divisible group {G} over {\mathrm{Spec}(k)}. Our moduli functor {\mathcal{M}} is a contravariant functor from {\mathrm{Nilp}_W} to the category of of pairs {(H, \rho)_S}, where {H} is a {p}-divisible group over {S} and {\rho} is a quasi-isogeny {G_{\overline{S}}\rightarrow H_{\overline{S}}}. Then we mod out by isomorphism where an isomorphism {(H_1, \rho_1)\rightarrow (H_2, \rho_2)} in this category is given by a lift to an isomorphism of {\rho_1\circ \rho_2^{-1}}, i.e. an isomorphism that commutes with the quasi-isogenies.

The theorem is that {\mathcal{M}} is representable by a formal scheme formally locally of finite type over {\mathrm{Spf}(W)}. The way to prove representability is the usual way of finding particularly nice open and closed subfunctors. In our case, part of the breakdown is already in this post. Since the definition of quasi-isogeny involves an integer {n} for which {p^n\rho} is an isogeny, you can break {\mathcal{M}} up using this integer. See the book Period Spaces for p-divisible Groups by Rapoport and Zink for more details.

It comes out in the proof that for the nice cases we were considering in the past two motivational posts we get that the functor is representable by {\mathrm{Spf}(W[[t_1, \ldots, t_d]])} where {d=\dim G \dim G^t}. One really interesting consequence of this is that if {G} has height {1}, then since {\mathrm{ht}(G)=\dim G + \dim G^t} we have that one of {\dim G} or {\dim G^t} is {1} and the other is {0}, so in either case the product is {0} and we get that the deformation functor of {G} is representable by {\mathrm{Spf}(W)}. In general, it is always smooth and unobstructed.

I’m not sure if I should continue on with {p}-divisible groups now that I’ve done this. Maybe I’ll go back to crystalline stuff or move on to something else altogether.


More Motivation for p-Divisible Groups

Perhaps the last post didn’t provide you with enough motivation to understand deformations of {p}-divisible groups. Today we’ll look at a much more general situation where it is again useful to know the deformation theory. Recall the general setup of trying to understand a class of objects. Say abelian varieties of dimension {g}. You probably want to form some sort of moduli space, so you usually say what the points of the space are and then try to prove that these points form some sort of scheme or stack that isn’t too horrible.

Often times it is horrible, though, so there is another trick to try to understand what is going on. You look at the deformations of some object. This will tell you what is happening in some formal local neighborhood of that object on the moduli space. This means that deformations actually are useful for understanding classification. For instance, all K3 surfaces can be deformed to eachother (analytically, but not algebraically), so this means that as manifolds they are all diffeomorphic, but they are not algebraically equivalent.

Here is the motivation. Recall that one of the easiest examples of a {p}-divisible group is that whenever you have an abelian variety {A} of dimension {g}, you can take the {p^\nu}-torsion points {A[p^\nu]=\mathrm{ker}(A\stackrel{p^\nu}{\rightarrow} A)}. Set {G_{\nu}=A[p^\nu]}, then you have a height {2g} {p}-divisible group {\lim G_{\nu}=A[p^\infty]} (as long as the characteristic of {k} is relatively prime to {p} otherwise the height is different, but you still get a {p}-divisible group).

Here is an absolutely beautiful result of Serre and Tate: Fix some Artin local {k}-algebra {R} and let AbSch({R}) be abelian schemes over {R} and BT({R}) the category with objects {(X,G,\epsilon)} where {X} is an abelian variety over {k}, {G} is a {p}-divisible group over {R} and {\epsilon} is a choice of isomorphism of the special fiber of {G} with the {p}-divisible group associated to {X}, so {\epsilon: G\otimes_R k\stackrel{\sim}{\rightarrow} X[p^\infty]}. The result is that there is an equivalence of categories {F:\mathrm{AbSch}(R)\rightarrow \mathrm{BT}(R)} given simply by {F(A)=(A\otimes_R k, A[p^\infty], \epsilon)}.

In particular, for an abelian variety {X} we get an isomorphism of deformation functors {Def_X\rightarrow Def_{X[p^\infty]}}. So if you want to understand the deformation theory of abelian varieties you could try to understand the deformations of {p}-divisible groups. This is the topic of the great book by Messing The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes. Might I point out that whenever I find a book great and useful it seems to always have been moved to math storage as something no one ever looks at.

A similar thing happens with ordinary K3 surfaces (you can’t quite take the Artin-Mazur formal group, but instead an “enlarged” version of it). Many people are interested in knowing whether certain varieties lift to characteristic {0}, and not only that but in a very nice way (maybe you remember the Hodge filtration or you want it to satisfy the Tate conjecture). These are called “canonical lifts” and a really beautiful way to construct them is by showing that the deformation functor is isomorphic to the deformations of some particular {p}-divisible group of height {1}.

If you know that the deformations of a {p}-divisible group are unobstructed, then this produces a lift. If you know even more, like the deformation functor is represented by a smooth formal group scheme (which in the height {1} case it is) you can take a canonical choice of lift, namely the one you get from the identity element. This is one way to get a canonical lift of ordinary abelian varieties and ordinary K3 surfaces.

Now hopefully everyone has at least a little interest in deformations of {p}-divisible groups. I have to say that I find this idea amazing and quite under-used. You want to know whether there is a lift to characteristic {0}, or maybe just in general you want to understand the deformations of some object. This in general is incredibly hard. Instead, you find a candidate {p}-divisible group associated to the object and show that the deformation functors are related or isomorphic. Then you can extrapolate information about the original deformation functor, since the deformations of {p}-divisible groups are well understood. I have to admit that this technique has not been very useful to me so far in my research, but I’m quite optimistic still…

What are Crystals?

Since {p}-divisible groups are pretty awesome, I’m coming back to them for a little bit. Today will tie into the crystalline stuff we’ve been looking at, but after that we’ll move on to the deformation theory of them. First, you’ll need to recall the posts about the Dieudonne module and the heights of {p}-divisible groups (in particular, you should at the very least remind yourself what a {p}-divisible group is).

Unless otherwise stated we will be working over a perfect field {k} of positive characteristic. Almost everything done in the next several posts could actually be done in more generality by working on a general base and taking the {p}-divisible group to be a sheaf of groups on the fppf site of Sch/S. Recall that a {p}-divisible group is a formal group {G=\lim_{\rightarrow}G_\nu} with the property that {|G_\nu|=p^{\nu h}} and we call {h} the height of {G} (this is NOT the definition, but merely a fact about them).

For any {p}-divisible group we can form the Dieudonne module {D(G)} which turns out to be a free {W(k)}-module of rank {h}. For our first new concept, {D(G)} is an example of something called an {F}-crystal, or just crystal sometimes. For a definition, an {F}-crystal is a free {W(k)}-module {M} together with an injective endomorphism {\phi: M\rightarrow M} that is Frobenius semi-linear, i.e. {\phi(\lambda m)=F(\lambda)\phi(m)}, where {F} is the lift of Frobenius to {W(k)}. The {F}-crystals over {k} form a category in the obvious way (maps have to respect the endomorphism) and more importantly this is the category for which we get an equivalence with {p}-divisible groups just by the functor {G\mapsto D(G)} (note {D(G)} is an {F}-crystal by using the standard Frobenius as the endomorphism).

Suppose {K} is the fraction field of {W(k)}. An {F}-isocrystal is a (finite dimensional) vector space over {K} equipped with a Frobenius semi-linear automorphism. We always get one of these from a {p}-divisible group as well by tensoring with {K}. So define {E(G)=D(G)\otimes K} and the automorphism is obtained by extending Frobenius linearly. It is a major theorem that every {F}-isocrystal has a direct sum decomposition called the slope decomposition {\bigoplus E_\lambda}. To see what this is, I typed it up here. The {\lambda} are called the slopes and they are a finite set of rational numbers. If {E} is an isocrystal we also write {E_{[a,b]}} to mean {\bigoplus_{\lambda\in [a,b]}E_\lambda}.

This brings us to an incredibly fascinating tie in to the heights of varieties. Recall that we have a {p}-divisible group representing {\Phi (S)=\mathrm{ker}(H_{et}^n(X\otimes S, \mathbb{G}_m)\rightarrow H_{et}^n(X, \mathbb{G}_m))} and the height of this is the height of the variety. But if we look at the crystalline cohomology {H^n_{crys}(X/W)\otimes K} we get an {F}-isocrystal, and the part {(H^n_{crys}(X/W)\otimes K)_{[0,1)}} with slopes strictly less than {1} is some finite dimensional vector space, and the dimension of this is the height of {X}! Maybe I’ll explain this more some other time, but it is very far off topic for now.

Grothendieck saw how nice the theory of crystals helped analyze {p}-divisible groups and other things, so he tried to generalize it to crystalline cohomology (note, I have no idea about the accuracy of this historical tidbit). Suppose {\mathcal{F}} is a Zariski stack on Sch. An {\mathcal{F}}-crystal is a Cartesian section of the fibered category {\mathcal{F}\times_{Sch} Crys(X)} where the map from the crystalline site {Crys(X)\rightarrow Sch} is given by {(U\hookrightarrow T, \gamma)\mapsto T} (a functor we’ve talked about already). If you unravel all this, then you find that an {\mathcal{F}}-crystal is always a sheaf {\mathcal{G}} on {Crys(X)} that satisfies another condition. We’ll always think of {\mathcal{O}_X}-crystals in which case the extra condition is just that for any map {u} in {Crys(X)} we get {u^*\mathcal{G}\rightarrow \mathcal{G}} is an isomorphism.

We see that this is some sort of “rigidity” condition. In Berthelot and Ogus, they claim that Grothendieck coined the term crystal because it has two properties, it is both “rigid” as we saw and it “grows” over PD thickenings (this is automatically satisfied because it is a sheaf). A trivial example of a crystal is {\mathcal{O}_X} itself. A nontrivial, but incredibly useful example of a crystal is given any closed immersion (over some fixed PD scheme) {i: Y\hookrightarrow X} the sheaf {i_{crys *}(\mathcal{O}_Y)} is an {\mathcal{O}_X}-crystal.

Suppose now we have some fixed (PD) base {S}. In order to extend Dieudonne theory to this new sense of crystals, we might want a functor {\mathbb{D}} that sends a {p}-divisible group to an {\mathcal{O}_S}-crystal. There is such a functor, but its construction requires us to know that the deformations of any {p}-divisible group are unobstructed. This brings me to the point of this post. We want to understand the deformation theory of {p}-divisible groups.

Stratification 3: The Definition

Last time we looked at the characteristic {0} case to figure out how our old definition of a connection on a sheaf could be rephrased in terms of a “parallel transport” rule. This took the form of giving an isomorphism {(p_1^*\mathcal{E})|_{X^{(2)}}\rightarrow (p_2^*\mathcal{E})|_{X^{(2)}}} that restricted to the identity on the diagonal. Moreover, if the connection is integrable you can lift these isomorphisms to all infinitesimal neighborhoods of the diagonal {X^{(n)}} so that the restrictions are all the previous ones (they are compatible).

Two times ago we looked at the n-th infinitesimal neighborhood of the PD-envelope of the diagonal in the {(\nu+1)} product and called it {D_{X/S}^n(\nu)}. A theorem that we won’t prove is that a connection on {\mathcal{E}} can be lifted compatibly to {D_{X/S}^n(1)} for all {n} if and only if it is integrable. This finally brings us to our definition of stratification.

If {\mathcal{E}} is an {\mathcal{O}_X}-module, a PD stratification on {\mathcal{E}} is a collection of isomorphisms {\epsilon_n: \mathcal{D}_{X/S}^n(1)\otimes \mathcal{E}\rightarrow \mathcal{E}\otimes \mathcal{D}_{X/S}^n(1)} such that each {\epsilon_n} is {\mathcal{D}_{X/S}^n(1)}-linear, the {\epsilon_n}‘s are compatible (they restrict to the previous one and the {\epsilon_0} is the identity) and they satisfy the standard cocycle condition at all levels.

Essentially, we took the intuition from the characteristic {0} case and just encoded it into a definition. A stratification is just a compatible choice of infinitesimal parallel transport at all levels. I don’t want to go too far down this road which will involve differential operators and things. I plan to come back to these ideas in the not-to-distant future, but for the next few weeks I want to change gears.

One thing that keeps coming up for me and I keep using is the deformation theory of {p}-divisible groups. Since we already have some groundwork on {p}-divisible groups done, I hope we can actually prove that the deformation functor over Artin {W}-algebras is formally smooth and prorepresentable by {W[[t_1, \ldots, t_d]]} where {d} is the dimension of the group times the dimension of its dual.

Stratification 2

Before defining stratification, we’ll look at what this notion is in what is hopefully a more familiar context. Let’s forget about all the PD stuff for today (but keep it in mind for later). Suppose {S} is a scheme and {X} is smooth of finite type over {S}. The diagonal is a closed immersion {\Delta: X\rightarrow X\times_S X}. Suppose it is defined by the quasi-coherent ideal sheaf {\mathcal{I}} (it is generated by things of the form {t\otimes 1 - 1 \otimes t}). So far all this should feel very familiar from our setup in the last post.

Rather than worry about PD things, we’ll define the {n}-th infinitesimal neighborhood of {\Delta} to be {X^{(n)}}, the subscheme defined by {\mathcal{I}^{n+1}}. There are natural inclusions {X\hookrightarrow X^{(2)}\hookrightarrow X^{(3)}\rightarrow \cdots}. and all of these sit inside {X\times X}. Define {p_1} and {p_2} to be the first and second projections {X\times X\rightarrow X}. Given a quasi-coherent sheaf {\mathcal{E}}, we already have a notion of connection. Recall that it is just linear map {\nabla: \mathcal{E}\rightarrow \mathcal{E}\otimes \Omega^1} that satsifies the Leibniz rule.

Grothendieck defined a connection on {\mathcal{E}} to be an isomorphism {(p_1^*\mathcal{E})|_{X^{(2)}}\stackrel{\sim}{\rightarrow} (p_2^*\mathcal{E})|_{X^{(2)}}} which restricts to the identity on {\Delta}. I’ll give you the punchline up front. This definition is equivalent to the other one! If you look at the last post, it should be clear that this is the definition that will extend easier in the PD case. Let’s try to understand what this definition is saying.

This definition is somehow related to parallel transport. How could we think about parallel transport? Suppose you have some pointed {S}-scheme, say {T} with {t}. Consider a closed immersion {f:(T,t)\rightarrow (X, x)}. Maybe this is like a path and you remember the starting point (but it doesn’t have to be). There is always the constant map {f_x: (T,t)\rightarrow (X,x)} which just sends all of {T} to {x}. Parallel transport along {f(T)} from {x} of {\mathcal{E}} should be an isomorphism {\mathcal{E}|_{f(T)}\stackrel{\sim}{\rightarrow} \mathcal{E}|_{f_x(T)}} which restricts to be the identity at {x}.

Strictly speaking what I just wrote is nonsense, so what do I mean? If we restrict {\mathcal{E}} to the image of {T} I want to be able to choose a trivialization to give a linear isomorphism with the trivial bundle having fiber {\mathcal{E}|_{x}}. If you are thinking in terms of vector bundles, you could also think of it as a compatible choice of isomorphisms of the fiber at all points of {F(T)} with the fiber at {x} and the isomorphism at {x} must be the identity.

Now the definition of connection we gave should probably more accurately be described as “first-order” parallel transport where for us we should be thinking that {T=\mathrm{Spec}(k[\epsilon])}. Let’s check that our new notion of connection gives parallel transport with this {T}. Choose a {k}-point on {x\in\Delta}, then {f: T\rightarrow X\times X} is just going to {x} and choosing a tangent vector there. Since we have by definition an isomorphism {p_1^*\mathcal{E}|_{X^{(2)}}\rightarrow p_2^*\mathcal{E}|_{X^{(2)}}} we can just restrict this further to {f(T)} which lies in {X^{(2)}} to get the parallel transport isomorphisms. This shows that connections give parallel transport along tangent vectors (i.e. along first-order infinitesimally short paths).

Now nothing is stopping us from continuing in this fashion. What happens if we take {X_3^{(2)}} as the first-order infinitesimal neighborhood of the diagonal in {X\times X\times X}. Then we have three projections {X\times X\times X\rightarrow X\times X} which we’ll call {p_{i,j}} for projecting onto the {i} and {j} factors. Using the definition of connection we can now obtain {\epsilon_{i,j}: p_i^*\mathcal{E}|_{X_3^{(2)}}\stackrel{\sim}{\rightarrow} p_j^*\mathcal{E}|_{X_3^{(2)}}} which just comes from first pulling back using {p_{i,j}} then restricting.

Let’s think back to when we talked about connections months ago. The next major definition was what it meant to be integrable. We defined the curvature {K(\nabla)} to be the composition {\mathcal{E}\rightarrow \mathcal{E}\otimes \Omega^1\rightarrow \mathcal{E}\otimes \Omega^2} and {\nabla} was integrable if the curvature was {0}. Or in other words, if the associated sequence was actually a complex.

Using our new definition we get that a connection is integrable if {\epsilon_{1,3}=\epsilon_{2,3}\circ \epsilon_{1,2}}. In other words, if the isomorphisms coming from the associated {X\times X\times X} satisfy some sort of cocycle condition. In characteristic {0} being integrable actually guarantees that you can lift the isomorphisms to all {n}-th order neighborhoods. This gives {n}-th order parallel transport, meaning we get parallel transport using {T=\mathrm{Spec}(k[x]/(x^n))}.

Now everything is compatible here (we’re lifting the isomorphisms, meaning they restrict to the previous one). Thus we actually have directed systems to take a limit. This gives us what could maybe be called formal local parallel transport. Believe it or not, this is exactly the type of thing we are after in the PD case. It is basically the purpose of building a notion of “{x^n/n!}” so that we can get some sort of formal power series. I think that is a good enough reminder of this “classical” case. Maybe if you are super motivated you can go to the previous post and work out how these definitions extend. The setup is all there.

Stratification 1

I’m back. I have several trains of thought started at the nlab, so I’ll probably be jumping back and forth for awhile, but I consider this blog to be of more importance at this time since I need to understand this material sooner. It’s been awhile, so I’ll recall that we talked about divided powers on ideals of rings for several posts. This was a way to get something that looked “power series-like” in positive characterstic where we can’t actually divide by some numbers.

Then we noticed that all these concepts sheafified, so we could define the category of sheaves of P.D. rings. This allowed us to define a PD scheme, as something locally isomorphic to {\mathrm{Spec}(A, I, \gamma)} where the spectrum of a PD ring is defined to be the locally ringed space {(|\mathrm{Spec}(A)|, \mathcal{O})} where {\mathcal{O}} is the sheaf of PD rings inherited from {\gamma}.

Then we defined crystalline cohomology, but instead of moving forward with more concepts along these lines, we’ll go backwards for a post or two to figure out what is happening with PD schemes better. Recall the setup for the crystalline site, we fix some PD scheme {(S, \mathcal{I}, \gamma)} and consider {X} an {S}-scheme. In general, when {i:X\rightarrow Y} is a closed immersion we can define the quasi-coherent {\mathcal{O}_X}-algebra {\mathcal{D}_{\mathcal{O}_Y, \gamma}(\mathcal{J})} as the sheaffified version of taking the PD envelope of {\mathcal{J}} where {\mathcal{J}} is the defining ideal. We’ll just use the shorthand {D_{X, \gamma}(Y)} for this. As a scheme we have {D_{X, \gamma}(Y)=\underline{Spec}_Y(\mathcal{D}_{X,\gamma}(Y))}.

If {\gamma} extends to {X}, then {\mathcal{D}_{X, \gamma}(Y)/\overline{\mathcal{J}}\simeq \mathcal{O}_X}. I.e. {i} factors through a PD immersion {j: X\rightarrow D_{X, \gamma}(Y)} defined by {\overline{\mathcal{J}}}. We define {D^n_{X, \gamma}(Y):=\mathcal{D}_{X, \gamma}(Y)/\overline{\mathcal{J}}^{[n+1]}} and it is called the {n}-th order divided power neighborhood of {X} in {Y}. Caution: This is NOT in general a subscheme of {Y}. This {n}-th order divided power neighborhood can be formed for the locally closed immersion case as well.

Example: If {X\rightarrow Y} is an immersion of smooth {S}-schemes and {m\mathcal{O}_Y=0}, then {\mathcal{D}_{X, \gamma}(Y)} is locally isomorphic to a PD polynomial algebra over {\mathcal{O}_X}.

One important thing we can look at is {X/S^{\nu+1}} defined to be the {(\nu+1)}-fold product of {X} with itself over {S}. Let {\Delta: X\rightarrow X/S^{\nu+1}} be the diagonal map. This is a locally closed immersion so we can apply the above construction. Suppose {\gamma} extends to {X} and {m\mathcal{O}_X=0} or {X/S} is separated. This allows us to define the divide power envelope {D_{X/S}(\nu)} of {X} in {X/S^{\nu+1}} and the {n}-th order divided power neighborhood {D^n_{X/S}(\nu)}.

By the example if {X/S} is smooth, {m\mathcal{O}_X=0} and we pick {x_1, \ldots, x_n} local coordinates on {X}, then the structure sheaf {\mathcal{D}_{X/S}(1)} of {D_{X/S}(1)} (the envelope of {\Delta: X\rightarrow X\times_S X}) is isomorphic to {\mathcal{O}_X\langle \xi_1, \ldots, \xi_n\rangle} where {\xi_i=1\otimes x_i- x_i\otimes 1}. Well, that was like a paragraph from Berthelot and Ogus. Next time we’ll move on to what a stratification is.