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.

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