# The Dieudonné Module

I’m not sure how much of this Witt sheaf stuff to keep talking about. There is this beautiful invariant associated to any variety in positive characteristic that doesn’t come up in characteristic ${0}$. It is called the height, and the way it is defined is by attaching a ${p}$-divisible (formal) group to your variety and looking at the height of that. This will tie together all these things we’ve been talking about since it turns out that the Dieudonné module of this formal group is exactly ${H^n(X, \mathcal{W})}$, and the non-finite generatedness of this module corresponds to the variety being “supersingular” which just means it has infinite height.

So anyway, this means at some point I should talk about formal groups, and ${p}$-divisible groups, and height, and Dieudonné modules if I ever want to get there, which means we should cut off the discussion on Witt sheaf cohomology soon. I’ve essentially been giving you the highlights of Serre’s paper on the topic and we’re only about half-way through, so it seems a shame to stop now.

One thing that we are consistently interested in is the projective system ${W_n(A)\rightarrow W_{n-1}(A)}$ formed by restriction or truncating. We seem to consistently ignore that we also have an inductive system formed by shifting ${W_n(A)\rightarrow W_{n+1}(A)}$, i.e. ${V(a_0, \ldots, a_{n-1})=(0, a_0, \ldots, a_{n-1})}$. Recall that we’ve already thought about several properties related to this like ${RFV=p}$, and it is how we know that ${W(k)/pW(k)\simeq k}$, since this is essentially just quotient by the shift.

We should be a bit more careful, though. Now let ${\Lambda=W(k)}$. We normally treat ${W(A)}$ as a ${\Lambda}$-module by taking ${\lambda\in \Lambda}$ and considering it as an element in ${W(A)}$ via the natural map ${W(k)\rightarrow W(A)}$ induced from the algebra map and then using Witt multiplication. When we do this, recall that ${V}$ is only a semi-linear map. It has the property that ${V(\lambda a)=\lambda^{1/p}V(a)}$. So it is not a map of ${\Lambda}$-modules, and hence the inductive system is not a system of ${\Lambda}$-modules.

We can alter the ${\Lambda}$-structure to make all of this work. Awhile ago we made the blanket assumption that ${k}$ was perfect. We will still need this here. For notation, we’ll say that ${\overline{\lambda}}$ is the image of ${\lambda}$ via ${W(k)\rightarrow W(A)\rightarrow W_n(A)=W(A)/V^n(W(A))}$. Let ${W_n(A)}$ be a ${\Lambda}$-module by the altered action ${\lambda \star x= \overline{\lambda}^{p^{1-n}}x}$, where two elements next to eachother means Witt multiplication.

Let’s check that this makes ${V}$ a ${\Lambda}$-linear map. ${\displaystyle V(\lambda\star a)=T(\overline{\lambda}^{p^{1-n}}a)=V(F(\overline{\lambda}^{p^{-n}})a)=\overline{\lambda}^{p^{-n}}V(a)=\lambda\star V(a)}$

Thus we have an inductive system of ${\Lambda}$-modules. Let’s shift categories for a second. If we have an affine commutative unitary group scheme over ${k}$, then we use this inductive system to define ${D(G)}$, the Dieudonne module of ${G}$ by taking ${\displaystyle D(G)=\lim_{\rightarrow} Hom(G, W_n(k))}$, and similarly in the Ind category so we have a Dieudonne module for formal group schemes and ${p}$-divisible groups.

Since all the ${V}$ operators are are monomorphisms, we get that ${Hom(G, W_n(k))\rightarrow Hom(G, W_{n+1}(k))}$ are all injective and hence we can identify ${Hom(G, W_n(k))}$ with a submodule of ${D(G)}$ or explicitly we know that ${Hom(G, W_n(k))=\{m \in D(G): V^n(m)=0\}}$. Thus every element of ${D(G)}$ is killed by a power of ${V}$.

If we introduce one more strange bit of abstraction we can see the beauty of all this. Let ${\mathbf{D}_k=\Lambda\{F, V\}}$ be the noncommutative polynomial ring over the Witt vectors on two indeterminates that satisfy the commutation laws ${Fw=w^pF}$, ${w^pV=Vw}$, and ${FV=VF=p}$. This is called the Dieudonné ring. We have a canonical way to consider ${D(G)}$ as a left ${\mathbf{D}_k}$-module. Thus ${G\mapsto D(G)}$ is a contravariant functor from affine unitary group schemes to the category of ${\mathbf{D}_k}$-modules with ${V}$ torsion. This turns out to be an anti-equivalence of categories.

We can get tons of information about these group schemes by studying their corresponding properties as ${\mathbf{D}_k}$-modules (which we probably won’t do). Maybe we’ll start thinking about ${p}$-divisible groups next time to try to work our way towards defining the height of a variety.