This is a note on the construction of Dieudonne modules (over a perfect field), prepared for a student seminar on $p$-adic Hodge theory at Harvard, Fall 2012. Our main references are [1] and [2].

TopDieudonne modules

Let $k$ be a perfect field of characteristic $p>0$. Our goal is to classify finite (commutative) $k$-groups of $p$-power order using (semi-)linear algebraic data. Recall every such $k$-group can be decomposed as $$G=G_\mathrm{et}\times G_\mathrm{loc}=\underbrace{G_{\mathrm{et}, \mathrm{loc}} \times  G_{\mathrm{loc}, \mathrm{loc}}}_{\text{unipotent}}\times \underbrace{G_{\mathrm{loc},\mathrm{et}}}_{\text{multiplicative}}.$$ Let $F: G\rightarrow G^{(p)}$ be the Frobenius and $V: G^{(p)}\rightarrow G$ be the Verschiebung, then

  1. on $G_{\mathrm{et}, \mathrm{loc}}$: $F$ is an isomorphism and $V$ is nilpotent.
  2. on $G_{\mathrm{loc}, \mathrm{loc}}$: $F$ is nilpotent and $V$ is nilpotent.
  3. on $G_{\mathrm{loc}, \mathrm{et}}$: $F$ is nilpotent and $V$ is an isomorphism.
Example 1 The constant group $\mathbb{Z}/p \mathbb{Z}$ is of etale-local type, $F$ acts as the identity and $V$ acts as 0. Dually, the multiplicative group $\mu_p$ is of local-etale type, $F$ acts as 0 and $V$ acts as the identity. The self-dual group $\alpha_p=\ker F(\mathbb{G}_a\rightarrow \mathbb{G}_a)$ is of local-local type and both $F$ and $V$ acts as 0.

The etale part is relatively easy to deal with: it is same as giving the data of a finite $G_k$-module. Over $\bar k$, the etale part is a constant group of $p$-power order and can be determined by its Pontryagin dual $M(G)= \Hom(G, \mathbb{C}^\times)=\Hom(G, \mathbb{Q}_p/\mathbb{Z}_p)$. Of course, the local part cannot be detected at the level of points: we would like some enhancement of $\mathbb{Q}_p/\mathbb{Z}_p$ to also detect the local part. A natural candidate is using the Witt ring scheme $W$. Let $W_n$ be the ring scheme of Witt vectors of length $n$. We have at the level of $\mathbb{F}_p$-points $$W_n(\mathbb{F}_p)=\mathbb{Z}/p^n \mathbb{Z},\quad \varinjlim W_n(\mathbb{F}_p)\cong \mathbb{Q}_p/\mathbb{Z}_p,$$ and in general, $$W_n(k)=W(k)/p^n W(k),\quad \varinjlim W_n(k)\cong \mathrm{Frac}(W(k))/W(k).$$ Motivated by this, we introduce the local group $W_n^m$ to be the kernel of $F^m$ on $W_n$. These are the replacements of $\mathbb{Z}/p^n \mathbb{Z}$ for local groups.

Remark 1 Recall that the Frobenius and Verschiebung acts on $W$ as $$F (a_0, a_1,\ldots)=(a_0^p, a_1^p,\ldots), \quad V (a_0,a_1,\ldots )=(0,a_0,a_1,\ldots).$$ Then $W_1=\mathbb{G}_a$ and $W_1^1=\alpha_p$. Moreover, we have exact sequences $$0\rightarrow W_{n_1}^m\xrightarrow{V^{n_2}} W_{n_1+n_2}^m\xrightarrow{\text{trunc}} W_{n_2}^m\rightarrow0,$$ $$0\rightarrow W_n^{m_1}\xrightarrow{\text{inc}}W_n^{m_1+m_2}\xrightarrow{F^{m_2}} W_n^{m_2}\rightarrow 0.$$ It follows that $W_n^m$ is a successive extension of $nm$ copies of $\alpha_p$.

Staring at the action of $W(k)$ and $F,V$ on $W$ motivates the following definition.

Definition 1 Let $\sigma:W(k)\rightarrow W(k)$ be the automorphism lifting $x\mapsto x^p$ on $k$. Let $D_k=W(k)[F,V]$ be the Dieudonne ring (noncommutative unless $k=\mathbb{F}_p$) subject to the relations $F x=\sigma(x)F$, $V\sigma(x)=xV$ and $FV=VF=p$.
Definition 2 Let $G$ be of local-local type. We define the (contravariant) Dieudonne module of $G$ to be $$M(G)=\varinjlim _{m,n} \Hom(G,W_n^m)=\varinjlim_n\Hom(G, W_n).$$ Notice $M(G)$ becomes a left $D_k$-module via the action of $D_k$ on $W_n^m$.
Remark 2 The action of $W(k)$ on $W_n^m$ needs to be modified so that it is compatible with the transition maps between the $W_n^m$'s. Namely, the action of $x\in W(k)$ on $W_n^m$ is modified to be multiplication by $\sigma^{-n}(x)$.
Theorem 1 The functor $G\mapsto M(G)$ gives an exact anti-equivalence of categories between {finite $k$-group schemes of local-local type} and {left $D_k$-modules of finite $W(k)$-length with $F$, $V$ nilpotent}.

We will not prove this theorem in detail but let us explain why the functor lies in the desired target.

Proposition 1 Suppose $G$ is of local-local type.
  1. If $F^m=0$ and $V^n=0$ on $G$, then $F^m$ and $V^n$ annihilate $M(G)$.
  2. $\mathrm{length}_{W(k)}M(G)=\log_p\# G$.
Proof The first part follows from the functoriality of $F$ and $V$. The second part follows from induction and the fact that $M(\alpha_p)=M(W_1^1)=\End(\alpha_p)\cong k$ has $W(k)$-length 1. Notice that $\End(W_n^m)\rightarrow M(W_n^m)$ is always injective. To show the surjectivity, one uses the fact that if $F^m=0$ and $V^n=0$ on $G$, then any homomorphism $\phi: G\rightarrow W_{n'}^{m'}$ for $m'\ge m$ and $n\ge n$ factors uniquely through $W_n^m$ (again by functoriality of $F$ and $V$).
Remark 3 One can further identify $D_k/(D_kF^m+D_kV^n)=\End(W_n^m)$. The injectivity is clear since $F^{m-1}V^{n-1}$ acts non-trivially on $W_n^m$. For the surjectivity, one computes the length of $D_k/(D_kF^m+D_kV^n)$ as a $D_k$ module to be $nm$, which is equal to $\log_p\# W_n^m=nm$.

For a general finite $k$-group, we define its Dieudonne module for the three parts in its decomposition separately.

Definition 3 Suppose $G$ is etale-local, then we define $M(G)=\varinjlim_n\Hom(G, W_n)$. Suppose $G$ is local-etale, then we define $M(G)=M(G^\vee)^\vee$, where the dual outside is given by $M^\vee=\Hom_{W(k)}(M, W(k)[1/p]/W(k))$. In general, we decompose $G=G_{\mathrm{et},\mathrm{loc}}\times G_{\mathrm{loc},\mathrm{loc}}\times G_{\mathrm{loc},\mathrm{et}}$ and define its (contravariant) Dieudonne module $$M(G)=M(G_{\mathrm{et},\mathrm{loc}})\oplus M(G_{\mathrm{loc},\mathrm{loc}})\oplus M(G_{\mathrm{loc},\mathrm{et}}).$$
Remark 4 The formation of Dieudonne module commutes with taking the dual, i.e., $M(G^\vee)\cong M(G)^\vee$. This reduces to the local-local case, which essentially reduces to the duality between $W_n^m$ and $W_m^n$.

Let us also mention an important property that recovers the cotangent space of $G$ from its Dieudonne module.

Proposition 2 There is an natural isomorphism of vector spaces $T_{G,0}\cong (M(G)/F M(G))^\vee$.
Proof Notice that the tangent space $$T_{G,0}:=\ker (G(k[\varepsilon])\rightarrow G(k))\cong \Hom(G^\vee, \mathbb{G}_a)=\Hom(G^\vee,W_1).$$ Since $W_1=\ker V$ on any $W_n$, we know that $$\Hom(G^\vee,W_1)=\ker V|M(G^\vee)=\ker V|M(G)^\vee=\mathrm{coker}(F|M(G))^\vee,$$ as desired.
Remark 5 It is amusing to compare it with the exact sequence (in any characteristic) associated to an abelian variety $A$: $$0\rightarrow H^0(A, \Omega^1_A)\rightarrow H^1_\mathrm{dR}(A)\rightarrow H^1(A,\mathcal{O}_A)\rightarrow 0.$$ Oda [3] proved that $H^1_\mathrm{dR}(A)$ is canonically isomorphic to $M=M(A[p])$ over any perfect field of characteristic $p>0$ and the Hodge filtration $H^0(A,\Omega_A^1)\subseteq H_\mathrm{dR}^1(A)$ can be identified with $\ker F|M=VM\subseteq M$.

We summarize the main theorem of Dieudonne theory as follows.

Theorem 2 There is an exact additive anti-equivalence of categories $G\mapsto M(G)$ between {finite $k$-group schemes of $p$-power order} and {left $D_k$-modules of finite $W(k)$-length}. satisfying:
  1. $\log_p\#G=\mathrm{length}_{W(k)}M(G)$.
  2. $M(G^\vee)=M(G)^\vee$.
  3. $M(G)/F M(G)$ is canonically isomorphic to the cotangent space of $G$.
  4. (extension of scalars) for any extension $k\rightarrow k'$ of perfect fields, $M(G_k')\cong W(k')_{W(k)}\otimes M(G)$.
Example 2 Suppose $G$ is of order $p$. Then $M(G)$ is a $W(k)$-module of rank 1 and $p=FV$ acts as 0 on $M(G)$. Hence $M(G)\cong k$ is a one dimensional $k$-vector space with the action of $F$ and $V$.
  1. For $G=\mathbb{Z}/p \mathbb{Z}$, $F$ acts as $\sigma$ and $V$ acts as 0.
  2. For $G=\mu_p$, $F$ acts as 0 and $V$ acts as $\sigma^{-1}$.
  3. For $G=\alpha_p$, both $F$ and $G$ acts as 0.
Remark 6 Taking limit in the main theorem gives us an anti-equivalence between $p$-divisible groups over $k$ (of height $h$) and left $D_k$-modules which are free as $W(k)$-modules (of rank $h$).

TopFontaine's uniform construction

One unsatisfying aspect of the above construction of the Dieudonne functor is that definition for the local-etale (multiplicative) part seems a bit artificial. In the second part of this talk, we shall describe a uniform construction due to Fontaine.

Definition 4 Suppose $R$ is a $k$-algebra. By shifting to the left, we see that every element of $\varinjlim_n W_n(R)$ is represented by a covector $(a_{-n})=(\ldots, a_{-n},\ldots,a_0)$, $a_{-n}\in R$. Let $S_m$ be the $m$-th universal addition polynomial for Witt vectors, then the addition rule on covectors $(c_{-n})=(a_{-n})+(b_{-n})$ is given by $c_n=S_m(a_{-m-n},\ldots,a_{-n},b_{-m-n},\ldots,b_{-n})$ for $m\gg0$ (which stabilizes). We denote the $k$-group scheme obtained this way by $CW^\mathrm{u}$, called the group of unipotent Witt covectors. This is simply a reformulation of what we used to detect the unipotent part of finite $k$-group schemes of power order.

In order to also detect the multiplicative part, Fontaine generalizes it to the following.

Definition 5 Suppose $R$ is a $k$-algebra. We define $CW(R)$ to consist of $(\ldots, a_{-n},\ldots,a_0)$, $a_{-n}\in R$ for which there exists $r\ge0$ such that the ideal generated by $\{a_{-n},n\ge r\}$ is nilpotent. Then $CW^\mathrm{u}(R)\subseteq CW(R)$ and indeed the addition rule also extends to $CW(R)$. We call the $k$-group scheme $CW$ the group of Witt covectors. In particular, $CW(k)=CW^u(k)=\mathrm{Frac}(W(k))/W(k)$.
Remark 7 We endow $R$ with the discrete topology and $CW(R)$ with the induced subspace topology from the product topology. Then $CW(R)$ becomes a complete and separated topological group. Endow $D_k$ with the $p$-adic topology from $W(k)$, then $W(k)$ acts on $CW(R)$ continuously and makes $CW(R)$ a topological $D_k$-module, which is torsion, complete and separated.
Theorem 3 The functor $CW_k$ from finite $k$-algebras to groups $R\mapsto CW(R)$ is pro-represented by a formal $p$-group $\widehat{CW}_k$.
Remark 8 More explicitly, $\widehat{CW}_k=\Spf A$, where $A$ is the profinite completion of $$k^0[{}[X]]=\varprojlim_{r,s} k[\ldots,X_{-n},\ldots,X_0]/\mathfrak{a}_r^s, \quad \mathfrak{a}_r=(X_{-n},n\ge r).$$

Now we can define the Dieudonne module for the larger category of formal $p$-groups over $k$ using $\widehat{CW}_k$.

Definition 6 For any formal $p$-group $G$ over $k$, we define its (contravariant) Dieudonne module $\underline{M}(G):=\Hom_\mathrm{formal-group}(G,\widehat{CW}_k)$.
Remark 9 Suppose $B_G$ is the affine algebra of $G$, then $\underline{M}(G)\subseteq \widehat{CW}_k(B_G)=CW(B_G)$ is closed under the topology defined above. So $\underline{M}(G)$ is also a topological $D_k$-module. Moreover, one can check it is $W(k)[F]$-profinite, i.e., as a $W(k)[F]$-module, the quotient by any of its open submodules has finite length.
Definition 7 For any $W(k)[F]$-profinite topological $D_k$-module $M$, we define $\underline{G}(M)(R)=\Hom_{D_k, \mathrm{cont}}(M,\widehat{CW}_k(R))$ for any finite $k$-algebra $R$. Then $\underline{G}(M)$ is a formal $p$-group over $k$ (as it is left exact).
Theorem 4 (Fontaine) The functor $G\mapsto \underline{M}(G)$ gives an exact anti-equivalence of categories between {formal $p$-group schemes over $k$} and {$W(k)[F]$-profinite topological $D_k$-modules}. $\underline{G}$ is its quasi-inverse.
Remark 10 When restricting to the subcategory of finite $k$-groups of $p$-power order and the subcategory of $p$-divisible groups over $k$, we recover the classical Dieudonne functor with all the desired properties.

References

[1]Richard Pink, Finite group schemes, 2004, www.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf .

[2]Fontaine, J.M., Groupes p-divisibles sur les corps locaux, Société mathématique de France, 1977.

[3]Oda, T., The first de Rham cohomology group and Dieudonné modules, Ann. Sci. Ecole Norm. Sup.(4) 2 (1969), no.1, 63--135.