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Demazure, lectures on p-divisible groups, III.6, Dieudonné modules (p-torsion finite k-groups)

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Recall from relation of certain classes of k-groups? the following:

1. ${\mathrm{Feu}}_k$ denotes the category of formal étale unipotent affine $k$-groups.

2. ${\mathrm{Fiu}}_k$ denotes the category of formal infinitesimal unipotent $k$-groups.

3. $W(k)$ denotes the Witt ring over $k$.

4. For $D_k$ see D_k-module? in III.5, Dieudonné modules (affine unipotent groups).

5. $$M:\{\begin{array}{lll}{\mathrm{Acu}}_k& \to & W(k)-\mathrm{Mod}\\ G& \mapsto & M(G)\end{array}$$M:\begin{cases} Acu_k&\to& W(k)-Mod \\ G&\mapsto&M(G) \end{cases}

is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules.

Recall moreover from III.5, Dieudonné modules (affine unipotent groups) that ${\mathrm{Tor}}_V{D}_k-\mathrm{Mod}:={\mathrm{Acu}}_k(G,W_{\mathrm{nk}})=\{m\in M(G)\mid V^{n}m=0\}$ is a submodule.

which are $W(k)$-modules of finite length, killed by a power of $V$, Definition Verschiebung morphism?, and on which $F$, Definition Frobenius morphism, is bijective (resp. and killed by a power of $F$).

Proposition

(formulation of the statement is unclear) The functor

$$M:\{\begin{array}{lll}{\mathrm{Acu}}_k& \to & W(k)-\mathrm{Mod}\\ G& \mapsto & M(G)\end{array}$$M:\begin{cases} Acu_k&\to& W(k)-Mod \\ G&\mapsto&M(G) \end{cases}

which is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules induces the following contravariant equivalences of categories:

1. ${\mathrm{Feu}}_k\to {\mathrm{Tor}}_V{D}_k-\mathrm{Mod}\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $V$ on which $F$ is bijective.

2. ${\mathrm{Fiu}}_k\to {\mathrm{Tor}}_F{D}_k-\mathrm{Mod}\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $F$ (and killed by a power of $V$ ?)on which $F$ is bijective.

(Demazure p.69)

Proof

This follows from the theorem, and the fact that if $G$ is finite, then G is étale (resp, infinitesimal) if and only if $F_G$ is an isomorphism (resp. $F_G^n=0$ for large $n$).

Example

1. If $G=(\mathbb{Z}/p\mathbb{Z}{)}_k\in {\mathrm{Feu}}_k$, then $M(G)=k$ with $F=1$, $V=0$.

2. If $G=p{\alpha }_k\in {\mathrm{Fiu}}_k$, then $M(G)=k$ with $F=0$, $V=0$.

Corollary

For $G\in {\mathrm{Feu}}_k$ or $G\in {\mathrm{Fiu}}_k$, we have

rk(G)= p^{length(M(G))rk(G)= p^{length(M(G))

Proposition

Let $m,n$ be two positive integers. Then

1. The canonical injection $m^{W_n}\to W_n$ defines an element $u\in M(m^{W_n})$ satisfying $V^{n}u=F^n=0$. This gives a map

$${\lambda }_{n,m}:D_k/(D_k{F}^m+D{V}^n)\to M(m_n^W)$$\lambda_{n,m}: D_k/(D_k F^m + D V^n)\to M(m^W_n)

which is bijective.

Theorem

There is an isomorphism

$$M(D(G))\to M(G{)}^{*}$$M(D(G))\to M(G)^*

In prose this means that the autoduality $G\mapsto D(G)$ of ${\mathrm{Fiu}}_k$ corresponds via the Dieudonné-functor $M$ to the autoduality $M\to M^{*}$ in the category $\mathrm{fin}{\mathrm{Tor}}_{V,F}{D}_k-\mathrm{Mod}$ of $D_k$-modules of finite length killed by a power of $V$ and $F$.

Definition

(Dieudonné-module of an infinitesimal multiplicative $k$-group)

Let $G\in {\mathrm{Fim}}_k$. Then the Dieudonné-module of $M(G)*$ is defined by

$$M(G)=M(D(G){)}^{*}$$M(G)=M(D(G))^*

It follows by the Cartier duality between ${\mathrm{Fim}}_k$ and ${\mathrm{Feu}}_k$ that the functor $G\mapsto M(G)$ induces a contravariant equivalence

$${\mathrm{Fim}}_k\to \mathrm{fin}{\mathrm{Tor}}_F{\mathrm{Bij}}_V{D}_k-\mathrm{Mod}$$Fim_k\to fin Tor_F Bij_V D_k-Mod

between ${\mathrm{Fim}}_k$ and the category of all $D_k$-modules of finite length on which $F$ is nilpotent and $V$ is bijective.

Remark

Let $G\in {\mathrm{Fimd}}_k$ (i.e. $G\in {\mathrm{Fim}}_k$ and $G$ diagonalizable).

$G=D({\Gamma }_k)$. Then $D(G)\simeq {\Gamma }_k$, and

$M(D(G))=\mathrm{colim}{\mathrm{Acu}}_k({\Gamma }_k,W_{\mathrm{nk}})=\mathrm{colimGr}(\Gamma ,W_n(k))=\mathrm{Gr}(\Gamma ,W_{\mathrm{\infty }})={\mathrm{Mod}}_{W(k)}(W(k){\otimes }_{\mathbb{Z}},W_{\mathrm{\infty }})={\mathrm{Mod}}_{W(k)}(W(k){\otimes }_{\mathbb{Z}}\Gamma ,W_{\mathrm{\infty }})$

and hence

$$M(G)\simeq (W(\overline{k}){\otimes }_{\mathbb{Z}}\Gamma {)}^{\pi }$$M(G)\simeq(W(\overline k)\otimes_\mathbb{Z}\Gamma)^\pi

where $W_{\mathrm{\infty }}=\mathrm{Quot}(W(k))/W(k)={\mathrm{colim}}_n{W}_n(k)=\underline{W}(k)$, see p.66.

For $F$ and $V$ we have

$$F(\lambda \otimes \chi )={\lambda }^{(p)}\otimes p\chi$$F(\lambda \otimes \chi)=\lambda^{(p)}\otimes p\chi

$$V(\lambda \otimes \chi )={\lambda }^{(p^{-1})}\otimes p\chi$$V(\lambda \otimes \chi)=\lambda^{(p^{-1})}\otimes p\chi

Theorem

a) The Dieudonné functor

$$\{\begin{array}{l}{F}_p{}_k={\mathrm{Fiu}}_k\times {\mathrm{Feu}}_k\times {\mathrm{Fim}}_k\to (\mathrm{fin}W(k)-\mathrm{Mod},F,V)\\ G\to M(G)\end{array}$$\begin{cases} F_p_k=Fiu_k\times Feu_k\times Fim_k\to (fin W(k)-Mod,F,V) \\ G\to M(G) \end{cases}

is a contravariant equivalence between all finite $k$-groups of $p$-torsion, and the category of all triples $(M,F_M,V_M)$ where $M$ is a finite length $W(k)$-module and $F_M$, $V_M$ two group endomorphisms of $M$ satisfying

$$F_M(\lambda m)={\lambda }^{(p)}{F}_M(m)$$F_M(\lambda m)=\lambda^{(p)} F_M(m)

$$V_M({\lambda }^{(p)}m)=\lambda V_M(m)$$V_M(\lambda^{(p)}m)=\lambda V_M(m)

$$F_M{V}_M=V_M{F}_M=p\cdot {\mathrm{id}}_M$$F_M V_M=V_M F_M=p\cdot id_M

b) $G$ is étale, infinitesimal, unipotent or multiplicative according as $F_M$ is isomorphic, $F_M$ is nilpotent, $V_M$ is nilpotent, or $V_M$ is isomorphic

c) For any $G\in {\mathrm{Fp}}_k$ we have $\mathrm{rk}(G)=p^{\mathrm{length}M(G)}$.

d) If $k$ is a perfect extension of $k$, there exists a functorial isomorphism

$$M(D(G))=M(G{)}^{*}$$M(D(G))=M(G)^*

References

Michel Demazure, lectures on p-divisible groups web