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Demazure, lectures on p-divisible groups, III.4, duality of finite Witt groups

This entry is about a section of the text

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

Remark

Let $m,n>1$ and define $m^{W_n}:=\ker (F^m:W_{\mathrm{nk}}\to W_{\mathrm{nk}})$. Then there are morphisms

$$\begin{array}{ccc}{m}^{W_n}& \stackrel{t}{\to }& m^{W_{n+1}}\\ {\downarrow }^f& & {\downarrow }^r\\ {m-1}^{W_n}& \stackrel{i}{\hookrightarrow }& m^{W_n}\end{array}$$\array{ m^{W_n} &\stackrel{t}{\to}& m^{W_{n+1}} \\ \downarrow^f&&\downarrow^r \\ {m-1}^{W_n} &\stackrel{i}{\hookrightarrow}& m^{W_n} }

where $i$ is the canonical inclusion, and $r,f,t$ are induced by $R,F,T$ where $F$ is Frobenius, $T$ is Verschiebung and $R:W\to W_n$ is restriction. $i$ and $t$ are monomorphisms, $f$ and $r$ are epimorphisms, and for $m^{W_n}$ we have $F=\mathrm{if}$ and $V=\mathrm{rt}$.

Definition and Remark

1. For $R\in M_k$, let $W^{\prime }(R)$ be the set of all $({\alpha }_0,{\alpha }_1,\dots )\in W_k(R)$ such that $a_n=0$ for large $n$, and $a_n$ nilpotent for all $n$.

2. $W^{\prime }(R)$ is an ideal in $W_k(R)$.

3. $E(w,t)$ is a polynomial for $w\in W^{\prime }(R)$.

4. In particular $E(w,1)$ is defined for $w\in W^{\prime }(R)$, and we have a morphism of groups

$$\tilde{E}:\{\begin{array}{l}{W}^{\prime }\to {\mu }_k\\ w\mapsto E(w,1)\end{array}$$\tilde E: \begin{cases} W^\prime\to \mu_k \\ w\mapsto E(w,1) \end{cases}

If $x\in W_k(R)$, $y\in W^{\prime }(R)$ then $\mathrm{xy}\in W^{\prime }(R)$ and $E(\mathrm{xy},1)\in R^{*}$. (…)

The morphism

$$\{\begin{array}{l}W\times W^{\prime }\to {\mu }_k\\ (x,y)\mapsto E(\mathrm{xy},1)\end{array}$$\begin{cases} W\times W^\prime\to \mu_k \\ (x,y)\mapsto E(xy,1) \end{cases}

is bilinear and hence gives a morphism of groups

$$W^{\prime }\to D(W_k)$$W^\prime\to D(W_k)

which is an isomorphism.

Definition

Let

$${\sigma }_n:\{\begin{array}{l}{W}_{\mathrm{nk}}\to W_k\\ ({\alpha }_0,\dots ,{\alpha }_{n-1})\mapsto ({\alpha }_0,\dots ,{\alpha }_{n-1},0,\dots )\end{array}$$\sigma_n:\begin{cases} W_{nk}\to W_k \\ (\alpha_0,\dots,\alpha_{n-1})\mapsto(\alpha_0,\dots,\alpha_{n-1},0,\dots) \end{cases}

be the section of $R_n:W_k\to W_{\mathrm{nk}}$.

${\sigma }_n$ is not a morphism of groups. ${\sigma }_n$ sends $m^{W_n}$ in $W^{\prime }$.

Theorem

For $x\in m^{W_n}(R)$, $y\in n^{W_m}(R)$, define

$$<x,y>:=E({\sigma }_n(x){\sigma }_m(y),1)$$\lt x,y\gt:=E(\sigma_n(x)\sigma_m(y),1)

then $<x,y>$ is bilinear and gives an isomorphism

$$m^{W_n}\simeq D(n^{W_m})$$m^{W_n}\simeq D(n^{W_m})

and satisfies

$$<x,ty>=<fx,y>$$\lt x,t y\gt=\lt f x,y\gt

$$<x,ry>=<ix,y>$$\lt x,r y\gt=\lt i x,y\gt