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

This entry is about a section of the text

###### Remark

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

$\begin{array}{ccc}{m}^{{W}_{n}}& \stackrel{t}{\to }& {m}^{{W}_{n+1}}\\ {↓}^{f}& & {↓}^{r}\\ {m-1}^{{W}_{n}}& \stackrel{i}{↪}& {m}^{{W}_{n}}\end{array}$\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 ${W}^{\prime }\left(R\right)$ be the set of all $\left({\alpha }_{0},{\alpha }_{1},\dots \right)\in {W}_{k}\left(R\right)$ such that ${a}_{n}=0$ for large $n$, and ${a}_{n}$ nilpotent for all $n$.

2. ${W}^{\prime }\left(R\right)$ is an ideal in ${W}_{k}\left(R\right)$.

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

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

$\stackrel{˜}{E}:\left\{\begin{array}{l}{W}^{\prime }\to {\mu }_{k}\\ w↦E\left(w,1\right)\end{array}$\tilde E: \begin{cases} W^\prime\to \mu_k \\ w\mapsto E(w,1) \end{cases}

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

The morphism

$\left\{\begin{array}{l}W×{W}^{\prime }\to {\mu }_{k}\\ \left(x,y\right)↦E\left(\mathrm{xy},1\right)\end{array}$\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

${W}^{\prime }\to D\left({W}_{k}\right)$W^\prime\to D(W_k)

which is an isomorphism.

###### Definition

Let

${\sigma }_{n}:\left\{\begin{array}{l}{W}_{\mathrm{nk}}\to {W}_{k}\\ \left({\alpha }_{0},\dots ,{\alpha }_{n-1}\right)↦\left({\alpha }_{0},\dots ,{\alpha }_{n-1},0,\dots \right)\end{array}$\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 $x\in {m}^{{W}_{n}}\left(R\right)$, $y\in {n}^{{W}_{m}}\left(R\right)$, define

$:=E\left({\sigma }_{n}\left(x\right){\sigma }_{m}\left(y\right),1\right)$\lt x,y\gt:=E(\sigma_n(x)\sigma_m(y),1)

then $$ is bilinear and gives an isomorphism

${m}^{{W}_{n}}\simeq D\left({n}^{{W}_{m}}\right)$m^{W_n}\simeq D(n^{W_m})

and satisfies

$=$\lt x,t y\gt=\lt f x,y\gt
$=$\lt x,r y\gt=\lt i x,y\gt
Revised on June 10, 2012 19:29:12 by Stephan Alexander Spahn (79.227.170.74)