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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

Remark

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

m W n t m W n+1 f r m1 W n i m W n\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 ii is the canonical inclusion, and r,f,tr,f,t are induced by R,F,TR,F,T where FF is Frobenius, TT is Verschiebung and R:WW nR:W\to W_n is restriction. ii and tt are monomorphisms, ff and rr are epimorphisms, and for m W nm^{W_n} we have F=ifF=if and V=rtV=rt.

Definition and Remark
  1. For RM kR\in M_k, let W (R)W^\prime(R) be the set of all (α 0,α 1,)W k(R)(\alpha_0,\alpha_1,\dots)\in W_k(R) such that a n=0a_n=0 for large nn, and a na_n nilpotent for all nn.

  2. W (R)W^\prime(R) is an ideal in W k(R)W_k(R).

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

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

E˜:{W μ k wE(w,1)\tilde E: \begin{cases} W^\prime\to \mu_k \\ w\mapsto E(w,1) \end{cases}

If xW k(R)x\in W_k(R), yW (R)y\in W^\prime(R) then xyW (R)xy\in W^\prime(R) and E(xy,1)R *E(xy,1)\in R^*. (…)

The morphism

{W×W μ k (x,y)E(xy,1)\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 D(W k)W^\prime\to D(W_k)

which is an isomorphism.

Definition

Let

σ n:{W nkW k (α 0,,α n1)(α 0,,α n1,0,)\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 kW nkR_n:W_k\to W_{nk}.

σ n\sigma_n is not a morphism of groups. σ n\sigma_n sends m W nm^{W_n} in W W^\prime.

Theorem

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

<x,y>:=E(σ n(x)σ m(y),1)\lt x,y\gt:=E(\sigma_n(x)\sigma_m(y),1)

then <x,y>\lt x,y\gt is bilinear and gives an isomorphism

m W nD(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
Revised on June 10, 2012 19:29:12 by Stephan Alexander Spahn (79.227.170.74)