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Demazure, lectures on p-divisible groups, III.3, the Witt rings over k

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Let kk be a file of prime characteristic pp. Let WW denote the Witt ring over Z?

Definition

The Witt ring over kk denoted by is defined by the coefficient extension W k:=W kW_k:=W\otimes_\mathbb{Z} k. and W nk:=W n kW_{nk}:=W_n\otimes_\mathbb{Z}k

The phantom-components? W kα kW_k\to \alpha_k reduce now to (a i) 0ina 0 p n(a_i)_{0\le i\le n}\mapsto a_0^{p^n}.

Since W k=W F p F pkW_k=W_{F_p}\otimes_{F_p}k we can identify W k (p)W_k^{(p)}, see Definition Frobenius morphism, and W kW_k and the Frobenius morphism becomes the endomorphism

F:{W kW k (a 0,,a n,)(a 0 p,,a n p,)F:\begin{cases}W_k\to W_k\\(a_0,\dots,a_n,\dots)\mapsto (a_0^p,\dots,a_n^p,\dots)\end{cases}

This is a ring morphism since since FF commutes with products. Similar statements are true for W nkW_{nk} and the affine kk-group Λ k\Lambda_k defined in Artin-Hasse exponential series?.

Proposition
  1. The Verschiebung morphism? of Λ k\Lambda_k is given by ϕ(t)ϕ(t p)\phi(t)\to \phi(t^p).

  2. The Verschiebung morphism of K kK_k is the translation? TT.

  3. The Verschiebung morphism of W nkW_nk is RT=TRR \cdot T=T\cdot R.

  4. If x,yW k(R)x,y\in W_k(R), RM kR\in M_k, then V(Fxy)=xVyV(F x\cdot y)=x\cdot V y.

Corollary
Corollary
Corollary

Let kk be perfect. Then

  1. W(k)W(k) is a discrete valuation ring.

  2. W(k)W(k) is complete.

  3. W(k)/pW(k)=kW(k)/p W(k)=k

Proposition

(Witt) Let kk be perfect, let AA be compete, noetherian local ring with residue field kk. Let π:Ak\pi:A\to k be the canonical projection. There exists a unique ring morphism

u:W(k)Au:W(k)\to A

which is compatible with the projections W(k)kW(k)\to k and π:Ak\pi:A\to k.

If moreover AA is a discrete valuation ring with p1 A¬0p\cdot 1_A\not 0, then AA is a free finite W(k)W(k)-module of rank [A/p:A][A/p : A].

In particular if pA=ApA=A, then uu is an isomorphism.

Revised on May 27, 2012 13:43:39 by Stephan Alexander Spahn (79.227.168.80)