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Demazure, lectures on p-divisible groups, III.2, the Witt rings over Z

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The Artin-Hasse exponential series? E can be written as

E((a 0,),t)=exp(Σ n0t p nΦ n/p n)E((a_0,\cdots),t)=exp(-\Sigma_{n\ge 0}t^{p^n}\Phi_n /p^n)

with Φ n(a 0,)=a 0 p n+pa 1 p n1+p na n

The S i in the multiplication formula E((a i),t)E((b i),t)=E((S i(a 0,,a i,b 0,,b i),t) where s i (p)[X 0,,X i,Y 0,,Y i] of the Artin-Hasse exponential series can also be written as Φ n(a 0,,a n)+Φ n(d 0,,b n)=Φ n(S 0,,S n)

Lemma

We have S n[X 0,,X n]

Definition and Theorem

There exists a unique commutative group law W on the -group scheme W:=O called -group of Witt-vectors of finite length relative to p satisfying the following properties:

  1. E:O (p)Λ (p) is a morphism of k-groups.

  2. Each Φ n:O α is a morphism of k-groups.

  3. (a n)+(b n)=(S n(a 0,,a n,b 0,,b n))

For w=(a n)W(R)=R , a n is called the n-th component of w and Φ n(w) is called the n-th phantom component of w. The phantom components of w define a group isomorphism

W [p 1]α [p 1] W\otimes_\mathbb{Z}\mathbb{Z}[p^{-1}]\to \alpha^\mathbb{N}_{\mathbb{Z}[p^{-1}]}
Remark and Definition

There is an endomorphism of the group of Witt vectors

T:{WW (a o,,a n,)(0,a o,,a n,)T:\begin{cases} W\to W \\ (a_o,\dots,a_n,\dots)\mapsto(0,a_o,\dots,a_n,\dots) \end{cases}

called translation.

We define the group W n of Witt vectors of length n by W n(R):=cokerT n(R) or equivalently by the exact sequence

(1)0WT nWR nW n00\to W\stackrel{T^n}{\to}W\stackrel{R_n}{\to}W_n\to 0

and we have

  1. (a 0,a 1,)=(a 0,,a n1,0,)+T n(a n,a n+1,)

  2. The underlying scheme of W n is O k n, the projection morphism W/toW n is (a 0,,)(a 0,,a n1.

  3. The group law on W n is (a 0,,a n1)+(b 0,,b n1)=(S 0(a 0,b 0),,S n1(a 0,,a n1,b 0,,b n1))

  4. In particular W 1=α

The snake lemma gives from diagram (1) translation morphisms T:W nW n+1 such that T(a 0,,a n1)=(0,a 0,,a n1), projection morphisms R Wn+1W n such that R(a 0,,a n)=(a 0,,a n1) and exact sequence

0W mT nW n+mR mW n00\to W_m\stackrel{T^n}{\to}W_{n+m}\stackrel{R^m}{\to}W_n\to 0

and the projections WW n give rise to an isomorphism

Wlim nW nW\simeq lim_n W_n
Definition and Lemma

Let ι:{O W a(a,0,) be an inclusion.

Then we have Φ nι(a)=a p n andE(ι(a),t)=F(at)).

Theorem and Definition

There is a unique ring-structure on the -group W such that either of the two following conditions is satisfied:

  1. Each Φ n:WO is a ring homomorphism.

  2. ι(ab)=ι(a)ι(b), a,bRM .

The -ring W is called Witt-Ring, each W n is a quotient ring of W. The canonical morphisms R n:WW n and R:W n+1W n are ring homomorphisms (but not T).

Revised on May 27, 2012 13:42:18 by Stephan Alexander Spahn (79.227.168.80)