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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? EE 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\Phi_n(a_0,\cdots)=a_0^{p^n} + p a_1^{p^{n-1}}+\dots p^n a_n

The S iS_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)E((a_i),t)E((b_i),t)=E((S_i(a_0,\dots,a_i,b_0,\dots,b_i),t) where s i (p)[X 0,,X i,Y 0,,Y i]s_i\in \mathbb{Z}_{(p)}[X_0,\dots, X_i,Y_0,\dots, 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)\Phi_n(a_0,\cdots,a_n)+\Phi_n(d_0,\dots,b_n)=\Phi_n(S_0,\dots,S_n)

Lemma

We have S n[X 0,,X n]S_n\in\mathbb{Z}[X_0,\dots,X_n]

Definition and Theorem

There exists a unique commutative group law WW on the \mathbb{Z}-group scheme W:=O W:=O_\mathbb{Z}^\mathbb{N} called \mathbb{Z}-group of Witt-vectors of finite length relative to pp satisfying the following properties:

  1. E:O (p)Λ (p)E:O_\mathbb{Z}^\mathbb{N}\otimes_\mathbb{Z} \mathbb{Z}_{(p)}\to \Lambda_{\mathbb{Z}_{(p)}} is a morphism of kk-groups.

  2. Each Φ n:O α \Phi_n:O_\mathbb{Z}^\mathbb{N}\to \alpha_\mathbb{Z} is a morphism of kk-groups.

  3. (a n)+(b n)=(S n(a 0,,a n,b 0,,b n))(a_n)+(b_n)=(S_n(a_0,\dots, a_n,b_0,\dots,b_n))

For w=(a n)W(R)=R w=(a_n)\in W(R)=R^\mathbb{N}, a na_n is called the nn-th component of ww and Φ n(w)\Phi_n(w) is called the nn-th phantom component of ww. The phantom components of ww 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 nW_n of Witt vectors of length nn by W n(R):=cokerT n(R)W_n(R):=co ker\, T^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,)(a_0,a_1,\dots)=(a_0,\dots,a_{n-1},0,\dots)\overset{\bullet}{+} T^n(a_n,a_{n+1},\dots)

  2. The underlying scheme of W nW_n is O k nO_k^n, the projection morphism W/toW nW/to W_n is (a 0,,)(a 0,,a n1(a_0,\dots,)\mapsto(a_0,\dots,a_{n-1}.

  3. The group law on W nW_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))(a_0,\dots,a_{n-1})\overset{\bullet}{+}(b_0,\dots,b_{n-1})=(S_0(a_0,b_0),\dots,S_{n-1}(a_0,\dots,a_{n-1},b_0,\dots,b_{n-1}))

  4. In particular W 1=αW_1=\alpha

The snake lemma gives from diagram (1) translation morphisms T:W nW n+1T:W_n\to W_{n+1} such that T(a 0,,a n1)=(0,a 0,,a n1)T(a_0,\dots,a_{n-1})=(0,a_0,\dots,a_{n-1}), projection morphisms R Wn+1W nR_W_{n+1}\to W_n such that R(a 0,,a n)=(a 0,,a n1)R(a_0,\dots,a_n)=(a_0,\dots,a_{n-1}) 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 nW\to W_n give rise to an isomorphism

Wlim nW nW\simeq lim_n W_n
Definition and Lemma

Let ι:{O W a(a,0,)\iota:\begin{cases}O_\mathbb{Z}\to W\\a\to(a,0,\dots)\end{cases} be an inclusion.

Then we have Φ nι(a)=a p n\Phi_n \iota(a)=a^{p^n} andE(ι(a),t)=F(at)E(\iota(a),t)=F(at)).

Theorem and Definition

There is a unique ring-structure on the \mathbb{Z}-group WW such that either of the two following conditions is satisfied:

  1. Each Φ n:WO \Phi_n:W\to O_\mathbb{Z} is a ring homomorphism.

  2. ι(ab)=ι(a)ι(b)\iota(ab)=\iota(a)\iota(b), a,bRM a,b\in R\in M_\mathbb{Z}.

The \mathbb{Z}-ring WW is called Witt-Ring, each W nW_n is a quotient ring of WW. The canonical morphisms R n:WW nR_n:W\to W_n and R:W n+1W nR:W_{n+1}\to W_n are ring homomorphisms (but not TT).

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