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The Artin-Hasse exponential series? $E$ can be written as
with $\Phi_n(a_0,\cdots)=a_0^{p^n} + p a_1^{p^{n-1}}+\dots p^n a_n$
The $S_i$ in the multiplication formula $E((a_i),t)E((b_i),t)=E((S_i(a_0,\dots,a_i,b_0,\dots,b_i),t)$ where $s_i\in \mathbb{Z}_{(p)}[X_0,\dots, X_i,Y_0,\dots, Y_i]$ of the Artin-Hasse exponential series can also be written as $\Phi_n(a_0,\cdots,a_n)+\Phi_n(d_0,\dots,b_n)=\Phi_n(S_0,\dots,S_n)$
We have $S_n\in\mathbb{Z}[X_0,\dots,X_n]$
There exists a unique commutative group law $W$ on the $\mathbb{Z}$-group scheme $W:=O_\mathbb{Z}^\mathbb{N}$ called $\mathbb{Z}$-group of Witt-vectors of finite length relative to $p$ satisfying the following properties:
$E:O_\mathbb{Z}^\mathbb{N}\otimes_\mathbb{Z} \mathbb{Z}_{(p)}\to \Lambda_{\mathbb{Z}_{(p)}}$ is a morphism of $k$-groups.
Each $\Phi_n:O_\mathbb{Z}^\mathbb{N}\to \alpha_\mathbb{Z}$ is a morphism of $k$-groups.
$(a_n)+(b_n)=(S_n(a_0,\dots, a_n,b_0,\dots,b_n))$
For $w=(a_n)\in W(R)=R^\mathbb{N}$, $a_n$ is called the $n$-th component of $w$ and $\Phi_n(w)$ is called the $n$-th phantom component of $w$. The phantom components of $w$ define a group isomorphism
There is an endomorphism of the group of Witt vectors
called translation.
We define the group $W_n$ of Witt vectors of length $n$ by $W_n(R):=co ker\, T^n(R)$ or equivalently by the exact sequence
and we have
$(a_0,a_1,\dots)=(a_0,\dots,a_{n-1},0,\dots)\overset{\bullet}{+} T^n(a_n,a_{n+1},\dots)$
The underlying scheme of $W_n$ is $O_k^n$, the projection morphism $W/to W_n$ is $(a_0,\dots,)\mapsto(a_0,\dots,a_{n-1}$.
The group law on $W_n$ is $(a_0,\dots,a_{n-1})\overset{\bullet}{+}(b_0,\dots,b_{n-1})=(S_0(a_0,b_0),\dots,S_{n-1}(a_0,\dots,a_{n-1},b_0,\dots,b_{n-1}))$
In particular $W_1=\alpha$
The snake lemma gives from diagram (1) translation morphisms $T:W_n\to W_{n+1}$ such that $T(a_0,\dots,a_{n-1})=(0,a_0,\dots,a_{n-1})$, projection morphisms $R_W_{n+1}\to W_n$ such that $R(a_0,\dots,a_n)=(a_0,\dots,a_{n-1})$ and exact sequence
and the projections $W\to W_n$ give rise to an isomorphism
Let $\iota:\begin{cases}O_\mathbb{Z}\to W\\a\to(a,0,\dots)\end{cases}$ be an inclusion.
Then we have $\Phi_n \iota(a)=a^{p^n}$ and$E(\iota(a),t)=F(at)$).
There is a unique ring-structure on the $\mathbb{Z}$-group $W$ such that either of the two following conditions is satisfied:
Each $\Phi_n:W\to O_\mathbb{Z}$ is a ring homomorphism.
$\iota(ab)=\iota(a)\iota(b)$, $a,b\in R\in M_\mathbb{Z}$.
The $\mathbb{Z}$-ring $W$ is called Witt-Ring, each $W_n$ is a quotient ring of $W$. The canonical morphisms $R_n:W\to W_n$ and $R:W_{n+1}\to W_n$ are ring homomorphisms (but not $T$).