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$ \mathbb{Z}_p$ , $ \mathbb{Q}_p$ , and the ring of Witt vectors

No.12: \fbox{The ring of Witt vectors and $\mathbb {Z}_p$}

DEFINITION 12.1   Let $ A$ be a ring of characteristic $ p$ . We call the ring

$\displaystyle \mathcal W^{(p)}_n(A)=
\mathcal W^{(p)}(A)/V_p^n (\mathcal W^{(p)}(A))

the ring of Witt vectors of length $ n$ . Its elements are called Witt vectors of lenth $ n$ .

Note that

$\displaystyle \mathcal W^{(p)}(A)/V_p^n (\mathcal W^{(p)}(A))

may be considered as a set $ A^n$ with an unusual ring structure.


$\displaystyle \mathbb{Z}_p \cong \mathcal W^{(p)}(\mathbb{F}_p).

PROOF.. Since $ \mathcal W^{(p)}$ is a unital commutative ring, there naturally exists a natural ring homomorphism

$\displaystyle \iota:\mathbb{Z}\to \mathcal W^{(p)}(\mathbb{F}_p).

Let us first fix a positive integer $ n$ and examine the kernel $ K_n$ of a map

$\displaystyle \pi_n\circ \iota: \mathbb{Z}
\mathcal W^{(p)}(\mathbb{F}_p)/V_p^n (\mathcal W^{(p)}(\mathbb{F}_p))

where $ \pi_n$ is the natural projection. Since

$\displaystyle \char93 \mathcal W^{(p)}(\mathbb{F}_p)/V_p^n (\mathcal W^{(p)}(\mathbb{F}_p))
=\char93 (\mathbb{F}_p^n)=p^n,

we have

$\displaystyle \char93  (\mathbb{Z}/K_n) \vert p^n.

In other words, $ K_n=p^{s} \mathbb{Z}$ for some integer $ s$ . On the other hand, we have

$\displaystyle \pi_n \circ\iota(p^s)=(1-T)^{p^s}=(1-T^{p^s})


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$\displaystyle \pi_n \circ\iota(p^s)\in K_n \iff s\geq n.

This implies that $ K_n=p^n \mathbb{Z}$ and therefore we have an inclusion

$\displaystyle \mathbb{Z}/p^n \mathbb{Z}\hookrightarrow W^{(p)}_n(\mathbb{F}_p)

which turns to be a bijection ( $ \because \char93 (\mathbb{Z}/p^n \mathbb{Z})=\char93 (\mathcal W^{(p)}_n(\mathbb{F}_p))$ ).

We then take a projective limit of the both hand sides and obtain the resired isomorphism.

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$ \qedsymbol$

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Yoshifumi Tsuchimoto 2016-06-18