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

No.10: \fbox{ The ring of $p$-adic Witt vectors revisited}

LEMMA 10.1   Let $ A$ be a commutative ring. Then:
  1. For any $ a,b \in A$ , we have

    $\displaystyle [a]\cdot [b]=[a b]
$

  2. If $ a\in A$ satisfies % latex2html id marker 614
$ a^q=a$ for some positive integer % latex2html id marker 616
$ q$ , then we have

    % latex2html id marker 618
$\displaystyle [a]^{\cdot q}=[a].
$

  3. Let % latex2html id marker 620
$ q$ be a positive integer. If $ b \in A$ satisfies

    $\displaystyle \forall n\in \mathbb{Z}_{>0} \exists b_n\in A$ such that % latex2html id marker 625
$\displaystyle b_n^{q^n}=b,
$

    then we have

    $\displaystyle \forall n \in \mathbb{Z}_{>0} \exists c_n\in \mathcal W_1(A)$ such that % latex2html id marker 628
$\displaystyle c_n^{q^n}=[b].
$

% latex2html id marker 713
$ \qedsymbol$

Recall that the ring of $ p$ -adic Witt vectors is a quotient of the ring of universal Witt vectors. We have therefore a projection $ \varpi:\mathcal W_1(A) \to \mathcal W^{(p)}(A)$ . But in the following we intentionally omit to write $ \varpi$ .

PROPOSITION 10.2   Let $ p$ be a prime number. Let $ A$ be a ring of characteristic. Then:
  1. Every element of $ \mathcal W^{(p)}(A) $ is written uniquely as

    % latex2html id marker 647
$\displaystyle \sum_{j=0}^\infty V_p^j ([x_j]) \qquad (x_j \in A).
$

  2. For any $ x,y \in A$ , we have

    $\displaystyle V_p^n([x]) \cdot V_p^m ([y])=V_p^{n+m}([x^{p^m} y^{p^n}]).
$

  3. A map

    $\displaystyle \varphi: \mathcal W^{(p)}(A) \ni
\sum_{j=0}^\infty V_p^n ([x_j])
\mapsto x_0 \in A
$

    is a ring homomorphism from $ (\mathcal W^{(p)},+,\cdot)$ to $ (A,+,\times)$ .
  4. $ \operatorname{Ker}(\varphi)=\operatorname{Image}(V_p)$ .
  5. An element $ x \in \mathcal W^{(p)}$ is invertible in $ \mathcal W^{(p)}$ if and only if $ \varphi(x)$ is invertible in $ A$ .

% latex2html id marker 714
$ \qedsymbol$

COROLLARY 10.3   If $ k$ is a field of characteristic % latex2html id marker 676
$ p\neq 0$ , then $ \mathcal W^{(p)}$ is a local ring with the residue field $ k$ . If furthermore the field $ k$ is perfect (that means, every element of $ k$ has a $ p$ -th root in $ k$ ), then every non-zero element of $ \mathcal W^{(p)}$ may be writen as

% latex2html id marker 692
$\displaystyle p^k \cdot x \qquad (k\in \mathbb{N}, x\in (\mathcal W^{(p)})^\cdot$    (i.e. $x$:invertible)$\displaystyle )
$

Since any integral domain can be embedded into a perfect field, we deduce the following

COROLLARY 10.4   Let $ A$ be an integral domain of characteristic % latex2html id marker 702
$ p\neq 0$ . Then $ \mathcal W^{(p)}(A) $ is an integral domain of characteristic 0 .

PROOF.. $ \mathcal W^{(p)}(\iota)$ is always an injection when $ \iota$ is. % latex2html id marker 707
$ \qedsymbol$

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Yoshifumi Tsuchimoto 2016-06-18