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

No.10:

LEMMA 10.1   Let $A$ be a commutative ring. Then:

1. For any $a,b \in A$ , we have
   $$[a]\cdot [b]=[a b]$$

2. If $a\in A$ satisfies $a^q=a$ for some positive integer $q$ , then we have
   $$[a]^{\cdot q}=[a].$$

3. Let $q$ be a positive integer. If $b \in A$ satisfies
   $$\forall n\in \mathbb{Z}_{>0} \exists b_n\in A$$ such that $$b_n^{q^n}=b,$$
   then we have
   $$\forall n \in \mathbb{Z}_{>0} \exists c_n\in \mathcal W_1(A)$$ such that $$c_n^{q^n}=[b].$$

$\qedsymbol$

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
   $$\sum_{j=0}^\infty V_p^j ([x_j]) \qquad (x_j \in A).$$

2. For any $x,y \in A$ , we have
   $$V_p^n([x]) \cdot V_p^m ([y])=V_p^{n+m}([x^{p^m} y^{p^n}]).$$

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

$\qedsymbol$

COROLLARY 10.3   If $k$ is a field of characteristic $p\neq 0$ , 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

$$p^k \cdot x \qquad (k\in \mathbb{N}, x\in (\mathcal W^{(p)})^\cdot$$    (i.e. $x$:invertible)$$)$$

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 $p\neq 0$ . Then $\mathcal W^{(p)}(A)$ is an integral domain of characteristic 0 .

PROOF.. $\mathcal W^{(p)}(\iota)$ is always an injection when $\iota$ is. $\qedsymbol$

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