The ring of $p$-adic Witt vectors (when the characteristic of the base ring $A$ is $p$)

Before proceeding further, let me illustrate the idea. Proposition 8.9 tells us an existence of a set $\{e_n; n \in \mathbb{Z}_{>0}, p \nmid n \}$ of idempotents in $\mathcal W_1(A)$ such that its order structure is somewhat like the one found on the set $\{ n \mathbb{N}; n \in \mathbb{Z}_{>0}, p \nmid n\}$. Knowing that the idempotents correspond to decompositions of $\mathcal W_1(A)$, we may ask:

PROBLEM 8.15   What is the partition of ${\mathbb{Z}_{>0}}$ generated by the subsets $\{n \mathbb{N}; n \in \mathbb{Z}_{>0}\}$?

To answer this problem, it would probably be better to find out, for given positive number $n$ which is coprime to $p$, what the set

$\displaystyle S_{n;p}=n \mathbb{N}\setminus (\bigcup_{\substack{n\vert m\\ n<m \\ p \mid m }} m \mathbb{N})
$

should be. The answer is given by a fact which we know very well: every positive integer may uniquely be written as

% latex2html id marker 2368
$\displaystyle p^s k \quad ( s \in \mathbb{Z}_{\geq 0}, \quad k \in \mathbb{Z}_{>0}, \quad
\gcd(p,k)=1),
$

Knowing that, we see that the set $S_{n;p}$ as above is equal to

% latex2html id marker 2372
$\displaystyle \{ p^s n ; s\in \mathbb{Z}_{\geq 0} \}.
$

The answer to the problem is now given as follows:

% latex2html id marker 2374
$\displaystyle \mathbb{Z}_{>0} = \coprod_{p\nmid n}
\{ p^s n ; s\in \mathbb{Z}_{\geq 0} \}.
$

The same story applies to the ring $\mathcal W_1(A)$ of universal Witt vectors for a ring $A$ of characteristic $p$. We should have a direct product expansion

$\displaystyle \mathcal W_1(A)=\prod_{p \nmid n} e_{n;p} \mathcal W_1(A)
$

where the idempotent $e_{n;p}$ is defined by

$\displaystyle e_{n;p}= e_n - \bigvee_{\substack{n\vert m \\ n<m \\ p \nmid m }} e_m
$

Of course we need to consider infimum of infinite idempotents. We leave it to an exercise:

EXERCISE 8.1   Show that the supremum

$\displaystyle \bigvee_{\substack{n\vert m \\ n<m \\ p \nmid m }} e_m
=e_n-\prod_{\substack{n\vert m \\ n<m \\ p \nmid m }} (e_n-e_m)
$

exists. In other words, show that the right hand side converges.

PROPOSITION 8.16   Let $p$ be a prime. Let $A$ be an integral domain of characteristic $p$. Let us define an idempotent $f$ of $\mathcal W_1(A)$ as follows.

$\displaystyle f=
\bigvee
_{\substack{
n>1\\
p \nmid n
}}
e_n
(=[1]-
\prod_
{\substack
{p \nmid n\\
n>1
}}
([1]- e_n))
$

Then $f$ defines a direct product decomposition

$\displaystyle \mathcal W_1(A)
\cong
\left (
f \cdot \mathcal W_1(A)
\right )
\times
\left(
([1]- f)\cdot \mathcal W_1(A)
\right).
$

We call the factor algebra $([1] - f)\cdot \mathcal W_1(A)$ the ring $\mathcal W^{(p)}(A)$ of $p$-adic Witt vectors.

The following proposition tells us the importance of the ring of $p$-adic Witt vectors.

PROPOSITION 8.17   Let $p$ be a prime. Let $A$ be a commutative ring of characteristic $p$. For each positive integer $k$ which is not divisible by $p$, let us define an idempotent $f_k$ of $\mathcal W_1(A)$ as follows.

$\displaystyle f_k=\bigvee_{
\substack{p\nmid n \\ n >1}} e_{k n }
(=e_k - \prod
_{\substack{p\nmid n \\ n >1}}
(e_k - e_{k n}))
$

Then $f_k$ defines a direct product decomposition

$\displaystyle e_k\mathcal W_1(A)
\cong
\left (
f_k \cdot \mathcal W_1(A)
\right )
\times
\left(
(e_k- f_k) \cdot \mathcal W_1(A)
\right).
$

Furthermore, the factor algebra $(e_k- f_k)\cdot \mathcal W_1(A)$ is isomorphic to the ring $\mathcal W^{(p)}(A)$ of $p$-adic Witt vectors. Thus we have a direct product decomposition

$\displaystyle \mathcal W_1(A) \cong \mathcal W^{(p)}(A)^{\mathbb{N}}.
$