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

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

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

$$p^s k \quad ( s \in \mathbb{Z}_{\geq 0}, \quad k \in \mathbb{Z}_{>0}, \quad \gcd(p,k)=1),$$

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

$$\{ p^s n ; s\in \mathbb{Z}_{\geq 0} \}.$$

The answer to the problem is now given as follows:

$$\mathbb{Z}_{>0} = \coprod_{p\nmid n} \{ p^s n ; s\in \mathbb{Z}_{\geq 0} \}.$$

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

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

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

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

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

EXERCISE 8.1   Show that the supremum

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

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.

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

Then $f$ defines a direct product decomposition

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

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.

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

Then $f_k$ defines a direct product decomposition

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

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

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