The ring of $p$-adic Witt vectors for general $A$

In the preceding subsection we have described how the ring $\mathcal W_1(A)$ of universal Witt vectors decomposes into a countable direct sum of the ring of $p$-adic Witt vectors. In this subsection we show that the ring $W^{(p)}(A)$ can be defined for any ring $A$ (that means, without the assumption of $A$ being characteristic $p$).

We need some tools.

DEFINITION 9.14   Let $A$ be any commutative ring. Let $n$ be a positive integer. Let us define additive operators $V_n,F_n$ on $\mathcal W_1(A)$ by the following formula.

$$V_n((f(T))_W)=(f(T^n))_W.$$

$$F_n((f(T))_W)=(\prod_{\zeta\in \mu_n} f(\zeta T^{1/n}))_W$$

(The latter definition is a formal one. It certainly makes sense when $A$ is an algebra over $\mathbb{C}$. Then the definition descends to a formal law defined over $\mathbb{Z}$ so that $F_n$ is defined for any ring $A$. In other words, $F_n$ is actually defined to be the unique continuous additive map which satisfies

$$F_n((1-a T^l)_)= ((1-a ^{m/l} T^{m/n})^{l n/m})_W \qquad(m= \operatorname{lcm}(n,l)).$$

)

LEMMA 9.15   Let $p$ be a prime number. Let $A$ be a commutative ring of characteristic $p$. Then:

1. We have
   $$F_p(f(T))=(f(T^{1/p}))^{p} \qquad (\forall f\in \mathcal W_1(A)).$$
   in particular, $F_p$ is an algebra endomorphism of $\mathcal W_1(A)$ in this case.

2. $$V_p(F_p((f)_W)=F_p (V_p((f)_W))=(f(T)^p)_W=p \cdot (f(T))_W$$

DEFINITION 9.16   Let $A$ be any commutative ring. Let $p$ be a prime number. We denote by

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

and define

$$\pi_p: \mathcal W_1(A) \to \mathcal W^{(p)}(A)$$

by

$$\pi_p \left ( \sum _{j=1}^{\infty} (1-x_j T^j) \right ) = (x_1,x_p,x_{p^2},x_{p^3}\dots).$$

LEMMA 9.17   Let us define polynomials $\alpha_j(X,Y)\in \mathbb{Z}[X,Y]$ by the following relation.

$$(1-x T)(1-y T)=\prod_{j=1}^\infty (1-\alpha_j(x,y) T^j).$$

Then we have the following rule for “carry operation”:

$$(1-x T^n)_W + (1-y T^n)_W =\sum_{j=1}^{\infty} (1-\alpha_j(x,y)T^{j n}).$$

PROPOSITION 9.18   There exist unique binary operators $+$ and $\cdot$ on $\mathcal W^{(p)}(A)$ such that the following diagrams commute.

$$\begin{CD} \mathcal W_1 (A)\times \mathcal W_1 (A) @>+>> \mathcal... ...mathcal W^{(p)}(A)\times \mathcal W^{(p)}(A) @>+>> \mathcal W^{(p)}(A) \end{CD}$$

$$\begin{CD} \mathcal W_1 (A)\times \mathcal W_1 (A) @>\cdot >> \ma... ...al W^{(p)}(A)\times \mathcal W^{(p)}(A) @>\cdot >> \mathcal W^{(p)}(A) \end{CD}$$

PROOF.. Using the rule as in the previous lemma, we see that addition descends to an addition of $\mathcal W^{(p)}(A)$. It is easier to see that the multiplication also descends.

$\qedsymbol$

DEFINITION 9.19   For any commutative ring $A$, elements of $W^{(p)}(A)$ are called $p$-adic Witt vectors over $A$. The ring $(W^{(p)}(A),+,\cdot )$ is called the ring of $p$-adic Witt vectors over $A$.

LEMMA 9.20   Let $p$ be a prime number. Let $A$ be a ring of characteristic $p$. Then for any $n$ which is not divisible by $p$, the map

$$\frac{1}{n} \cdot V_n : \mathcal W_1(A) \to \mathcal W_1(A)$$

is a “non-unital ring homomorphism". Its image is equal to the range of the idempotent $e_n$. That means,

$$\operatorname{Image}(\frac{1}{n}\cdot V_n) =e_n \cdot \mathcal W_1(A) =\{\sum_j (1-y_j T^{n j})_W; y_j \in A\}.$$

PROOF.. $V_n$ is already shown to be additive. The following calculation shows that $\frac{1}{n} \cdot V_n$ preserves the multiplication: for any positive integer $a,b$ with lcm $m$ and for any element $x,y \in A$, we have:

$$(\frac{1}{n}\cdot V_n((1-x T^a)_W)) \cdot (\frac{1}{n}\cdot V_n((1-y T^b)_W))$$

$$= (\frac{1}{n}\cdot (1-x T^{a n})_W) \cdot (\frac{1}{n}\cdot (1-y T^{b n})_W)$$

$$= \frac{1}{n^2}\cdot\frac{an \cdot bn}{nm} \left((1-x^{m/a}y^{m/b} T^{nm} )^d\right)_W$$

$$= \frac{1}{n} \cdot V_n(((1-x T^a )_W\cdot (1-y T^b)_W)$$

We then notice that the image of the unit element $[1]$ of the Witt algebra is equal to $\frac{1}{n}V_n ([1] )= e_n$ and that $\frac{1}{n} V(e_n f)=e_n f$ for any $f \in \mathcal W_1(A)$. The rest is then obvious. $\qedsymbol$

In preparing from No.7 to No.10 of this lecture, the following reference (especially its appendix) has been useful:

http://www.math.upenn.edu/~chai/course_notes/cartier_12_2004.pdf