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

No.08:

In the following, we use infinite sums and infinite products of elements of $\mathcal W_1(A)=1+T A[[T]]$ . They are defined as limits of sums and products with respect to the filtration topology defined in the usual way.

LEMMA 08.1   Let $A$ be any commutative ring. Then every element of $1+T A[[T]]$ is written uniquely as

$$\prod_{j=1}^\infty (1-x_j T^j) \qquad(x_j \in A).$$

PROOF.. We may use an expansion

$$\prod_{j=1}^\infty (1-x_j T^j) \equiv -x_n T^n+poly(x_1,\dots,x_{n-1}, T) \pmod {T^{n+1}}$$

to inductively determine $x_j$ . More precisely, for each $n\in \mathbb{Z}_{>0}$ , let us define a polynomial $f_n(X_1,X_2,\dots,X_{n-1})$ in the following way:

$$f_n(X_1,\dots,X_{n-1})=\operatorname{coeff}(\prod_{j=1}^{n-1} (1-X_j T^j) , T^n)$$

Then for any element $1+\sum_{j=1}^\infty y_j T^j \in 1+T A[[T]]$ , we define

$$x_1=-y_1,\quad x_n=- y_n + f_n(x_1,\dots,x_{n-1}) \quad(\forall n>1).$$

Then it is easy to verify that an equation

$$1+\sum_{j=1}^\infty y_j T^j = \prod_{j=1}^\infty (1-x_j T^j)$$

holds. $\qedsymbol$

COROLLARY 08.2   ${\mathcal W}_1(A)=1+T A[[T]]$ is topologically generated by

$$\{(1-x_j T^j); \quad x_j \in A,\quad j=1,2,3,\dots \}.$$

LEMMA 08.3   Let $d,e$ be positive integers. Let $m$ be the least common multiple of $d,e$ . Then we have

$$(1-x_d T^d) *_{\mathcal L}(1-x_e T^e)= (1-x_d^{m/d} x_e^{m/e} T^m... ...(= (\frac{d e}{m}) \cdot_{\mathcal L}\left(1-x_d^{m/d} x_e^{m/e} T^m\right)) .$$

(Note that $m\geq d,e$ .)

PROOF.. let $d,e$ be positive integers. Let $m$ be the least common multiple of $d,e$ . We have,

$${\mathcal L}(1-x_d T^d)* {\mathcal L}(1-x_e T^e)= \frac{d x_d T^d... ..._e T^e} =d e (\sum_{i=1}^{\infty} (x_d T^d)^i *\sum_{j=1}^{\infty} (x_e T^e)^j)$$

$$= d e \sum_{u=1}^{\infty} x_d^{m u/d } x_e^{m u/e} T^{m u} = \frac{... ...}x_e^{m/e} T^m} =- \frac{d e}{m} \frac{d}{d T} \log(1- x_d^{m/d} x_e^{m/e} T^m)$$

$$= {\mathcal L}((1-x_d^{m/d} x_e^{m/e} T^m)^{d e/m}).$$

$\qedsymbol$

DEFINITION 08.4   Let $A$ be any commutative ring. Then we define an addition $\boxplus$ and a multiplication $\boxtimes$ on $\mathcal W_1(A)$ who satisfy the following requirements:

1. $f\boxplus g=fg$ .
2. For any positive integer $d,e$ , Let $m$ be the least common multiple of $d,e$ . Then we have
   $$(1-x_d T^d) \boxtimes (1-x_e T^e) = \left(1-x_d^{m/d} x_e^{m/e} T^m\right)^ {\frac{d e}{m}} .$$

3. for general $f,g$ , the multiplication $f\boxtimes g$ is defined by first expressing $f,g$ as a formal $\boxplus$ -sum as in Lemma 8.3 and then applying the rule 2 formally to each ``$\boxplus$ -summand''.

(Note that Lemma 8.1 guarantees the existence and the uniqueness of such multiplication $\boxtimes$ .)

THEOREM 08.5   Let $A$ be any commutative ring. Then:

1. Any element of $\mathcal W_1(A)$ is written uniquely as
   $$\sideset{}{^{\boxplus }}\sum _{j=1}^{\infty} (1-x_j T^j) .$$

2. $\mathcal W_1(A)$ forms a commutative ring under the binary operations $\boxplus$ and $\boxtimes$ . More precisely,
   1. $(\mathcal W_1(A),\boxplus )$ is an additive group with the zero element $1$ .
   2. The multiplication $\boxtimes$ is an associative commutative product on $\mathcal W_1(A)$ with the unit element $1-T$ .
   3. The distributive law holds.
3. When $A \supset \mathbb{Q}$ , the ring $(\mathcal W_1(A),\boxplus ,\boxtimes )$ is isomorphic to $(\mathcal W_0(A),+,*)$ via the map ${\mathcal L}_A=-T \frac{d}{d T}\log(\bullet)$ .

PROOF.. When $A \supset \mathbb{Q}$ , the statements trivially hold. This implies in particular that rules such as distributivity and associativity hold for universal cases (that means, for formal power series with indeterminate coefficients). Thus we conclude by specialzation arguments that the rule also hold for any ring $A$ .

$\qedsymbol$

DEFINITION 08.6   For any commutative ring $A$ , elements of $\mathcal W_1(A)$ are called universal Witt vectors over $A$ . The ring $(\mathcal W_1(A),\boxplus ,\boxtimes )$ is called the ring of universal Witt vectors over $A$ .

PROPOSITION 08.7   $(\mathcal W_1(\bullet),\boxplus,\boxtimes)$ is uniquely determined by the following properties.

1. $f \boxplus g =f g \qquad(\forall f,g \in \mathcal W_1(A)$ .
2. The multiplication $\boxtimes$ is $\boxplus$ -biadditive. (That means, $\mathcal W_1(A),\boxplus,\boxtimes)$ obeys the distributive law.)
3. $(1-x T) \boxtimes (1-y T)=(1-(xy)T) \qquad(\forall x,y \in A)$ .
4. $\boxtimes$ is continuous.
5. $\boxtimes$ is functorial.

PROOF.. We only need to prove the requirement (2) of Definition 8.4. With the help of distributive law, the requirement is satisfied if an equation

$$(1-x T^a)\boxtimes (1-y T^b) =(1-x^{m/a} y^{m/b} T^m)^{a b/m} \quad (m=l.c.m(a,b))$$ (#)

holds for each $(a,b)\in (\mathbb{Z}_{>0})^2$ .

To that aim, we first deal with a special case where $x=\alpha^a,y=\beta^b$ , $A=\mathbb{C}[\alpha,\beta]$ , $\alpha,\beta$ algebraically independent over $\mathbb{C}$ . In that case we may easily decompose the polynomials $(1-x T^a)$ and $(1-y T^b)$ and then we use the distributive law to see that the requirement actually holds. Indeed, let us put

$$\zeta_k=\exp(2 \pi \sqrt{-1}/k)$$

and compute as follows.

$$(1-x T^a)\boxtimes (1-y T^b)$$

$$= {\sideset{}{^\boxplus}{\sum}_{j,l}} (1-\zeta_a^j(\alpha)T) \boxtimes (1-\zeta_b^l(\beta)T)$$

$$= {\sideset{}{^\boxplus}{\sum}_{j,l}} (1-\zeta_a^j \zeta_b^l \alpha \beta T)$$

$$= \prod_l (1- \zeta_b^{a l} \alpha^a \beta^a T^a)$$

$$= \prod_{l'} (1-x \beta^a T^a \zeta_{b/d}^{l'})^d \qquad(d=g.c.d(a,b))$$

$$= (1-x^{a/d} y^{b/d} T^{ab/d} )^d.$$

We second deal with a case where $A=\mathbb{Z}[x,y]$ , $x,y$ algebraically independent over $\mathbb{C}$ . In that case we take a look at an inclusion

$$\iota:\mathbb{Z}[x,y ] \hookrightarrow \mathbb{C}[a,b].$$

and consider $\mathcal W_1(\iota)$ . It is easy to see that $\mathcal W_1(\iota)$ is injection so that the equation (#) is also true in this case. The general case now follows from specialization argument. $\qedsymbol$