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

No.08:

From here on, we make use of several notions of category theory. Readers who are unfamiliar with the subject is advised to see a book such as [#!MacLane2!#] for basic definitions and first properties.

Let $p$ be a prime number. For any commutative ring $k$ of characteristic $p\neq 0$, we want to construct a ring $W(k)$ of characteristic 0 in such a way that:

1. $W(\mathbb{F}_p)=\mathbb{Z}_p$.
2. $W(\bullet)$ is a functor. That means,
   1. For any ring homomorphism $\varphi: k_1\to k_2$ between rings of characterisic $p$, there is given a unique ring homomorphism $W(\varphi):W(k_1)\to W(k_2)$.
   2. $W(\bullet)$ should furthermore commutes with compositions of homomorphisms.

To construct $W(k)$, we construct a new addition and multiplication on a $k$-module $\prod_{j=1}^\infty k$. The ring $W(k)$ will then be called the ring of Witt vectors. The treatment here essentially follows the treatment which appears in [#!Lang1!#, VI,Ex.46-49], with a slight modification (which may or may not be good-it may even be wrong) by the author.

We first introduce a nice idea of Witt.

DEFINITION 08.1   Let $A$ be a ring (of any characteristic). Let $T$ be an indeterminate. We define the following copy of $A^{\mathbb{Z}_{>0}}$.

$$\mathcal W_1(A)= 1+ TA[[T]] = \left\{ 1+\sum_{j=1}^\infty y_j T^j \ ;\ x_n \in A(\forall n) \right\}$$

(as a set.)

For each element $a(T) \in 1+T A[[T]]$, we will denote by $(a(T))_W$ the corresponding element in $W_1(A)$.

We will equip $\mathcal W_1 (A)$ with a ring structure. To do so we first make use of “log”. 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 08.2   There is an well-defined map

$${\mathcal L}_A=-T\frac{d}{dT}\log(\bullet): 1+T A[[T]] \to T A[[T]].$$

If $A$ contains an copy of $\mathbb{Q}$, then the map is a bijection. The inverse is given by

$$T g(T) \mapsto \exp \left ( -\int_0^T g(s)d s \right).$$

PROOF.. To see that ${\mathcal L}$ is well defined (that is, “defined over $\mathbb{Z}$”), we compute as follows.

$$-T \frac{d}{d T}\log(1+T f_1) =-T (f_1'+f_1)(1+T f_1)^{-1} =-T (f_1'+f_1)\sum_{j=1}^\infty(-T f_1)^{j}$$

The rest should be obvious.

Note: the condition $A\supset \mathbb{Q}$ is required to guarantee exictence of exponential

$$\exp(\bullet)=\sum_{j=0}^\infty \frac{1}{j!} \bullet^j$$

and existence of the integration $\int_0^T g(s)d s$. $\qedsymbol$

DEFINITION 08.3   We equip $T A[[T]]$ with the usual addition and the following (unusual) “coefficient-wise" multiplication:

$$\left( \sum_{j=1}^\infty a^{(j)} T^j \right) * \left( \sum_{j=1}^\infty b^{(j)} T^j \right) =\sum_{j=1}^\infty (a^{(j)} b^{(j)}) T^j$$

It is easy to see that $T A[[T]]$ forms a (unital associative) commutative ring with these binary operations.

DEFINITION 08.4   Let $A$ be a ring which contains a copy of $\mathbb{Q}$. Then we define ring structure on $\mathcal W_1 (A)$ by putting

$$(f)_W + (g)_W = {\mathcal L}_A^{-1}(... ... (f)_W \cdot (g)_W = {\mathcal L}_A^{-1}({\mathcal L}_A(f)*{\mathcal L}_A(g)).$$

LEMMA 08.5   Let $A$ be a ring which contains a copy of $\mathbb{Q}$. For any $f,g\in \mathcal W_1 (A)$, we have

$$(f)_W + (g)_W = (f g)_W .$$

In particular, addition in $\mathcal W_1 (A)$ is defined over $\mathbb{Z}$.

PROOF.. easy $\qedsymbol$

We may thus extend the definition $+_{\mathcal L}$ on $\mathcal W_1 (A)$ to cases where the condition $A\supset \mathbb{Q}$ is no longer satisfied.

We next see that the multiplication of $\mathcal W_1 (A)$ is also defined over $\mathbb{Z}$. To do so, we need the following lemma.

LEMMA 08.6   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 08.7   ${\mathcal W}_1(A)=1+T A[[T]]$ is topologically generated by

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

LEMMA 08.8   Let $d,e$ be positive integers. Let $m$ be the least common multiple of $d,e$. Then for any $x,y \in A$, we have

$$(1-x T^d)_W \cdot (1-y T^e)_W = \left( (1-x^{m/d} y^{m/e} T^m)^{d e/m} \right)_W$$

$$= (\frac{d e}{m}) \cdot \left(1-x^{m/d} y^{m/e} T^m\right)_W .$$

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

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

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

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

$\qedsymbol$

DEFINITION 08.9   Let $A$ be any commutative ring. Then we define an addition $+$ and a multiplication $\cdot$ on $\mathcal W_1 (A)$ who satisfy the following requirements:

1. $(f)_W + (g)_W=(fg)_W$.
2. For any positive integer $d,e$, Let $m$ be the least common multiple of $d,e$. Then for any $x,y \in A$, we have
   $$(1-x T^d)_W \cdot (1-y T^e)_W = \left( \left(1-x^{m/d} y^{m/e} T^m \right)^ {\frac{d e}{m}} \right)_W .$$

3. the summation and the multiplication operations are continuous.

(Note that Lemma 8.6 guarantees the existence and the uniqueness of such multiplication.)

THEOREM 08.10   Let $A$ be any commutative ring. Then:

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

2. $\mathcal W_1 (A)$ is a commutative ring.
3. When $A\supset \mathbb{Q}$, the ring $\mathcal W_1 (A)$ is isomorphic to the ring $(T A[[T]],+,*)$ 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 08.11   For any commutative ring $A$, elements of $\mathcal W_1 (A)$ are called universal Witt vectors over $A$. The ring $\mathcal W_1 (A)$ is called the ring of universal Witt vectors over $A$.

PROPOSITION 08.12   $\mathcal W_1(\bullet)$ is uniquely determined by the following properties.

1. $(f)_W+ (g)_W =(f g)_W \qquad(\forall f,g \in 1+T A[[T]])$.
2. $(1-x T)_W (1-y T)_W=(1-(xy)T)_W \qquad(\forall x,y \in A)$.
3. The multiplication map is continuous.
4. The multiplication map is functorial.

PROOF.. We only need to prove the requirement (2) of Definition 8.9. With the help of distributive law, the requirement is satisfied if an equation

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

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.

$$(1-x T^a)_W (1-y T^b)_W$$

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

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

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

$$= \left( \prod_{l'} (1-x \beta^a T^a \zeta_{b/d}^{l'})^d \right)_W \qquad (d=\operatorname{gcd}(a,b))$$

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

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

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

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. $\qedsymbol$