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

No.07:

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 [1] 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 [2, 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 07.1   Let $A$ be a ring (of any characteristic). Let $T$ be an indeterminate. We define the following copies of $A^{\mathbb{Z}_{>0}}$ .

$$\mathcal W_0(A)= T A[[T]] = \left\{ \sum_{j=1}^\infty x^{(j)}T^j ; x^{(n)} \in A(\forall n) \right \}$$

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

LEMMA 07.2   $\mathcal W_0$ and $\mathcal W_1$ are functors from the category of rings to the category of sets. They are represented by ``polynomial rings in infinite indeterminates''

$$A_{\mathcal W_0} =\mathbb{Z}[X^{(1)},X^{(2)},X^{(3)},\dots]$$

and

$$A_{\mathcal W_1}= \mathbb{Z}[Y_1,Y_2,Y_3,\dots].$$

That means, there are functorial bijections

$$\operatorname{Hom}_{\operatorname{ring}}(A_{\mathcal W_0},A)\cong \mathcal W_0(A)$$

and

$$\operatorname{Hom}_{\operatorname{ring}}(A_{\mathcal W_1},A)\cong \mathcal W_1(A).$$

DEFINITION 07.3   We define the following ``universal elements''.

$$v_0= \sum_{j=1}^\infty X^{(j)} T^j \in \mathcal W_0(A_{\mathcal W_0}),$$

$$v_1= 1+\sum_{j=1}^\infty Y_j T^j \in \mathcal W_1(A_{\mathcal W_1}).$$

LEMMA 07.4   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 07.5   We equip $\mathcal W_0(A)=T A[[T]]$ with the usual addition and the following (unusual) 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 $\mathcal W_0(A)$ forms a (unital associative) commutative ring with these binary operations.

DEFINITION 07.6   Let $A$ be a ring which contains a copy of $\mathbb{Q}$ . Then we define ring structure on $\mathcal W_1(A)$ by putting

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

LEMMA 07.7   Let $A$ be a ring which contains a copy of $\mathbb{Q}$ . For any $f,g\in \mathcal W_1 (A)$ , we have

$$f+_{\mathcal L}g= f g.$$

In particular, addition $+_{\mathcal L}$ 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.