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

Yoshifumi Tsuchimoto

No.03:

DEFINITION 03.1   An ordered set $\Lambda$ is said to be directed if for all $i,j\in \Lambda$ there exists $k\in \Lambda$ such that $i\leq k$ and $j\leq k$ .

DEFINITION 03.2   Let $\Lambda$ be a directed set. Let $\{X_\lambda\}_{\lambda \in \Lambda}$ be a family of topological rings. Assume we are given for each pair of elements $(\lambda,\mu) \in \Lambda^2$ such that $\lambda \geq \mu$ , a continuous homomorphisms

$$\phi_{\mu\lambda} :X_\lambda \to X_\mu.$$

We say that such a system $(\{X_\lambda\}, \{\phi_{\mu\lambda}\})$ is a projective system of topological rings if it satisfies the following axioms.

1. $\phi_{\nu\mu}\phi_{\mu\lambda}=\phi_{\nu\lambda}$     ( $\forall \lambda,\forall \mu \forall \nu$ such that $\lambda \geq \mu \geq \nu$ ).
2. $\phi_{\lambda \lambda}=\operatorname{id}$     ( $\forall \lambda \in \Lambda$ ).

DEFINITION 03.3   Let $\mathcal X=(\{X_\lambda\}, \{\phi_{\mu\lambda}\})$ be a projective system of topological rings. Then we say that a projective limit $(X,\{\phi_\lambda\})$ of $\mathcal X$ is given if

1. $X$ is a topological ring.
2. $\phi_\lambda : X \to X_\lambda$ is a continuous homomorphism.
3. $\phi_{\mu\lambda }\circ\phi_\lambda =\phi_\mu$ for $\forall \mu,\lambda$ such that $\lambda \geq \mu$ .)
4. $(X,\{\phi_\lambda\})$ is a universal object among objects which satisfy (1)-(3).

The ``universal'' here means the following: If $(Y,\psi_\lambda)$ satisfies

1. $Y$ is a topological ring.
2. $\psi_\lambda : Y \to X_\lambda$ is a continuous homomorphism.
3. $\phi_{\mu\lambda }\circ\psi_\lambda =\psi_\mu$ for $\forall \mu,\lambda$ such that $\lambda \geq \mu$ .)

Then there exists a unique continuous homomorphism

$$\Phi: Y\to X$$

such that

$$\psi_\lambda=\phi_\lambda \circ \Phi (\forall \lambda \in \Lambda).$$

PROPOSITION 03.4   For any projective system of topological rings, a projective limit of the system exists. It is unique up to a unique isomorphism. (Hence we may call it the projective limit of the system.)

DEFINITION 03.5   For any projective system $(X,\{\phi_\lambda\})$ of topological rings, We denote the projective limit of it by

$$\varprojlim_\lambda X_\lambda.$$

Note: projective limits of systems of topological spaces, rings, groups, modules, and so on, are defined in a similar manner.

THEOREM 03.6  

$$\mathbb{Z}_p \cong \varprojlim_{k\to \infty} (\mathbb{Z}/p^k \mathbb{Z})$$

as a topological ring.

COROLLARY 03.7   $\mathbb{Z}_p$ is a compact space.

Note: There are several ways to prove the result of the above corollary. For example, the ring $\mathbb{Z}$ with the metric $d_p$ is easily shown to be totally bounded.

PROPOSITION 03.8   Each element of $\mathbb{Z}_p$ is expressed uniquely as

$$[0.a_1a_2a_3a_4\dots]_p \qquad(a_i\in \{0,1,\dots,p-1\} \quad(i=1,2,3,\dots)).$$

EXERCISE 03.1   Is $-4=1-5$ invertible in $\mathbb{Z}_5$ ? (Hint: use formal expansion

$$(1-x)^{-1}=1+x+x^2+\dots$$

is it possible to write down a correct proof to see that the result is true?)