next up previous
Next: About this document ...

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

Yoshifumi Tsuchimoto

No.03: \fbox{$\mathbb {Z}_p$\ as a projective limit of \{$\mathbb {Z}/p^k\mathbb {Z}$\}}

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 % latex2html id marker 656
$ i\leq k$ and % latex2html id marker 658
$ 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 % latex2html id marker 671
$ \lambda \geq \mu$ , a continuous homomorphisms

$\displaystyle \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 % latex2html id marker 681
$ \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 % latex2html id marker 706
$ \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 % latex2html id marker 720
$ \lambda \geq \mu$ .)
Then there exists a unique continuous homomorphism

$\displaystyle \Phi: Y\to X
$

such that

$\displaystyle \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

$\displaystyle \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  

$\displaystyle \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

% latex2html id marker 765
$\displaystyle [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

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

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


next up previous
Next: About this document ...
Yoshifumi Tsuchimoto 2016-05-10