Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

For each prime number $p$ the ring of $p$-adic integers $\mathbb{Z}_p$ is the formal completion of the ring $\mathbb{Z}$ at the prime ideal $(p)$. Geometrically this means that $\mathbb{Z}_p$ is the ring of functions on a formal neighbourhood of $p$ inside Spec(Z) (this is discussed in more detail below). Algebraically it means that the elements in $\mathbb{Z}_p$ look like formal power series where the formal variable is the prime number $p$.

## Definition

For any prime number $p$, the ring of $p$-adic integers $\mathbb{Z}_p$ (which, to avoid possible confusion with the ring $\mathbb{Z}/(p)$ used in modular arithmetic, is also written as $\widehat{\mathbb{Z}}_p$) may be described in one of several ways:

1. To the person on the street, it may be described as (the ring of) numbers written in base $p$, but allowing infinite expansions to the left. Thus, numbers of the form

$\sum_{n \geq 0} a_n p^n$

where $0 \leq a_n \lt p$, added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.

2. More abstractly, it is the limit $\underset{\leftarrow}{\lim} \mathbb{Z}/(p^n)$, in the category of (unital) rings, of the diagram

$\ldots \to \mathbb{Z}/(p^{n+1}) \to \mathbb{Z}/(p^n) \to \ldots \to \mathbb{Z}/(p) .$

This is also a limit in the category of topological rings, taking the rings in the diagram to have discrete topologies.

3. Alternatively, it is the metric completion of the ring of integers $\mathbb{Z}$ with respect to the $p$-adic absolute value. Since addition and multiplication of integers are uniformly continuous with respect to the $p$-adic absolute value, they extend uniquely to a uniformly continuous addition and multiplication on $\mathbb{Z}_p$. Thus $\mathbb{Z}_p$ is a topological ring.

4. Also $\mathbb{Z}[ [ x ] ]/(x-p)\mathbb{Z}[ [ x ] ]$, see at analytic completion.

Hence one also speaks of the $p$-adic completion of the integers. See completion of a ring (which generalizes 2&3).

There is also this characterization:

###### Lemma

There is a short exact sequence

$0 \to \mathbb{Z}_{p} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z}_{p} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.$
###### Proof

Consider the following commuting diagram

$\array{ \vdots && \vdots && \vdots \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^3\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^4 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^2\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^3 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^2 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ 0 &\longrightarrow& \mathbb{Z}/p\mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} } \,.$

Each horizontal sequence is exact. Taking the limit over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.

## Properties

### Topology

The ring of $p$-adic integers has the following properties:

### Relation to profinite completion of the integers

###### Example
$\widehat {\mathbb{Z}} \coloneqq \underset{\leftarrow}{\lim}_{n \in \mathbb{N}} (\mathbb{Z}/n\mathbb{Z}) \,.$

This is isomorphic to the product of the $p$-adic integers for all $p$

$\widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p \,.$
###### Definition

The ring of integral adeles $\mathbb{A}_{\mathbb{Z}}$ is the product of the profinite completion $\widehat{\mathbb{Z}}$ of the integers, example , with the real numbers

$\mathbb{A}_{\mathbb{Z}} \coloneqq \mathbb{R} \times \widehat{\mathbb{Z}} \,.$

The group of units of the ring of adeles is called the group of ideles.

### Pontrajgin duality to Prüfer $p$-group

Under Pontryagin duality, the abelian group underlying $\mathbb{Z}_p$ maps to the Prüfer p-group $\mathbb{Z}[p^{-1}]/\mathbb{Z}$, see at Pontryagin duality for torsion abelian groups.

$\array{ &\mathbb{Z}[p^{-1}]/\mathbb{Z} &\hookrightarrow& \mathbb{Q}/\mathbb{Z} &\hookrightarrow& \mathbb{R}/\mathbb{Z} \\ {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow \\ &\mathbb{Z}_p &\leftarrow& \hat \mathbb{Z} &\leftarrow& \mathbb{Z} }$

### As the formal neighbourhood of a prime

The formal spectrum $Spf(\mathbb{Z}_p)$ of $\mathbb{Z}_p$ may be understood as the formal neighbourhood of the point corresponding to the prime $p$ in the prime spectrum $Spec(\mathbb{Z})$ of the integers. The inclusion

$\{p\} \hookrightarrow Spf(\mathbb{Z}_p) \hookrightarrow Spec(\mathbb{Z})$

is the formal dual of the canonical projection maps $\mathbb{Z}\to \mathbb{Z}_p\to \mathbb{Z}/(p)$.

This plays a central role for instance in the function field analogy. It is highlighted for instance in (Hartl 06, 1.1, Buium 13, section 1.1.3). See also at arithmetic jet space and at ring of Witt vectors.

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Introductions and surveys include

• Dennis Sullivan, pp. 9 of Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springer (pdf)

• Bernard Le Stum, One century of $p$-adic geometry – From Hensel to Berkovich and beyond talk notes, June 2012 (pdf)

• Hendrik Lenstra, Profinite groups (pdf)

The synthetic differential geometry-aspect of the $p$-adic numbers is highlighted for instance in