Tags: ? Refs: Some formulas Witt vector

p-adic integers

Defined as an inverse imits (not a colimit) \begin{align*} {\mathbb{Z}}_p := \varprojlim_{n} {\mathbb{Z}}/p^n \end{align*}

The \(p{\hbox{-}}\)adic integers are metric space completions of \({\mathbb{Z}}\) that are not \({\mathbb{Q}}\). Two numbers are “close” in the \(p{\hbox{-}}\)adic metric exactly when they differ by a large power of \(p\).

The isomorphism \(\mathbb{Z}_{p} \simeq \lim \mathbb{Z} / p^{n} \mathbb{Z}\) gives us a canonical way to represent elements of \(\mathbb{Z}_{p}\) : we can write \(a \in \mathbb{Z}_{p}\) as a sequence \(\left(a_{n}\right)\) with \(a_{n+1} \equiv a_{n} \bmod p^{n}\), where each \(a_n \in {\mathbb{Z}}/p^n{\mathbb{Z}}\) is uniquely represented by an integer in \([0, p^{n-1}]\).

Examples

Redundant representations in \({ {\mathbb{Z}}_p }\) for \(p=7\):
The more canonical \(p{\hbox{-}}\)adic expansions:

Lifts

It almost never happens that a variety in characteristic p can be lifted to characteristic zero together with its Frobenius endomorphism.

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regular ring

The p-adic integers \({ {\mathbb{Z}}_p }\).
mixed characteristic

Completion: \({ {\mathbb{Z}}_p }\) the p-adic integers with the ideal \(\left\langle{p}\right\rangle\) is \((0, p)\)
localization

p-adic integers
limit

Build new objects by “imposing equations” on existing ones. - Ex: Construction of the p-adic integers as the limit of the sequence of quotient homomorphisms: \(\cdots \rightarrow \mathbb{Z}/p^n \rightarrow \cdots \rightarrow \mathbb{Z}/p^2 \rightarrow \mathbb{Z}/p\)
arithmetic geometry

Using p-adic integers,
adic completion

\({\mathbb{Z}}{ {}_{ \widehat{\left\langle{p}\right\rangle} } } = {\mathbb{Z}}{ {}^{ \wedge }_{p} }\) is the p-adic integers.
absolute value

p-adic integers as an adic completion and a Cauchy completion