---
date: 2022-04-27 22:16
modification date: Wednesday 27th April 2022 22:16:22
title: "disambiguating completion and localization"
aliases: [disambiguating completion and localization]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [localization of rings](Unsorted/localization%20of%20rings.md)
	- [adic completion](Unsorted/adic%20completion.md)

---

# disambiguating completion and localization


**Inversion**: adjoin a formal inverse.
$$\ZZ\invert{p} = \ts{{a\over b}\in \QQ \st b=p^k}
.$$
**Localization** at $p$: invert all primes $\ell\neq p$. Geometrically: invert all functions which are no in the ideal $\gens{p}$, so function that do not vanish at $p$, by adjoining inverses to *all* elements in $\ZZ\sm\gens{p}$
$$\ZZ_p \da
\ZZplocal = 
\ZZ\plocalize{\gens{p}} = \ts{{a\over b}\in \QQ \st b\not\in\gens{p}} = \ts{x\in \QQ \st v_p(x)\geq 0} = \ZZ\adjoin{\ts{\ell\inv \st \ell\neq p }}
$$
These are functions that are defined **locally** on a neighborhood of the point $p$.


**Completion (adically)** at $p$: compatible systems of lifts:
$$
\ZZpcomplete = \lim\qty{ \cdots \from^{\mod p^3} \ZZ/p^2 \ZZ\from^{\mod p^2} \ZZ/p\ZZ  } \subseteq \prod_{n\geq 0} \ZZ/p^n \ZZ
$$
These are functions defined on a [formal neighborhood](formal%20neighborhood) of the point $p$, and the projections $\ZZ \to \ZZpadic \to \ZZ/p\ZZ$ correspond to $\ts{p} \injects \Spf \ZZpadic \injects \Spec \ZZ$.

**Completion (profinite)**: a limit over all finite quotients (normal subgroups with finite index) over the subgroup lattice:
$$
\ZZprof = \lim\ts{\ZZ/n\ZZ \to \ZZ/m\ZZ \st m\mid n } \cong \prod_{p} \ZZpadic = \AA_\ZZ^{<\infty} 
$$
where $\AA_Z = \RR \times \prod_p \ZZpadic$.

**Prufer groups**, which are the Pontrayagin duals of $\ZZpadic$:
$$\ZZ/p^\infty \da \colim(\ZZ/p \mapsvia p \ZZ/p^2 \mapsvia p \cdots) \cong \gens{\zeta_{p}, \zeta_{p^2}, \cdots} = \ZZ\invert{p}/\ZZ = \ts{(x_i) \st x_1^p = 1, x_2^p = x_1, x_3^p = x_2,\cdots}.$$


Ideal localization: $L_I \ZZ$ is localization at the monoid $\ZZ\sm I$ and allows inverting all functions that do not vanish on $V(I) \subseteq \spec \ZZ$.

Prime localization: $L_{p^c} \ZZ$ is localization at $\ts{1, p, p^2,\cdots}$ allows inverting all functions that vanish on $\gens{p} \in \spec \ZZ$, so $\spec L_{p^c} \ZZ = (\spec \ZZ) \sm \gens{p}$ is a punctured $\spec \ZZ$.

# Questions

- What does $\spec \ZZ\invert{p} \to \spec \ZZ$ represent?
- What does $\spec \ZZpadic \to \spec \ZZ$ represent?
- What does $\spec \ZZplocal \to \spec \ZZ$ represent?
- What does $\Spec \ZZprof \to \spec \ZZ$ represent?