---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: mixed characteristic
aliases: [mixed characteristic]

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


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- Tags 
	- #arithmetic-geometry/p-adic-hodge-theory 
- Refs:
	- #todo/add-references
- Links:
	- [perfectoid MOC](Unsorted/perfectoid%20MOC.md)
	- [prismatic cohomology](Unsorted/prismatic%20cohomology.md) is a cohomology theory for mixed characteristic rings.
	- [prism](Unsorted/prismatic%20cohomology.md)

---

# mixed characteristic

![](attachments/Pasted%20image%2020220515001030.png)
![](attachments/Pasted%20image%2020220515001053.png)

For rings: $R$ is of mixed characteristic $(0, p)$ when $\characteristic R = 0$ but for some $I\normal R$, $\characteristic R/I = p>0$.

For fields $K$ equipped with a [valuation](Unsorted/Valuations.md): the [valuation ring](Unsorted/Valuations.md) $R$ is local with maximal ideal $\mfm$ inducing a [residue field](residue%20field), so $K$ is mixed characteristic when $\characteristic K = 0$ but $\characteristic R/\mfm = p$,

$K$ of mixed characteristic $(0,p)$  means that $K$ has characteristic 0, but its residue field $\kappa$ has characteristic $p$)

Example: schemes defined over $\ZZpadic$?

# Motivations

## Why non-Noetherian rings?

In mixed characteristic algebraic geometry, the basic geometric objects are smoot [p-adic formal schemes](p-adic%20formal%20schemes.md) over the [ring of integers](ring%20of%20integers.md) $\mathcal{O}_{K}$ of a complete algebraically closed nonarchimedean field $K / \QQpadic$.These rings are often non-Noetherian, e.g. the [value group](value%20group) of $\OO_K$ is a [divisible group](divisible%20group). Replacing $\OO_K$ with a [DVR](Unsorted/DVR.md) like $\ZZpadic$ is not ideal.
Applications of these non-Noetherian rings: [perfectoid MOC](Unsorted/perfectoid%20MOC.md) geometry, [descent](Unsorted/descent.md) for fine topologies( [pro-etale](pro-etale), [quasi-syntomic](quasi-syntomic), [v topology](v%20topology), [arc topology](arc%20topology)), and the theory of [delta rings](delta%20rings).

## Why derived/higher geometry?

Given a pair $(R, I)$ where $I\normal R$ is a finitely generated ideal, the subcategory of $I\dash$adically complete $R\dash$modules is not abelian, while the category of *derived* $I\dash$adically complete modules is abelian.

The functor $R\mapsto \derivedcat{\rmod}\complete{I}$ into the derived category of $I\dash$complete $R\dash$complexes is a [stacks MOC](Unsorted/stacks%20MOC.md) for the [flat topology](flat%20topology), which does not hold at the level of [triangulated categories](Unsorted/triangulated%20categories.md).

# Examples

- $\ZZ$ and the ideal $\gens{p}$ is $(0, p)$.
- $\OO_K$ the [ring of integers](Unsorted/ring%20of%20integers.md) of any $K\in \NF$.
- Localization: $L_{\gens p} \ZZ$ the [p-local integers](Unsorted/localization%20of%20rings.md) with the ideal $\gens{p}$ is $(0, p)$
- Completion: $\ZZpadic$ the [p-adic](Unsorted/p-adic.md) with the ideal $\gens{p}$ is $(0, p)$