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date: 2022-05-14 22:41
modification date: Saturday 14th May 2022 22:41:31
title: "almost purity"
aliases: [almost purity theorem, "Faltings' almost purity theorem"]
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# almost purity

Generally, *purity* means something happens in a particular codimension.
A foundational result for [perfectoid spaces](Unsorted/perfectoid%20MOC.md).
Reminiscent of Zariski–Nagata's purity theorem: the branch locus on a nonsingular algebraic variety, the branch locus (where the morphism ramifies) is a [Weil divisor](Unsorted/Weil%20divisor.md), i.e. is comprised purely of codimension 1 subvarieties.
This describes the locus on which a morphism can fail to be [etale](Unsorted/etale%20morphism.md).

![](attachments/Pasted%20image%2020220514233110.png)