---
date: 2022-03-14 09:58
modification date: Monday 14th March 2022 09:58:30
title: valuative criterion of separatedness
aliases: [valuative criterion of separatedness]

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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# valuative criterion of separatedness

A functor of points POV: when $f:X\to Y$ is quasi-separated, every rational point extends in at most one way to an integral point.

\begin{tikzcd}
	{\spec \ff(R)} && X && {X_Y(R)} \\
	\\
	{\spec R} && Y && {X_Y(\ff(R))}
	\arrow[from=3-1, to=3-3]
	\arrow[""{name=0, anchor=center, inner sep=0}, from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["{\exists \leq 1}", dashed, from=3-1, to=1-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=3-5, to=1-5]
	\arrow[shorten <=15pt, shorten >=15pt, Rightarrow, 2tail reversed, from=0, to=1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJcXHNwZWMgXFxmZihSKSJdLFswLDIsIlxcc3BlYyBSIl0sWzIsMCwiWCJdLFsyLDIsIlkiXSxbNCwwLCJYX1koUikiXSxbNCwyLCJYX1koXFxmZihSKSkiXSxbMSwzXSxbMiwzXSxbMCwyXSxbMCwxXSxbMSwyLCJcXGV4aXN0cyBcXGxlcSAxIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzUsNCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNywxMSwiIiwyLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfSwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn19fV1d)