---
date: 2022-03-23 12:43
modification date: Wednesday 23rd March 2022 12:43:03
title: formally unramified
aliases: [formally unramified]

---

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


---

- Tags 
	- #AG 
- Refs:
	- #todo/add-references
- Links:
	- [scheme](Unsorted/scheme.md)
	- [formally smooth](formally%20smooth)
	- [formally etale](Unsorted/formally%20etale.md)

---

# formally unramified

Idea: meant to look like an [immersion](immersion).

Similar to [formally etale](Unsorted/formally%20etale.md), except *at most* one extension should exist:

\begin{tikzcd}
	{\in \Algs{R}} &&&&&& {\in \Sch\slice{S}} \\
	\textcolor{rgb,255:red,92;green,92;blue,214}{R} && {\forall B} && {\forall T} && \textcolor{rgb,255:red,92;green,92;blue,214}{X} \\
	\\
	\textcolor{rgb,255:red,92;green,92;blue,214}{A} && {\forall B/I,\quad I^2=0} && {T_1 {\scriptsyle \text{first order thickening}}} && \textcolor{rgb,255:red,92;green,92;blue,214}{S} \\
	{} &&&& {}
	\arrow[draw={rgb,255:red,92;green,92;blue,214}, from=2-1, to=4-1]
	\arrow["\forall", from=2-1, to=2-3]
	\arrow[""{name=0, anchor=center, inner sep=0}, two heads, from=2-3, to=4-3]
	\arrow["{\forall }", from=4-1, to=4-3]
	\arrow["{\exists \leq 1}"{description}, dashed, from=4-1, to=2-3]
	\arrow[from=4-5, to=4-7]
	\arrow[draw={rgb,255:red,92;green,92;blue,214}, from=2-7, to=4-7]
	\arrow[from=2-5, to=2-7]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{\forall }", from=2-5, to=4-5]
	\arrow["{\exists !}"{description}, dashed, from=4-5, to=2-7]
	\arrow[shorten <=26pt, shorten >=26pt, Rightarrow, squiggly, from=0, to=1]
\end{tikzcd}

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

TFAE:

- $f\in \Sch(X, S)$ is formally unramified
- $\Omega_{X/S} = 0$ (i.e. the [relative differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md) vanish).
- $\globsec{X; \OO_X}\to \globsec{Y; \OO_Y}$ is a formally unramified morphism of rings.