---
date: 2022-03-22 21:25
modification date: Tuesday 22nd March 2022 21:25:56
title: formally etale
aliases: [formally etale]

---

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


---

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

---

# formally etale

Idea: meant to look like a local isomorphism by inducing a pointwise isomorphism on tangent spaces.

Idea: $X\to S$ is etale when for all $T\to X$, a first-order thickening $T_1 \to S$ lifts uniquely to $T_1\to X$.

\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[color={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 !}"{description}, dashed, from=4-1, to=2-3]
	\arrow[from=4-5, to=4-7]
	\arrow[color={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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)

Equivalent characterizations:

- The [relative differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md) $\Omega_{X/S} = 0$ (the [relative differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md) vanish) and the [conormal sheaf](conormal%20sheaf) vanishes, $\mathcal{C}_{X/S} = 0$.
- $\globsec{X; \OO_X}\to \globsec{Y; \OO_Y}$ is a formally etale morphism of rings.

Trivially implies [formally unramified](formally%20unramified.md).
If $f$ is additionally [locally of finite presentation](locally%20of%20finite%20presentation), then $f$ is an [etale morphism](Unsorted/etale.md).