---
date: 2022-12-26 18:11
modification date: Monday 26th December 2022 18:11:05
title: "nonsingular"
aliases: [nonsingular, regular, smooth]
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Last modified date: NaN

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

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# nonsingular

- $x$ is nonsingular iff $\OO_{X, x}$ is a [regular ring](Unsorted/regular%20ring.md).
- Equivalently, for $I\da \mfm_{X, x}$, the $I\dash$adic completion $\widehat\OO_{X, x}$ is isomorphic to $\widehat\OO_{\AA^n, \vector 0} \cong k\fps{x_1,\cdots x_n}$.
- If $k$ is perfect and $X = V(f)$ is a hypersurface with $f$ squarefree, $x$ is singular iff $\nabla f(x) \da \tv{\dd{f}{x_1}(x), \cdots, \dd{f}{x_n}(x)} = \vector 0$.