---
tag: Z1CGB483J6
kind: concept
name: Cotangent space of a local ring
msc: [13H99, 14B05]
sources:
  - url: https://en.wikipedia.org/wiki/Zariski_tangent_space
---

## Summary

The cotangent space of a [[local ring|MZ02BKPHJA]] $(R, \mathfrak m, k)$ is the [[vector space|000AYPA9AJ]] $\mathfrak m / \mathfrak m^2$ over the [[residue field|NJ3APQ8BHH]] $k$.
Its dual is the Zariski tangent space at the point that $R$ describes.
Its [[dimension|03GSZ8RTZJ]] is the [[embedding dimension|DP3KSCRKFP]] of $R$, which is used to define [[regular local rings|63XXTF6B5M]], that is, smooth points.

## Explanations

### L1 human

Near a point of a curve or a surface, the elements of $\mathfrak m$ are the functions that vanish at the point.
Two such functions have the same image in $\mathfrak m / \mathfrak m^2$ when they agree to first order, so the quotient records only their linear parts.
For the plane at the origin, the linear parts are spanned by $x$ and $y$, so the cotangent space has dimension $2$.
The [[definition|TCRZV021XR]] gives the construction.