---
tag: JX20MNDDXT
kind: concept
name: Multiplicity of a local ring
msc: [13H15, 13D40]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert–Samuel_function
---

## Summary

The multiplicity of a Noetherian [[local ring|MZ02BKPHJA]] $(R, \mathfrak m)$ of dimension $d$ is the positive integer $e(R)$ such that the [[Hilbert–Samuel function|BNQVS83GZV]] $\ell(R/\mathfrak m^n)$ grows like $e(R) n^d / d!$.
It measures how singular the corresponding point is.
A regular local ring has multiplicity $1$, and a plane curve singularity has multiplicity equal to the lowest degree of a term in its equation.

## Explanations

### L1 human

At a smooth point a curve looks like a single line, and at a node it looks like two lines crossing.
The multiplicity counts how many sheets come together at the point, in an algebraic way.
For the cusp $y^2 = x^3$ the lowest degree term is $y^2$, of degree $2$, and the multiplicity is $2$.
The [[definition|B7CV5DZPRK]] makes this precise.