---
tag: 95QTEPMDA4
kind: concept
name: System of parameters
msc: [13C15, 13H05]
sources:
  - url: https://en.wikipedia.org/wiki/System_of_parameters
same_as:
  wikidata: Q7663733
  wikipedia: System of parameters
---

## Summary

A system of parameters of a Noetherian [[local ring|MZ02BKPHJA]] of [[Krull dimension|MVGBS3J1P7]] $d$ is a list of $d$ elements of the maximal ideal $\mathfrak m$.
The ideal that they generate must be primary to $\mathfrak m$.
The [[dimension theorem for local rings|KW3HY77V6D]] says that such a list always exists and that no shorter list can do the same job.
Systems of parameters play the role of local coordinates, and they are the starting point of the theory of Cohen–Macaulay rings and multiplicities.

## Explanations

### L1 human

Near a point of a $d$-dimensional space, $d$ equations are needed to cut the space down to that single point.
A system of parameters is such a list of $d$ equations, read in the local ring of the point.
For the plane at the origin, the coordinates $x$ and $y$ work, and so do $x^2$ and $y^3$, since $x^2 = y^3 = 0$ still only allows the origin.
The [[definition|XSZ9C68SPG]] makes this precise.