---
tag: 69R5R2FCS5
kind: concept
name: Regular sequence
msc: [13D02, 13C15]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_sequence
same_as:
  wikidata: Q7309617
  wikipedia: Regular sequence
---

## Summary

A regular sequence is a list of ring elements in which each element is a non-zero-divisor modulo the earlier ones.
It is the algebraic version of a set of independent equations, like the coordinates $x_1, \ldots, x_n$ of a polynomial ring.
Regular sequences define the [[depth|M2RADBFW99]] of a module, and the [[Koszul complex|YZY11W08SD]] of a regular sequence is a free resolution.

## Explanations

### L1 human

Cut a space by one equation, then cut the result by a second equation, and so on.
The sequence of equations is regular when each new equation genuinely cuts down what is left, and never vanishes on a whole piece of it.
In $K[x, y]$ the sequence $x, y$ is regular: $x$ is a non-zero-divisor in $K[x,y]$, and $y$ is a non-zero-divisor in $K[x,y]/(x) \cong K[y]$.
The sequence $x, xy$ is not regular, because $xy$ is already zero modulo $x$.