---
tag: EPM2HKGMZS
kind: concept
name: Complete intersection ring
msc: [13H10, 14M10, 13C40]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_intersection_ring
same_as:
  wikidata: Q5156503
  wikipedia: Complete intersection ring
---

## Summary

A complete intersection ring is a local ring that is, after completion, a [[regular local ring|63XXTF6B5M]] divided by an ideal generated by a [[regular sequence|69R5R2FCS5]].
It is the algebraic version of a space cut out by exactly as many equations as its codimension.
Every complete intersection ring is [[Gorenstein|T6HCX959PJ]], and hence [[Cohen–Macaulay|N1QJQEGBFC]].

## Explanations

### L1 human

In three-dimensional space, a surface is usually cut out by one equation and a curve by two.
When a space of codimension $r$ is cut out by exactly $r$ equations, it is a complete intersection.
The plane curve $xy = 0$, the union of two axes, is a complete intersection: one equation in the plane.
The three coordinate axes in space form a curve that needs three equations, $xy = yz = zx = 0$, so they are not a complete intersection.