---
tag: N3PGYJJQ7W
kind: concept
name: Ideal sheaf
msc: [14F06, 14A15]
specializes: [S33B2H3MYD]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_sheaf
same_as:
  wikidata: Q5988009
  wikipedia: Ideal sheaf
---

## Summary

An ideal sheaf on a ringed space $X$ is a subsheaf of the structure sheaf $\mathcal{O}_X$ whose sections over each open set form an ideal.
It is the sheaf version of an ideal of a ring.
On a scheme, quasi-coherent ideal sheaves correspond exactly to closed subschemes, the subscheme being cut out by the functions in the ideal.

## Explanations

### L1 human

An ideal of a ring of functions is a set of functions closed under addition and under multiplication by any function.
An ideal sheaf picks such an ideal on every open set, in a way compatible with restriction.
The typical example is the sheaf of functions that vanish on a fixed closed subset, such as a point or a curve.
The ideal sheaf then remembers that subset, and even multiplicities, such as functions vanishing to order two at a point.