---
tag: 22M7FHJ3N8
kind: definition
name: Ideal sheaf
msc: [14F06, 14A15]
defines: N3PGYJJQ7W
uses: [KNDXDE7K8M, S33B2H3MYD, 096Z9MP2ME]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_sheaf
---

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].
An *ideal sheaf* on $X$ is a sub-$\mathcal{O}_X$-module $\mathcal{I} \subseteq \mathcal{O}_X$ in the sense of [[sheaves of modules|S33B2H3MYD]].
Equivalently, $\mathcal{I}$ is a subsheaf of $\mathcal{O}_X$ such that $\mathcal{I}(U)$ is an [[ideal|096Z9MP2ME]] of $\mathcal{O}_X(U)$ for every open set $U \subseteq X$.

## Remarks

On $X = \operatorname{Spec} A$, an ideal $I \subseteq A$ gives the ideal sheaf $\widetilde{I}$, with $\widetilde{I}(D(f)) = I_f \subseteq A_f$.
It is a sub-$\mathcal{O}_X$-module of $\widetilde{A} = \mathcal{O}_X$, since localization preserves ideals.
On $X = \mathbb{A}^1_k = \operatorname{Spec} k[t]$, the subsheaf of $\mathcal{O}_X$ of constant functions is not an ideal sheaf.
Its sections over $X$ form $k \subseteq k[t]$, which is not an ideal, since $t \cdot 1 = t$ is not constant.
For a closed subscheme $Z \subseteq X$ of a scheme, the kernel of $\mathcal{O}_X \to \mathcal{O}_Z$ is a quasi-coherent ideal sheaf $\mathcal{I}_Z$.
This gives a bijection between closed subschemes of $X$ and quasi-coherent ideal sheaves on $X$.