---- > [!definition] Definition. ([[sheaf of ideals]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]], $\mathcal{F}$ a [[sheaf of modules|sheaf of]] $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. An **ideal sheaf** $\mathcal{I}$ in $\mathcal{O}_{X}$ is a [[subsheaf]] of [[abelian group|abelian groups]] of $\mathcal{O}_{X}$ such that for each $U \subset X$, $\mathcal{I}(U) \subset \mathcal{O}_{X}(U)$ is an [[ideal]]: $\mathcal{O}_{X}(U) \cdot \mathcal{I}(U) \subset \mathcal{I}(U)$. ^definition > [!basicexample] Example. (Kernel of a closed immersion is a sheaf of ideals) > Let $i:Z \to X$ be a [[subscheme|closed immersion]] of [[scheme|schemes]], so that $i$ is a [[homeomorphism]] of $Z$ onto a [[closed set|closed subset]] of $X$, and the map $i^{\sharp}:\mathcal{O}_{X} \to i_{*}\mathcal{O}_{Z}$ is [[surjection|surjective]]. We write $\mathcal{I}_{Z / X}=\operatorname{ker }i^{\sharp}$. > > $\mathcal{I}_{Z / X}$ is a [[sheaf of ideals]] of $\mathcal{O}_{X}$. > > To see this, first note that $i_{*}\mathcal{O}_{Z}$ naturally carries $\mathcal{O}_{X}$-module structure as follows: for $f \in \mathcal{O}_{X}(U)$, $r \in (i_{*}\mathcal{O}_{X})(U)=\mathcal{O}_{Z}\big( i ^{-1} (U) \big)$, define $f \cdot r := i^{\sharp}_{U}(f) \cdot r, \ \ (*)$ > where the latter multiplication occurs in the [[ring]] $\mathcal{O}_{Z}\big( i ^{-1} (U) \big)$. This makes $i^{\sharp}$ naturally into a [[morphism of sheaves of modules|morphism of]] $\mathcal{O}_{X}$-[[sheaf of modules|modules]].[^1] Indeed, $i^{\sharp}_{U}$ is clearly an $\mathcal{O}_{X}(U)$-[[linear map]], as for all $s \in \mathcal{O}_{X}(U)$ and $f \in \mathcal{O}_{X}(U)$ [[module|acting on]] on $s$: $\begin{align} > i^{\sharp}_{U}(f \cdot s)&= i^{\sharp}_{U}(f) \cdot i^{\sharp}_{U}(s) \text{ (}i^{\sharp} \text{ is a ring hom.)} \\ > &= f \cdot i _{U}^{\sharp}(s) \text{ (put } r:= i_{U}^{\sharp}(s) \text{ in } (*) \text{ )}. > \end{align}$ > As a consequence, if $i^{\sharp}_{U}(s)=0$, then $i_{U}^{\sharp}(f \cdot s)=f \cdot i_{U}^{\sharp}(s)=f\cdot 0=0$ for all $f \in \mathcal{O}_{X}(U)$, making $\operatorname{ker }i^{\sharp}$ stable under the action of $\mathcal{O}_{X}$. > (Or maybe I am going about this oddly) ---- #### [^1]: Viewing $\mathcal{O}_{X}$ as an $\mathcal{O}_{X}$-module over itself. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```