---- > [!definition] Definition. ([[reduced scheme]]) > A [[scheme]] $X$ is said to be **reduced** if for every open $U \subset X$, $\mathcal{O}_{X}(U)$ is [[nilpotent element of a ring|reduced]]: $\text{Nil }\mathcal{O}_{X}(U)=(0)$. ^definition > [!basicexample] > - Any [[integral scheme]] is reduced. >- If $X=\text{Spec }A$ is an [[affine scheme]], then $X=\text{Spec }A$ is reduced if and only if the [[nilradical of a ring|nilradical]] $\text{Nil }A=(0)$, that is, iff $A$ is [[nilpotent element of a ring|reduced]]. > > [!proof]- Proof. > > > > Will use properties in [[structure sheaf on a ring spectrum]] and [[Zariski topology on a ring spectrum]]. > > > > $\to$. If $X$ is reduced, then in particular $\mathcal{O}_{\text{Spec }A}(\text{Spec }A)=A$ has no (nonzero) [[nilpotent element of a ring|nilpotents]], i.e., $\text{Nil }A=(0)$, i.e., $A$ is reduced. > > > > $\leftarrow$. If $\text{Nil }A=(0)$, then $A=\mathcal{O}_{\text{Spec }A}(\text{Spec }A)$ has no nonzero nilpotents. If $U$ is any open subset of $X$, then $\mathcal{O}_{X}(U)$ consists of sections $s:U \to \coprod_{\mathfrak{p} \in U} A_{\mathfrak{p}}$. Because $A$ is reduced, $A_{\mathfrak{p}}$ is reduced for each $\mathfrak{p}$ ([[localization commutes with taking radicals|see the corollary here]]). The result follows. ---- #### ---- #### 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 > ```