---- > [!definition] Definition. ([[irreducible scheme]]) > A [[scheme]] $X$ is said to be **irreducible** if it is [[irreducible topological space|irreducible as a]] [[topological space]]. ^definition > [!basicexample] > If $X=\text{Spec }A$ is an [[affine scheme]], then $X$ is irreducible if and only if the [[nilradical of a ring|nilradical]] $\text{Nil }A$ is [[prime ideal|prime]]. > > > > [!proof]- Proof. > > We will use the equivalence in [[irreducible topological space]] and the properties in [[Zariski topology on a ring spectrum]]. > > > > $\to$. Suppose $X$ is irreducible. Let $fg \in \text{Nil }A$. Then $D(fg)=\emptyset$. But $D(fg)=D(f) \cap D(g)$, and since $X$ is irreducible this intersection cannot be empty if $D(f) \neq \emptyset$ and $D(g) \neq \emptyset$. Hence $D(f) = \emptyset$ (implying $f \in \text{Nil }A$) or $D(g) = \emptyset$ (implying $g \in \text{Nil }A$). Thus $\text{Nil }A$ is [[prime ideal|prime]]. > > > > $\leftarrow$. Suppose $\text{Nil }A$ is [[prime ideal|prime]]. Let $U \ni \mathfrak{p}$ and $V \ni \mathfrak{q}$ be arbitrary nonempty open subsets of $\text{Spec }A$. Using that the $D(f)$ form a [[basis for a topology|basis]] for the topology on $\text{Spec }A$, obtain $f,g \in A$ such that $\mathfrak{p} \in D(f) \subset U$ and $\mathfrak{q} \in D(g) \subset V$. *Claim: $D(f) \cap D(g) \neq \emptyset$.* Indeed, if $D(f) \cap D(g)=\emptyset$ then $D(fg)=\emptyset$ and hence $fg \in \text{Nil }A$. But since $\text{Nil }A$ is [[prime ideal|prime]], this would mean $f \in \text{Nil }A$ or $g \in \text{Nil }A$ and in turn contradict the fact that $D(f)$ and $D(g)$ are nonempty. > > ---- #### ---- #### 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 > ```