----- > [!proposition] Proposition. ([[schemes are sober]]) > The [[topological space]] $X$ underlying any [[scheme]] $(X, \mathcal{O}_{X})$ is a [[sober topological space]]: for any [[irreducible topological space|irreducible]] [[closed set|closed subset]] $Z \subset X$, there exists a unique generic (i.e., [[dense]]) point $\eta \in Z$ [[closure|satisfying]] $\overline{\{ \eta \}}=Z$. ^proposition > [!note] Remark. > $\eta$ is 'generic' in the sense that it is maximal with respect to the [[the specialization preorder on a topological space|specialization preorder]], i.e., in the sense that it is contained in every open subset. ^note > [!proof]- Proof. ([[schemes are sober]]) > (stacks: https://stacks.math.columbia.edu/tag/01IS) > > **The affine case.** First assume $X=\text{Spec }A$ is [[affine scheme|affine]]. Let $Z \subset X$ be [[irreducible topological space|irreducible]] and [[closed set|closed]], [[irreducible closed subspaces of Spec are precisely the vanishing of primes|so]] $Z=V(\mathfrak{p})$ for some $\mathfrak{p} \in X$. Then the only closed subset of $Z$ containing $\mathfrak{p}$ is $Z$ itself, so trivially $\overline{\{ \mathfrak{p} \}}=Z$. As for uniqueness: If $\mathfrak{q} \in Z$ (i.e., $\mathfrak{q} \supset \mathfrak{p}$) also satisfied $\overline{\{ \mathfrak{q} \}}=Z$, then the only closed subset of $Z$ containing $\mathfrak{q}$ is $Z=V(\mathfrak{p})$ itself. Since $V(\mathfrak{q}) \subset V(\mathfrak{p})$ this means $V(\mathfrak{q})=V(\mathfrak{p})$. Recalling [[Zariski topology on a ring spectrum|the general fact that]] $V(I)=V(J) \iff \sqrt{ I }=\sqrt{ J }$ > and that $\mathfrak{p},\mathfrak{q}$ are both [[radical of an ideal|radical]], it follows that $\mathfrak{p}=\mathfrak{q}$. > > **The general case.** Now assume $X$ is any [[scheme]], say, with [[affine scheme|open affine]] [[cover]] $\{ U_{i} \}$, $U_{i} \cong \text{Spec }A_{i}$ via some [[morphism of locally ringed spaces|isomorphism]] $\tilde{\alpha}_{i}: \text{Spec }A_{i} \to U_{i}$. Let $Z \subset X$ be a [[closed set|closed subset]]. > > We have $Z=\bigcup_{i} Z \cap U_{i}.$ > Now, $Z \cap U_{i}$ is [[closed set|closed]] as a [[subspace topology|subspace]] of $U_{i}$. It is open as a [[subspace topology|subset]] of $Z$, and therefore [[irreducible topological space|irreducible]]. Since $U_{i}$ affine, we know there is a unique $\mathfrak{p}_{i} \in \text{Spec }A_{i}$ such that $\overline{\tilde{\alpha}_{i}(\mathfrak{p}_{i})}=Z \cap U_{i}$. > > The goal is to glue the 'local generic points' $\eta_{i}:=\tilde{\alpha}_{i}(\mathfrak{p}_{i})$ into a global generic point $\eta \in Z$. > > Agreement on overlaps (maybe this could be more precise): consider an overlap $(Z \cap U_{i}) \cap (Z \cap U_{j})=Z \cap(U_{i} \cap U_{j})$. This is a closed irreducible subset of $U_{i} \cap U_{j}$, and $U_{i} \cap U_{j}$ is sober as an open subset of a sober space, hence there is a unique element in the overlap whose closure equals $Z \cap (U_{i} \cap U_{j})$. The points $\eta_{i}=\tilde{a}_{i}(\mathfrak{p}_{i})$ and $\eta_{j}=\tilde{\alpha}_{j}(\mathfrak{p}_{j})$ are *both* candidates for such a point, so they coincide. > > > Now, define $\eta \in Z$ to be the point that corresponds to $\eta_{i}$ in any $U_{i}$ containing it (well-defined because of agreement on overlaps). Want to show $\overline{\{ \eta \}}=Z$. The containment $\subset$ is immediate. For $\supset:$ let $z \in Z$. Then $z \in Z \cap U_{i}=\overline{\{ \eta_{i} \}}=\overline{\{ \eta \}} \cap U_{i}$ for some $i$, hence $z \in \overline{\{ \eta \}}$. As for uniqueness: if $\xi \in Z$ also satisfies $\overline{\{ \xi \}}=Z$, then $\xi \in U_{i}$ for some $i$ and $\overline{\{ \xi \}}\cap U_{i}=Z \cap U_{i}=\overline{\{ \eta_{i} \}}$. By uniqueness in the affine case, $\xi=\eta_{i}=\eta$. ----- #### ---- #### 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 > ```