---- > [!definition] Definition. ([[generic point of an integral scheme]]) > Let $X$ be an [[integral scheme]]. There is a unique point $\eta \in X$ such that $\{ \eta \}$ is [[dense]] in $X$ (i.e., $\overline{\{ \eta \}}=X$); this is called the **generic point** of $X$. > > The [[(pre)sheaf stalk|stalk]] of $\mathcal{O}_{X}$ at $\eta$ is a [[field]], called the **function field of $X$** and denoted by $K(X)$. In particular, $K(X)=\text{Frac }A$, where $A$ is the [[ring]] corresponding to any choice $U=\text{Spec }A$ of [[affine scheme|open affine]] subset of $X$. ^definition > [!justification] > ^justification - [ ] Probably a better way to proceed when reviewing this: use that $\mathfrak{p} \in \text{Spec }A$, $\overline{\{ \mathfrak{p} \}}=\{ \mathfrak{q} \in \text{Spec }A: \mathfrak{q} \supset \mathfrak{p} \}=V(\mathfrak{p})$ ---- Let $\{ U_{i} \}=\{ \text{Spec }A_{i} \}$ be an [[cover]] of $X$ by [[affine scheme|open affines]]. Integrality of $X$ means each $\text{Spec }A_{i}$ is an [[integral domain]]; in particular, $(0_{A_{i}}) \in \text{Spec }A_{i}$. The claim is that each zero [[ideal]] $(0_{A_{i}})$ corresponds to the *same* point $\eta \in X$, that the [[closure]] $\overline{\{ \eta \}}=X$, and that $\eta$ is unique with respect to such a property. We will use the fact that (I think) there is a basis of $X$ consisting of distinguished affine opens $D(f_{i})$ in the affine pieces $\text{Spec }A_{i}$ of $X$. Note that $(0_{A_{i}}) \in D(f_{i})$ for all $f_{i} \neq 0$. **$\eta$ is well-defined.** We want to show, for all $i,j$, that $(0_{A_{i}})=(0_{A_{j}})$ in $X$. Wrap each in an open affine neighborhood $D(f_{i}) \ni (0_{A_{i}})$, $D(f_{j}) \ni (0_{A_{j}})$, then consider $D(f_{i}) \cap D(f_{j})$. Because $X$ is [[irreducible topological space|irreducible]] (it is [[integral scheme|integral]]), this set is nonempty, and therefore may be covered (per [[Nike's lemma]]) by nonempty distinguished open affines $U_{i}=D(g_{i})=D(g_{j})$ for $g_{i} \in A_{i} - \{ 0 \}$, $g_j \in A_{j} - \{ 0 \}$. In particular, $\underbrace{(0_{A_{i}})}_{\in D(g_{i})}=(\underbrace{0_{A_{j}}}_{ \in D(g_{j})})$. **$\overline{\{ \eta \}}=X$.** To see $\overline{\{ \eta \}}=X$, [[neighborhood-basis characterization of set closure|it is equivalent]] to show $B \cap \{ \eta \} \neq \emptyset$ for any nonempty basic open set $B$ in $X$, i.e., that $\eta \in B$ for every nonempty basic open set $B$ of $X$. Since for all $i$ and for all $0 \neq f_{i} \in A_{i}$, $D(f_{i})\ni (0_{A_{i}})=\eta$, this holds. **$\eta$ is unique.** Let $x \in X$, $x \neq \eta$. Identify $x$ with some $0 \neq \mathfrak{q} \in \text{Spec } A_{i}$. then for any $0 \neq f_{i} \in \mathfrak{q}$ one has $D(f_{i}) \not \ni \mathfrak{q}$ and thus there exists a basic open set $B$ satisfying $B \cap \{ x \}=\emptyset$. So $\overline{\{ x \}} \neq X$. **Re: the stalk of $\mathcal{O}_{X}$ at $\eta$ and the function field $K(X)$.** Letting $U=\text{Spec }A$ be any open affine subset of $X$, [[structure sheaf on a ring spectrum|we have]] $K(X)=\mathcal{O}_{X, \eta}=A_{\eta}=A_{(0_{A})}=\text{Frac }A,$ where we've recalled that [[localization|localizing]] an [[integral domain]] at the [[ideal|zero ideal]] recovers precisely the [[field of fractions]] definition. ---- #### ---- #### 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 > ```