----- Recall our default assumption in this chapter: Assume $(X, \mathcal{O}_{X})$ is a [[locally Noetherian scheme|Noetherian]], [[integral scheme|integral]] [[scheme]] [[scheme over a field|over]] a [[field]] $k$ which is [[regular scheme|regular]] in [[codimension of a closed subspace|codimension]] $1$. > [!proposition] Proposition. ([[almost all prime divisors vanish on a given function]]) > For all $f \in K(X)^{*}$, $V_{Y}(f)=0$ for all but finitely many [[prime divisor in a scheme|prime divisors]] $Y$. ^proposition > [!proof]- Proof. ([[almost all prime divisors vanish on a given function]]) > Let $f \in K(X)^{*}$, where $K(X)^{*}$ represents [[unit|the nonzero]] elements of the [[generic point of an integral scheme|function field]] $K(X)$ of $X$. > > First note that we can find an [[affine scheme|open affine]] $U=\text{Spec }A$ where $f$ is a regular function: $f \in \Gamma(U, \mathcal{O}_{X})$.[^1] $Z:= X- U$ is a proper [[closed set|closed subset]] of a [[Noetherian topological space]], hence [[decomposing Noetherian topological spaces|decomposes]] into a finite union of [[irreducible topological space|irreducible]] [[closed set|closed subsets]] of $X$. Each has [[codimension of a closed subspace|codimension]] of at least $1$ (else would equal $X$). Thus, $Z$ contains only finitely many [[irreducible topological space|irreducible]] [[codimension of a closed subspace|codimension]]-1 [[closed set|closed components]], and hence $Z$ contains only finitely many [[prime divisor in a scheme|prime divisors]]. > > We can split the prime divisors $Y$ of $X$ into two categories: > - Those $Y$ in $Z=X-U$ (finitely many) > - The rest, i.e., those $Y$ satisfying $Y \cap U \neq \emptyset$. > > > This means it is enough to show the result (that $\nu_{Y}(f)=0$ for all but finitely many $Y$) under the assumption $Y \subset U$. Indeed, only finitely many prime divisors live in $Z$, and every prime divisor $Y$ not living in $Z$ intersects $U$ nontrivially and thus has [[generic point of an integral scheme|generic point]] $\eta_{Y}$ in $U$.[^2] > > So assume $X=U=\text{Spec } A$ is [[affine scheme|affine]] and $f \in \Gamma (\text{Spec }A, \mathcal{O}_{\text{Spec } A})=A$. > > > Observe that $\nu_{Y}(f) \geq 0$ for any [[prime divisor in a scheme|prime divisor]] $Y \subset U$, since $f \in \Gamma(U, \mathcal{O}_{X})$ [[(pre)sheaf stalk|induces a germ]] $[U \ni \eta_{Y}, f] \in \mathcal{O}_{X, \eta_{Y}}=\mathcal{O}_{U, \eta_{Y}}$ and the latter [[subring]] (of $K(X)$) is a [[DVR]] . Note that at this point we are in a way viewing $f$ from three perspectives: > 1. $f$ as a nonzero element of the [[generic point of an integral scheme|function field]] $K(X)$ of the [[integral scheme]] $X$ ; > 2. $f$ as a section of $\mathcal{O}_{X}$ over $U$, $f \in \Gamma(U, \mathcal{O}_{X})$, for some affine $U=\text{Spec }A$. Equivalently, $f$ as an element of $A$; > 3. $f$ as an element of the stalk $\mathcal{O}_{X, \eta_{Y}}$. > > [[every DVR is a Noetherian local domain of dimension 1|Also]] $\nu_{Y}(f) >0$ iff $\frac{f}{1} \in \mathfrak{m}_{\mathcal{O}_{X}, \eta_{Y}}=\mathfrak{p}A_{\mathfrak{p}}$, where $\mathfrak{p}$ denotes the [[prime ideal]] of $A$ identified with $\eta_{Y} \in U$. This happens in turn iff $f \in \mathfrak{p}$, equivalently $\mathfrak{p} \in V(\langle f \rangle)$. Now, $\mathfrak{p} \in V(\langle f \rangle)$ iff $Y \subset V(\langle f \rangle)$, since $Y$ is identified with $\overline{\{ \mathfrak{p} \}}$ and $V(\langle f \rangle)$ is closed. Put together: $\nu_{Y}(f) > 0 \iff Y \subset V(\langle f \rangle ).$ > $V(\langle f \rangle)$ is a proper closed subset, so the same reasoning as before says it only finitely many prime divisors $Y$. ----- #### [^1]: For instance: [[generic point of an integral scheme|recall]] that $K(X)$ is the [[field of fractions]] $\text{Frac }B$ for $\text{Spec }B$ any [[affine scheme|open affine piece]] of $X$. Take $f \in \text{Frac }B$; write $f=\frac{b}{s}$ for $b, s \in B$, $s \neq 0$. Then $f \in B_{s} \cong \mathcal{O}_{\text{Spec } B}\big( D(s) \big) \cong \Gamma(U, \mathcal{O}_X)$ for $U\cong D(s) \cong \text{Spec } \underbrace{ B_{s} }_{ := A }$ open affine. [^2]: This follows from some [[topological space|point-set topology]]: suppose $\eta \notin U$. Then $\overline{\{ \eta \}} \cap U = \emptyset$. But $\overline{\{ \eta \}}=Y$ and we've assumed $Y \cap U \neq \emptyset$. Hence $\eta \in U$. ---- #### 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 > ```