----- > [!proposition] Proposition. ([[Untitled 542]]) > Let $A$ be a ([[commutative ring|commutative]]) [[ring]]. For $X=\text{Spec }A$ an [[affine scheme]], there is a [[bijection]] $\begin{align} > \{ \text{height-}1 \text{ prime ideals } \mathfrak{p} \subset A \} &\leftrightarrow \{ \text{prime divisors }Y \subset X \} \\ > \mathfrak{p} & \mapsto V(\mathfrak{p}) > \end{align}$ > > > [!proof]- Proof. ([[Untitled 542]]) > **Forward map.** prime First have to check we really have a map $\{ \text{height-}1 \text{ prime ideals of }A \} \xrightarrow{\varphi} \{ \text{prime divisors of }X \} $, i.e., that $V(\mathfrak{p})$ is a [[prime divisor in a scheme|prime divisor]] on $X=\text{Spec }A$ for any [[height of a prime ideal|height]]-$1$ [[prime ideal]] $\mathfrak{p} \subset A$. > > [[irreducible closed subspaces of Spec are precisely the vanishing of primes|It is always true that]] the [[subscheme|closed subscheme]] $V(\mathfrak{p})$ is [[irreducible topological space|irreducible]] and [[closed set|closed]] as a subset of $X$. It is [[integral scheme|integral]] because $A / \mathfrak{p}$ is an [[integral domain]] (as $\mathfrak{p}$ is prime). Finally, $V(\mathfrak{p})$ has [[codimension of a closed subspace|codimension]] $1$ because $\text{codim}\big( V(\mathfrak{p}), \text{Spec }A \big)=\text{ht }\mathfrak{p}=1$. So $V(\mathfrak{p})$ is indeed a [[prime divisor in a scheme|prime divisor]] on $X$. > > **Inverse map.** Next let $Y \subset X$ be a [[prime divisor in a scheme|prime divisor]] on $X$. Because $Y$ is [[closed set|closed]] and [[irreducible topological space|irreducible]] as a subset of $X$, [[irreducible closed subspaces of Spec are precisely the vanishing of primes|necessarily]] $Y=V(\mathfrak{p})$ for a unique [[prime ideal]] $\mathfrak{p} \in \text{Spec }A$. Hence $\underbrace{ \text{codim}(Y,X) }_{ =1 }=\text{codim}(V(\mathfrak{p}), \text{Spec }A)=\text{ht }\mathfrak{p}$. Thus $\text{ht }\mathfrak{p}=1$, giving us a [[well-defined]] map $\{ \text{height-}1 \text{ prime ideals } \text{of }A \} \xleftarrow{\psi} \{ \text{prime divisors of }X\} $ > defined by taking $Y \mapsto \mathfrak{p}$. > > **Showing mutual inverses.** This is immediate. ----- #### ---- #### 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 > ```