----- > [!proposition] Proposition. ([[lying over]]) > Let $A \subset_{\iota} B$ be an [[integral algebra|integral extension]] of [[ring|rings]]. Let $\mathfrak{p} \in \text{Spec }A$. Then there exists $\mathfrak{q} \in \text{Spec }B$ such that $\mathfrak{q} \cap A=\mathfrak{p}$.[^1] > > $\mathfrak{q}$ is 'lying over' $\mathfrak{p}$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRRkAjFVqMWbAELdeIDNjwEiI8uPrNWiEAB1dMAB5Y4OOAAJ9AWzo4AFgDMATnQDWwAI5dLurGB84RjjAAMpoMADG5t6yPHxKgqqkYtSaUjrWto4u7mje+n4BQaHhUd6cXOIwUADm8ESgzhBWSGQgOBBIau10WAxsdhAQrow4IKmS2nq+EDh04yAMdABGMAwACvzKQiBOWDV2Y3EgTS2IAEzUHUgAzBNabAWzdAB6AFTcFFxAA > \begin{tikzcd} > A \arrow[d, "\iota"', hook'] & \exists \mathfrak{q} \in \text{Spec } B \arrow[d, "\iota^*"] \\ > B & \mathfrak{p} \in \text{Spec } A > \end{tikzcd} > \end{document} > ``` > > [!proof]- Proof. ([[lying over]]) > Summary: >- Draw the commutative diagram below. >- Let $\mathfrak{m}$ be any maximal ideal of $B_{\mathfrak{p}}$. Contract it to $\mathfrak{p} \subset A$. >- By commutativity, contracting $\mathfrak{m}$ along $\text{Spec }B_{\mathfrak{p}} \to \text{Spec } B \to \text{Spec } A$ also gives $\mathfrak{p}$. Result follows. > > Recall that the notation $B_{\mathfrak{p}}=(A-\mathfrak{p})^{-1}B$ is, strictly speaking, not a [[localization]] of $B$ at a [[prime ideal]] ($\mathfrak{p}$ is likely not a [[prime ideal]] of $B$). > > Observe that the following square commutes: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQBGclVqMWbAELdeIDNjwEiZYePrNWiDgH0AOvoC2dHAAsAZgCc6Aa2BoucvksFFR66pqk7pB46aWNvaO3OIwUADm8ESg1hBGSGQgOBBIoil0WAxsZhAQtiBektoghvg4dM4g8YmIyalIAEw8cVYJ6dSNiADMrTXtdU1dab1cFFxAA > \begin{tikzcd} > A \arrow[r, "\iota", hook] \arrow[d] & B \arrow[d] \\ > A_\mathfrak{p} \arrow[r] & B_\mathfrak{p} > \end{tikzcd} > \end{document} > ``` > > Take a [[maximal ideal]] $\mathfrak{m} \in \text{mSpec }B_{\mathfrak{p}}$. [[integral algebra|Now]], $A_{\mathfrak{p}} \subset B_{\mathfrak{p}}$ is an [[integral algebra|integral extension]]. Since [[contraction of an ideal|contractions]] under [[integral algebra|integral extensions]] preserve [[maximal ideal|maximality]] ([[integral extensions, units, and fields#^corollary|per this corollary]]), $\mathfrak{m} \cap A_{\mathfrak{p}}$ is [[maximal ideal|maximal]]. But $A_{\mathfrak{p}}$ is a [[local ring]]; in particular, it has unique maximal ideal[^2] $\mathfrak{p}A_{\mathfrak{p}}$. Hence $\mathfrak{m} \cap A_{\mathfrak{p}}=\mathfrak{p}A_\mathfrak{p}$. > > In light of the [[bijection]] given by [[extension and contraction under localization|extension and contraction under the localization]] $A \to A_{\mathfrak{p}}$ (or really just by definition) > $\{ \mathfrak{q} \in \text{Spec }A: \mathfrak{q} \subset \mathfrak{p} \} \leftrightarrow \text{Spec } A_{\mathfrak{p}},$ > we see that $\mathfrak{p}A_{\mathfrak{p}} \in \text{Spec }A_{\mathfrak{p}}$ contracts under localization to $\mathfrak{p} \in \text{Spec } A$. We have just shown that $\mathfrak{m} \subset B_{\mathfrak{p}}$ contracts to $\mathfrak{p}$ about the $\downarrow_ \to$ part of the diagram. Since the diagram commutes, it also contracts to $\mathfrak{p}$ about the $\to_\downarrow$ part. In particular, if we denote by $\mathfrak{q}$ the contraction $\mathfrak{m}^{c}$ of $\mathfrak{m}$ under the localization map $B \to B_{\mathfrak{p}}$, then $\iota^{*}(\mathfrak{q})=\mathfrak{q} \cap A=\mathfrak{p}$. ----- #### [^1]: That is, all fibers of the [[contraction of an ideal|contraction map]] $\iota^{*}=\text{Spec }\iota:\text{Spec } B \to \text{Spec } A$ are nonempty, i.e., $\iota^{*}$ is a [[surjection]]. [^2]: Recall that $\mathfrak{p}A_{\mathfrak{p}}$ denotes the [[extension of an ideal|extension]] of $\mathfrak{p}$ under the [[localization|localization map]] $A \to A_{\mathfrak{p}}$, $\mathfrak{p}A_{p}=\mathfrak{p}^{e}=\left\{ \frac{p}{f}: p \in \mathfrak{p}, f \notin \mathfrak{p} \right\}$. ---- #### 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 > ```