---- Let $A$ be a ([[commutative ring|commutative]]) [[ring]]. > [!definition] Definition. ([[height of a prime ideal]]) > The **height** $\text{ht} \ \mathfrak{p}$ of a [[prime ideal]] $\mathfrak{p}$ of $A$ is the maximal length $d$ of a chain of [[prime ideal|prime ideals]] of the form $\underbrace{ \mathfrak{p}_{d} }_{ \mathfrak{p} } \supsetneq \mathfrak{p}_{d-1} \supsetneq \cdots \supsetneq \mathfrak{p}_{0}$ > (there are $d+1$ [[prime ideal|prime ideals]] in a chain of length $d$). > [[irreducible closed subspaces of Spec are precisely the vanishing of primes|Equivalently]], $\text{ht }\mathfrak{p}=\text{codim}(V(\mathfrak{p}), \text{Spec }A)$ where the [[codimension of a closed subspace|codimension]] is that of an [[irreducible closed subspaces of Spec are precisely the vanishing of primes|irreducible]] [[closed set|closed]] [[subspace topology|subspace]] of the [[topological space|topological]] [[Zariski topology on a ring spectrum|space]] $\text{Spec }A$. > [!generalization] > This notion is used to define the [[height of an ideal|height of an (arbitrary) ideal]]. ^generalization > [!basicproperties] > - $\text{ht }\mathfrak{p}=\text{dim }A_{\mathfrak{p}}$. > > > [!proof]- > > [[extension and contraction under localization|Using the bijection]] $\{ \mathfrak{q} \in \text{Spec }A : \mathfrak{q} \subset \mathfrak{p} \} \leftrightarrow \text{Spec }A_{\mathfrak{p}}$ > > Indeed, we know $\text{dim }A_{\mathfrak{p}}=\text{ht }\mathfrak{m}=\text{ht } \mathfrak{p}A_{\mathfrak{p}}$; a maximal chain $\mathfrak{p}A_{\mathfrak{p}} \supsetneq \dots$identifies under [[contraction of an ideal|contraction]] with a maximal chain $\mathfrak{p} \supsetneq \dots$ > > > > ^71f2b0 ^properties ---- #### ---- #### 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 > ```