----- > [!proposition] Proposition. ([[dimension of a finite-type algebra-domain over a field]]) > Let $k$ be a [[field]], and let $B$ be a $k$-[[algebra]] [[subalgebra generated by a subset|of finite type]] that is also an [[integral domain]]. Then for any [[prime ideal]] $\mathfrak{p} \subset B$: $\text{ht }\mathfrak{p} + \dim B / \mathfrak{p}=\dim B,$ where $\text{ht }\mathfrak{p}$ denotes the [[height of a prime ideal|height]] of $\mathfrak{p}$. ^proposition > [!proposition] Corollary. > Noting that > - $\text{ht }\mathfrak{p}= \text{codim}\big( V(\mathfrak{p}), \text{Spec } B \big)$ [^1] > - $\dim B / \mathfrak{p}=\dim V(\mathfrak{p})$ [^2] > > we have: $\text{codim}\big(V(\mathfrak{p)}, \text{Spec } B\big)+\dim V(\mathfrak{p})=\dim \text{Spec } B$ > as an equality of [[topological space|topological]] [[codimension of a closed subspace|(co)]][[dimension of a topological space|dimensions]]. ^proposition ----- #### [^1]: [[codimension of a closed subspace#^basic-example|See here.]] [^2]: Because $\dim B / \mathfrak{p}=\dim \text{Spec } B / \mathfrak{p}$ and $\text{Spec }B / \mathfrak{p}$ is [[homeomorphism|homeomorphic]] to $V(\mathfrak{p})$ per [[the correspondence theorem for rings]]. ---- #### 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 > ```