----- > [!proposition] Proposition. ([[codimension is well-behaved for irreducible closed subspaces of varieties over a field]]) > Let $k$ be a [[field]]. Let $X$ be a [[variety]] ([[integral scheme|integral]] + [[scheme morphism locally of finite type|finite type]] [[scheme]]) [[scheme over a field|over]] $k$. > > Then [[dimension of a topological space|dimension]] and [[codimension of a closed subspace|codimension]] interact nicely: for any [[irreducible topological space|irreducible]] [[subscheme|closed]] subset $Z$ of $X$, $\dim Z + \text{codim}(Z, X)=\text{dim }X.$ > [!proof]- Proof. ([[codimension is well-behaved for irreducible closed subspaces of varieties over a field]]) > $X$ can be covered with [[affine scheme|open affines]] $U_{i}=\text{Spec }A_{i}$ such that each $A_{i}$ is a [[subalgebra generated by a subset|finitely generated]] $k$-[[algebra|algebras]] and an [[integral domain]]. Now the corollary in [[dimension of a finite-type algebra-domain over a field]] says > $\text{dim }V(\mathfrak{p}) + \text{codim}\big( V(\mathfrak{p}), \text{Spec }A_{i} \big) =\text{dim }U_{i}$ > for any $\mathfrak{p}\in \text{Spec }A_{i}$. > > Now let $Z$ be any irreducible closed subset of $X$. In what follows we employ properties/examples in [[irreducible topological space]] and [[dimension of a topological space]] liberally. > > For a given $i$, > - $Z \cap U_{i}$ is an *open subset of $Z$,* and $Z$ is irreducible, hence $\text{dim }Z \cap U_{i}=\text{dim }Z$; > - $Z \cap U_{i}$ is a *closed irreducible subset of $U_{i}$,* hence of the form $V(\mathfrak{p})$ for some $\mathfrak{p} \in \text{Spec }A_{i}$. > Conclude: $\text{dim }Z=\text{dim }V(\mathfrak{p})$ for some [[prime ideal]] $\mathfrak{p} \subset A_{i}$. > > > [[dimension of a topological space|Recall]] that taking closures preserves chain maximality. We know that a maximal chain determining the [[codimension of a closed subspace|codimension]] of $\underbrace{ Z \cap U_{i} }_{ V(\mathfrak{p}) } \subset U_{i}$ takes the form $V(\mathfrak{p}) \subsetneq \dots \subsetneq \text{Spec } A_{i}$ > (note $\text{Spec }A_{i}$ is irreducible as an open subset of the irreducible space $X$, and $\overline{U_{i}}=X$). Now, $V(\mathfrak{p})=Z \cap U_{i}$ is open in the irreducible closed set $Z$, hence its [[closure]] in $Z$ equals $Z$ itself. Now, it is generally true for topological spaces that if $Z \subset X$ is [[closed set|closed in]] $X$ and $A \subset Z$ is dense in $Z$ (i.e., the closure of $A$ in $Z$ equals $Z$), then the closure of $A$ in $X$ equals $Z$. Hence $\overline{V(\mathfrak{p})}=Z$ in $X$. So taking closures we get a maximal chain $\overline{V(\mathfrak{p})} \subsetneq \dots \subsetneq \overline{U_{i}}$ > which is precisely a maximal chain $Z \subsetneq \dots \subsetneq X$ > hence $\text{codim}(V(\mathfrak{p}), \text{Spec } A_{i})=\text{codim}(Z, X)$. (I have since abstracted this reasoning into a property in [[dimension of a topological space]]). > > > Finally, because $X$ is [[irreducible scheme|irreducible]] ([[integral scheme|by the characterization of integral schemes]]), and since $U_{i}$ is open in $X$, we have $\dim U_{i}=\text{dim } X$. > > Thus $\underbrace{ \text{dim }V(\mathfrak{p}) }_{ = \text{dim }Z } + \underbrace{ \text{codim}\big( V(\mathfrak{p}), \text{Spec }A_{i} \big) }_{=\text{codim}(Z, X)} =\underbrace{ \text{dim }U_{i} }_{ =\text{dim }X }$ > and we are done. ----- #### ---- #### 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 > ```