---- > [!definition] Definition. ([[dimension of a topological space]]) > The **dimension** of a [[topological space]] $X$ is the length of the longest chain $Z_{0} \subsetneq Z_{1} \subsetneq \dots \subsetneq Z_{d}$ of [[closed set|closed]] [[irreducible topological space|irreducible]] [[subspace topology|subspaces]] of $X$.[^1] > The **dimension of a scheme** $(X , \mathcal{O}_{X})$ is just $\dim X$. ^definition > [!basicproperties] > - Recall: [[closure|closure]] preserves irreducibility of a subset ([[irreducible topological space|see properties here]]). > - Let $Y \subset X$. In general, if $Z_{0} \subsetneq Z_{1} \subsetneq \dots \subsetneq Z_{d}$ > is a maximal chain of irreducible closed subsets of $Y$, then $\overline{Z_{0}} \subsetneq \overline{Z_{1}} \subsetneq \dots \subsetneq \overline{Z_{d}}$ > is a maximal chain of irreducible closed subsets of $\overline{Y}$. > > - A corollary of the above: if $X$ is [[irreducible topological space|irreducible]] and $U \subset X$ is a nonempty open subset, then *$\dim U=\text{dim }X$*.[^2] Indeed, recall $U$ must be [[dense]] in $X$, and hence if we put $Y=U$ above then $\overline{Y}=\overline{U}=X$. > - The analogous statement for codimension: if $Z$ is an irreducible closed subset of $X$ and $U$ is any open subset of $X$, then *$\text{codim}(Z,X)=\text{codim}(Z \cap U, U)$*, assuming $Z \cap U$ is nonempty. > - If $Y \subset X$ is a [[closed set|closed subset]] of $X$, then ([[codimension of a closed subspace|proof]]) $\text{codim}(Y, X)=\inf\{ \text{dim }\mathcal{O}_{X, y} : y \in Y \}.$ > So computing dimension (put $Y=\emptyset$) reduces to computing the dimensions of [[local ring|local rings]]. **Closure preserves irreducibility.** If $Z$ is an irreducible subset of $X$, then so is $\overline{Z}$. Indeed, if $\overline{Z}=C_{1} \cup C_{2}$ for $C_{1},C_{2}$ closed in $\overline{Z}$, then write $Z=Z \cap \overline{Z}=(Z \cap C_{1}) \cup (Z \cap C_{2})$ witnessing $Z$ to be the union of two relatively closed sets in the [[subspace topology]] on $Z$. If $C_{1}$ and $C_{2}$ are both nonempty, this contradicts irreducibility of $Z$. So $C_{1}$ or $C_{2}$ is empty, showing $\overline{Z}$ is irreducible. **Maximality passes to closures.** We are given a maximal chain of irreducible closed subsets $Z_{0} \subsetneq Z_{1} \subsetneq \dots \subsetneq Z_{d}$ of a subspace $Y \subset X$. The chain $\overline{Z_{0}} \subsetneq \overline{Z_{1}} \subsetneq \dots \subsetneq \overline{Z_{d}}$ certainly consists of closed + irreducible subsets of $Y$; just need to show maximality. Suppose we could extend to $\overline{Z_{d}} \subsetneq W$ for some closed + irreducible $W \subset \overline{Y}$. Now, $W \cap Y$ is irreducible (it is a nonempty open subset of an irreducible set $W$). And $Z_{d} \subsetneq W \cap Y$: there is an inclusion because intersections respect containment, and this inclusion is proper because the closures are distinct — $W \cap Y$ is dense in $W$, while $Z_{d}$ cannot be since $\overline{Z_{d}} \subsetneq W$ be assumption. ---- #### [^1]: The dimension does not need to be finite. [^2]: Compare to $n$-[[topological manifold|manifolds]], where any open subset is also an $n$-manifold. ---- #### 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 > ```