Lemma 32.20.3. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.

The topological space of $X$ is catenary (Topology, Definition 5.11.4).

For $x \in X$ closed, we have $\dim _ x(X) = \dim (\mathcal{O}_{X, x})$.

For $X$ irreducible we have $\dim (X) = \dim (U)$ for any nonempty open $U \subset X$ and $\dim (X) = \dim _ x(X)$ for any $x \in X$.

For $X$ irreducible any chain of irreducible closed subsets can be extended to a maximal chain and all maximal chains of irreducible closed subsets have length equal to $\dim (X)$.

For $x \in X$ we have $\dim _ x(X) = \max \dim (Z) = \min \dim (\mathcal{O}_{X, x'})$ where the maximum is over irreducible components $Z \subset X$ containing $x$ and the minimum is over specializations $x \leadsto x'$ with $x'$ closed in $X$.

If $X$ is irreducible with generic point $x$, then $\dim (X) = \text{trdeg}_ k(\kappa (x))$.

If $x \leadsto x'$ is an immediate specialization of points of $X$, then we have $\text{trdeg}_ k(\kappa (x)) = \text{trdeg}_ k(\kappa (x')) + 1$.

The dimension of $X$ is the supremum of the numbers $\text{trdeg}_ k(\kappa (x))$ where $x$ runs over the generic points of the irreducible components of $X$.

If $x \leadsto x'$ is a nontrivial specialization of points of $X$, then

$\dim _ x(X) \leq \dim _{x'}(X)$,

$\dim (\mathcal{O}_{X, x}) < \dim (\mathcal{O}_{X, x'})$,

$\text{trdeg}_ k(\kappa (x)) > \text{trdeg}_ k(\kappa (x'))$, and

any maximal chain of nontrivial specializations $x = x_0 \leadsto x_1 \leadsto \ldots \leadsto x_ n = x$ has length $n = \text{trdeg}_ k(\kappa (x)) - \text{trdeg}_ k(\kappa (x'))$.

For $x \in X$ we have $\dim _ x(X) = \text{trdeg}_ k(\kappa (x)) + \dim (\mathcal{O}_{X, x})$.

If $x \leadsto x'$ is an immediate specialization of points of $X$ and $X$ is irreducible or equidimensional, then $\dim (\mathcal{O}_{X, x'}) = \dim (\mathcal{O}_{X, x}) + 1$.

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