A point is locally cut out by dim X functions

Given: The objects and hypotheses in the statement.

If $X$ is an irreducible classical variety and $x$ is a closed point, then $\dim {\mathcal{O}}_{X,x}=\dim X={\mathrm{codim}}_X\{x\}$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Closed-point local dimension equals ambient irreducible dimension).

Let $R$ be a Noetherian commutative ring and let $\mathfrak{p}\in \mathrm{Spec}(R)$ have finite height $n$. Then in the local ring $R_{\mathfrak{p}}$ there exist elements $x_1,\dots ,x_n\in \mathfrak{p}$ such that the maximal ideal $\mathfrak{p}{R}_{\mathfrak{p}}$ is minimal over $(x_1\mathrm{/}1,\dots ,x_n\mathrm{/}1)$. Equivalently, $\mathfrak{p}$ is minimal over an $n$-generated ideal after localizing at $\mathfrak{p}$. (Converse to Krull's height theorem in localised form).

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).