r equations lower dimension by at most r

Given: The objects and hypotheses in the statement, including the Axiom of Choice (The Axiom of Choice).

For a nonempty affine algebraic set $X$, $\dim X=\dim k[X]$, where the right side is Krull dimension. For this comparison only, extend ring dimension to the zero ring by $\dim (0)=-\mathrm{\infty }$; then the equality also holds for $X=\mathrm{\varnothing }$. 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. (Affine geometric dimension equals ring dimension).

If $U$ is a nonempty open of an irreducible classical variety $X$, then $\dim U=\dim X$. Every proper closed subvariety $Z⊊X$ has $\dim Z<\dim X$. 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. (Nonempty opens preserve irreducible dimension).

Under the Axiom of Choice, if $R$ is Noetherian, $I=(x_1,\dots ,x_n)$ has $n\ge 1$ generators, and $\mathfrak{p}$ is minimal over $I$, then $\mathrm{ht}(\mathfrak{p})\le n$ (Krull's height theorem).

Under the Axiom of Choice, if $k$ is a field, $A$ is a finite-type $k$-domain, and $\mathfrak{p}\in \mathrm{Spec}(A)$, then $\mathrm{ht}(\mathfrak{p})+\dim (A\mathrm{/}\mathfrak{p})=\dim A$ (Height plus quotient dimension equals ambient dimension in an affine domain).