= Principal hypersurface dimension lemma
{title2=$\dim V_X(f)_j=\dim X-1$}
For an <irreducible variety> which is an <affine variety> of dimension $d$ and a nonzero nonunit <regular function> $f$, each <irreducible component> of $V(f)$ has dimension $d-1$. Choose a closed point on just the component in question. In its <local ring> $R$, $\sqrt{(f)}$ is that component's <prime ideal>. A <system of parameters> of $R/(f)$, lifted and supplemented by $f$, generates a maximal-primary ideal; the <Krull height theorem> gives $\dim R/(f)\ge d-1$. Extending any chain above $(f)$ by the zero prime of the domain gives the reverse inequality. This proof does not identify dimension with <transcendence degree>.
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