Indeterminacy of a rational map into an affine scheme is of pure codimension one

Let $Y=\mathrm{Spec}B$ and choose generators $b_1,\dots ,b_n$ of $B$ over $R$ as in [F1]. On the dense open where $u$ is represented, the pullbacks $u^{\ast }(b_i)$ are rational functions $f_i$ on $Z$. On an integral affine chart $U=\mathrm{Spec}A\subseteq Z$ with fraction field $K$, the $f_i$ lie in $K$, and a morphism $U\to Y$ is given exactly by an $R$-algebra map $B\to A$, i.e. by elements $f_1,\dots ,f_n\in A$ satisfying every defining relation of $B$. Since those relations vanish on the dense open where $u$ is defined, they vanish as rational functions; hence $u$ is defined at a point $z\in U$ if and only if $f_1,\dots ,f_n$ all lie in the local ring ${\mathcal{O}}_{Z,z}$.

On an integral affine chart $U=\mathrm{Spec}A$, the nonregular locus of $f\in \mathrm{Frac}A$ is $V(J_f)$, where $J_f=\{a\in A:af\in A\}$: membership in $A_{\mathfrak{q}}$ is equivalent to $J_f$ containing an element outside $\mathfrak{q}$. If $\mathfrak{q}$ is a prime minimal over $J_f$, then $f\mathrm{\notin }{A}_{\mathfrak{q}}$. Apply [F2] to the normal Noetherian local domain $A_{\mathfrak{q}}$: there is a height-one prime $\mathfrak{p}{A}_{\mathfrak{q}}$ at which $f$ is not regular, with $\mathfrak{p}\subseteq \mathfrak{q}$. All chains below $\mathfrak{p}$ survive localization, so $\mathfrak{p}$ has height one in $A$. Nonregularity implies $J_f\subseteq \mathfrak{p}$, and minimality of $\mathfrak{q}$ therefore gives $\mathfrak{p}=\mathfrak{q}$. Every irreducible component of $V(J_f)$ thus has codimension one.

By step 1.1 the indeterminacy locus of $u$ on $U$ is the union of the pole loci of $f_1,\dots ,f_n$. If all $f_i$ are regular on $U$, this locus is empty and $u$ is a morphism on $U$. Otherwise it is the union of finitely many closed subsets each of which is of pure codimension one by step 2.1; a finite union of pure-codimension-one closed subsets of a Noetherian scheme has all its irreducible components of codimension one, so the indeterminacy locus is of pure codimension one.

If $u$ is defined at every point of height at most one, then by step 2.1 no pole locus meets the height-one points of $U$, so each pole locus is empty; thus all $f_i$ are regular on every affine chart, and the local morphisms $U\to Y$ glue to an $R$-morphism $Z\to Y$ extending $u$, unique because $Y$ is separated over $R$ and two extensions agree on the dense domain of $u$ by [F2]. Finite generation of $B$ over $R$ is used to have finitely many $b_i$; the relation ideal need not be finitely generated, so that a common regular locus can be exhibited.