Birational morphisms restrict to isomorphisms between principal affine opens

Since $f$ is locally of finite type, [F3] applied at the point ${\eta }_X$ provides a nonempty affine open $U=\mathrm{Spec}A\subseteq X$ and an affine open $V=\mathrm{Spec}B\subseteq Y$ with $f(U)\subseteq V$ and the induced ring map $\phi :B\to A$ of finite type.

By [F7] the rings $A$ and $B$ are domains, since $U$ and $V$ are nonempty affine opens of the integral schemes $X$ and $Y$. By [F2] the stalks at the generic points are canonically identified with the fraction fields, $K(X)={\mathcal{O}}_{X,{\eta }_X}\cong \mathrm{Frac}A$ and $K(Y)={\mathcal{O}}_{Y,{\eta }_Y}\cong \mathrm{Frac}B$.

The stalk map of [F1] is the localisation of the chart map $\phi$ at the generic point, so under the identifications of step 1.2 it is the map $\mathrm{Frac}B\to \mathrm{Frac}A$ induced by $\phi$. By [F1] this map is an isomorphism; in particular $\phi$ is injective, because its composite with the injection $A\hookrightarrow \mathrm{Frac}A$ is the injection $B\hookrightarrow \mathrm{Frac}B$ of [F6].

By [F4] there are finitely many elements $a_1,\dots ,a_n\in A$ with $A=B[a_1,\dots ,a_n]$, the case $n=0$ meaning $A=B[]$ is generated by the empty list over $B$; in particular $\phi$ is then surjective.

Since $\mathrm{Frac}B\to \mathrm{Frac}A$ is an isomorphism by step 2.1, and $A\subseteq \mathrm{Frac}A$, each $a_i$ has the form $a_i=b_i\mathrm{/}{s}_i$ with $b_i\in B$ and $s_i\in B\setminus \{0\}$, by [F5]. Multiplying by $s_i$ and using that $A\hookrightarrow \mathrm{Frac}A$ is injective gives the equality $s_i{a}_i=b_i$ in $A$. Put $\sigma =s_1\cdots s_n$, an element of $B\setminus \{0\}$; when $n=0$ this is $\sigma =1$.

In the localisation $A_{\sigma }$ of $A$ at the image of $\sigma$ the class of every $s_i$ is a unit, because $s_i$ divides $\sigma$; hence the image of $a_i$ equals $\phi (b_i)\phi (s_i{)}^{-1}$, an element in the image of the localised map $B_{\sigma }\to A_{\sigma }$. Since $A=B[a_1,\dots ,a_n]$ and localisation of an algebra presentation is generated by the localised generators, [F11] shows that $B_{\sigma }\to A_{\sigma }$ is surjective.

The composite $B_{\sigma }\to A_{\sigma }\to \mathrm{Frac}(A_{\sigma })$ is injective: the localisation $A_{\sigma }$ of the domain $A$ at the nonzero element $\phi (\sigma )$ is a subring of $\mathrm{Frac}A$ with $\mathrm{Frac}(A_{\sigma })=\mathrm{Frac}A$ by [F5] and [F6], and under these identifications the composite is the localisation map $B_{\sigma }\to \mathrm{Frac}B=\mathrm{Frac}A$ of the domain $B$ at the nonzero element $\sigma$, which is injective by [F6].

Steps 4.1 and 4.2 show that ${\phi }_{\sigma }:B_{\sigma }\to A_{\sigma }$ is an isomorphism. By [F8] it corresponds to an isomorphism $\mathrm{Spec}{A}_{\sigma }\to \mathrm{Spec}{B}_{\sigma }$ of affine schemes. The localisation maps $B\to B_{\sigma }$ and $A\to A_{\sigma }$ induce open immersions $\mathrm{Spec}{A}_{\sigma }\to \mathrm{Spec}A$ and $\mathrm{Spec}{B}_{\sigma }\to \mathrm{Spec}B$ whose images are $D(\phi (\sigma ))$ and $D(\sigma )$ by [F9] and [F10]; the composite $\mathrm{Spec}{A}_{\sigma }\to \mathrm{Spec}A\to \mathrm{Spec}B$ is $f$ restricted to the open subscheme $\mathrm{Spec}{A}_{\sigma }\subseteq U$ and factors through the isomorphism $\mathrm{Spec}{A}_{\sigma }\cong \mathrm{Spec}{B}_{\sigma }\to \mathrm{Spec}B$, so by [F12] the restriction of $f$ to $\mathrm{Spec}{A}_{\sigma }$ is an isomorphism onto $\mathrm{Spec}{B}_{\sigma }$, and it is invertible after restriction to the open subschemes $D(\phi (\sigma ))\subseteq U$ and $D(\sigma )\subseteq V$.

Preimages of principal opens under the chart map are principal opens: a prime $\mathfrak{p}\in \mathrm{Spec}A$ lies over $D(\sigma )$ exactly when $\sigma \mathrm{\notin }{\phi }^{-1}(\mathfrak{p})$, that is $\phi (\sigma )\mathrm{\notin }\mathfrak{p}$, so $f^{-1}(D(\sigma ))\cap U=D(\phi (\sigma ))=\mathrm{Spec}{A}_{\sigma }$. Setting $V_0=D(\sigma )$ gives a nonempty open subscheme of $V$, because $\sigma \mathrm{\ne }0$ in the domain $B$ makes $(0)\in D(\sigma )$ by [F10], and the restriction $f^{-1}(V_0)\cap U\to V_0$ is an isomorphism by step 5.1. The construction used only the finitely many generators of a finite-type presentation and the finitely many denominators of step 3.2; no infinite choice is made, and for $n=0$ the element is $\sigma =1$, so that $U\to V$ is itself the required isomorphism onto $V_0=V$. ∎