Sections of a sheaf flat over the base are flat over affine opens

Set $M:=\mathcal{F}(U)$. By [F1] the restriction $\mathcal{F}{\mathrm{\mid }}_U$ is quasi-coherent and is isomorphic to $\tilde{M}$, and for every prime $\mathfrak{q}\subseteq B$ the stalk of $\mathcal{F}$ at $\mathfrak{q}$ is $M_{\mathfrak{q}}$.

We prove that $M$ is flat over $A$. Let $I\subseteq A$ be a finitely generated ideal and let $K$ be the kernel of the multiplication map $I{\otimes }_{A}M\to M$, so that $K$ is a $B$-submodule. Suppose $K\mathrm{\ne }0$. By the zero criterion of [F6] applied to the $B$-module $K$ there is a maximal ideal $\mathfrak{m}\subseteq B$ with $K_{\mathfrak{m}}\mathrm{\ne }0$; by [F6] the maximal ideal $\mathfrak{m}$ is prime, and we put ${\mathfrak{p}}_0=\mathfrak{m}\cap A$.

Let $\mathfrak{q}\subseteq B$ be a prime and put $\mathfrak{p}=\mathfrak{q}\cap A$. By [F2] the point $\mathfrak{q}$ lies in $U\subseteq X$ with $f(\mathfrak{q})=\mathfrak{p}$ and ${\mathcal{O}}_{S,f(\mathfrak{q})}=A_{\mathfrak{p}}$. Hypothesis [F3] therefore states that $M_{\mathfrak{q}}={\mathcal{F}}_{\mathfrak{q}}$ is a flat $A_{\mathfrak{p}}$-module.

Localizing the exact sequence of $A$-modules $0\to K\to I{\otimes }_{A}M\to M$ at $\mathfrak{m}$, the module $K_{\mathfrak{m}}$ is the kernel of the localized multiplication map $(I{\otimes }_{A}M{)}_{\mathfrak{m}}\to M_{\mathfrak{m}}$. By the localizations and regroupings of [F5], $(I{\otimes }_{A}M{)}_{\mathfrak{m}}\cong I{\otimes }_A{M}_{\mathfrak{m}}\cong I_{{\mathfrak{p}}_0}{\otimes }_{A_{{\mathfrak{p}}_0}}{M}_{\mathfrak{m}},$ where $I_{{\mathfrak{p}}_0}=I{A}_{{\mathfrak{p}}_0}$ is the extension of $I$ to $A_{{\mathfrak{p}}_0}$, and under these isomorphisms the localized map is the multiplication map $I_{{\mathfrak{p}}_0}{\otimes }_{A_{{\mathfrak{p}}_0}}{M}_{\mathfrak{m}}\to M_{\mathfrak{m}}$.

By step 2.1 applied to the prime $\mathfrak{m}\subseteq B$ the module $M_{\mathfrak{m}}$ is flat over $A_{{\mathfrak{p}}_0}$, so by the ideal criterion [F4] the map $I_{{\mathfrak{p}}_0}{\otimes }_{A_{{\mathfrak{p}}_0}}{M}_{\mathfrak{m}}\to M_{\mathfrak{m}}$ is injective. By step 2.2 its kernel is $K_{\mathfrak{m}}$, hence $K_{\mathfrak{m}}=0$, contradicting the choice of $\mathfrak{m}$ in step 1.2. Therefore $K=0$ for every finitely generated ideal $I\subseteq A$.

Since every finitely generated ideal $I\subseteq A$ gives an injective multiplication map $I{\otimes }_{A}M\to M$, the criterion [F4] shows that $M=\mathcal{F}(U)$ is a flat $A$-module.

Boundary and choice accounting. If $U=\mathrm{\varnothing }$ then $B=0$ and $M=0$, the zero module being flat over $A$; if $\mathcal{F}=0$ then $M=0$; if $A=0$ then also $M=0$, and if $X=\mathrm{\varnothing }$ or $S=\mathrm{\varnothing }$ there is no nonempty affine open to consider. By [F7] the Axiom of Choice is available exactly as the zero criterion [F6] consumes it, namely to supply the maximal ideal $\mathfrak{m}$ of step 1.2; the localizations used are those of [F1], [F2] and [F5], and no further selection is made. The assertion of the statement is exactly the flatness of $M=\mathcal{F}(U)$ over $A$ established in step 4.1, so the lemma is proved.