Finite-stage descent of relative flatness for a finitely presented sheaf

Choose a finite affine cover $X_i={\bigcup }_{a=1}^m{U}_{a,i}$, with $U_{a,i}=\mathrm{Spec}{B}_{a,i}$. By [F1], every $B_{a,i}$ is a finitely presented $A_i$-algebra and ${\mathcal{F}}_i{\mathrm{\mid }}_{U_{a,i}}$ corresponds to a finitely presented $B_{a,i}$-module $M_{a,i}$. The charts and modules at later stages and at the limit are their base changes.

Fix one chart and a prime $\mathfrak{q}\subseteq B_a$ at the limit, and set $\mathfrak{p}=\mathfrak{q}\cap A$. Its module stalk $(M_a{)}_{\mathfrak{q}}$ is flat over $A_{\mathfrak{p}}$: the given $A$-flatness localizes, and the stalk is naturally an $A_{\mathfrak{p}}$-module. Let ${\mathfrak{p}}_j,{\mathfrak{q}}_j$ be the contracted primes at later stages. Each $A_j$ is Noetherian because it is finite type over $\mathbb{Z}$, and $B_{a,j}$ is Noetherian because it is finite type over $A_j$. The localized pairs $(A_j{)}_{{\mathfrak{p}}_j}\to (B_{a,j}{)}_{{\mathfrak{q}}_j}$ with modules $(M_{a,j}{)}_{{\mathfrak{q}}_j}$ have the localization/base-change transitions and colimit stated in [F2]. Apply its eventual-flatness conclusion to obtain $j(\mathfrak{q})$ at which $(M_{a,j(\mathfrak{q})}{)}_{{\mathfrak{q}}_{j(\mathfrak{q})}}$ is flat over $(A_{j(\mathfrak{q})}{)}_{{\mathfrak{p}}_{j(\mathfrak{q})}}$. This is the relative flatness condition at that stage point.

By [F3], the flat locus of $M_{a,j(\mathfrak{q})}$ in $\mathrm{Spec}{B}_{a,j(\mathfrak{q})}$ contains a principal open $D(g_{\mathfrak{q}})$ around ${\mathfrak{q}}_{j(\mathfrak{q})}$. Its pullback to $\mathrm{Spec}{B}_a$ contains $\mathfrak{q}$ and is flat. As $\mathfrak{q}$ varies, these pulled-back principal opens cover the entire limit chart.

The limit chart is quasi-compact, so choose finitely many ${\mathfrak{q}}_1,\dots ,{\mathfrak{q}}_r$ whose pulled-back principal flats cover it, and pass to a common stage $j$ where their corresponding stage principal opens are defined. Let $U_j$ be their union and let $I_j$ be the ideal generated by their finitely many defining functions. The pullback of $U_j$ is all of $\mathrm{Spec}{B}_a$, so $I_j{B}_a=B_a$. Choose a finite identity $1={\sum }_{k=1}^n{c}_k{s}_k$ in $B_a$ with $s_k\in I_j$. Each coefficient and this one equality occur at some later stage $k\ge j$; hence $I_j{B}_{a,k}=B_{a,k}$ and the pullback of $U_j$ is all of the stage chart $\mathrm{Spec}{B}_{a,k}$. Flatness holds on all of that chart, since it held on $U_j$ and survives base change. If the limit chart is empty, the same argument with the empty open gives $B_{a,k}=0$, so the stage chart becomes empty.

Repeat step 4.1 for the finitely many affine charts and choose one common later stage. The resulting ${\mathcal{F}}_j$ is flat over $A_j$ at every point of $X_j$. If ${\mathcal{F}}_i=0$, it is flat at every stage without enlargement. The only directed choices were finite; AC is declared for the local supplier framework. No nonflat transition was treated as a faithfully flat map. The Stacks 05LY/02JO references verify the scope of this local proof but are not proof premises.