Noetherian approximation of proper flat finitely presented sheaf data

Filtered system. Let $\mathcal{T}$ be the set of finite subsets $T\subseteq A$ and for $T\in \mathcal{T}$ let $A_T\subseteq A$ be the $\mathbb{Z}$-subalgebra generated by $T$, so that $T\subseteq T^{\mathrm{\prime }}$ implies $A_T\subseteq A_{T^{\mathrm{\prime }}}$, the system is filtered, ${\bigcup }_T{A}_T=A$, and each $A_T$ is a finitely generated algebra over $\mathbb{Z}$, hence Noetherian by [F3].

Descent of the scheme and morphism (D1). The given $f$ is of finite presentation, and the bases $\mathrm{Spec}{A}_T$ form an inverse system of affine quasi-compact and quasi-separated schemes with affine transition maps and limit $\mathrm{Spec}A$. Apply the scheme-stage result (D1) of [F4], including its morphism and equality clauses, to obtain $T\in \mathcal{T}$ and a morphism $f_T:X_T\to \mathrm{Spec}{A}_T$ of finite presentation with an isomorphism $X\cong X_T{\times }_{\mathrm{Spec}{A}_T}\mathrm{Spec}A$ compatible with $f$. Its local proof includes the gluing of affine charts.

Descent of the sheaf (D2). The schemes $X_T$ of step 1.2 are quasi-compact and quasi-separated and their transition maps are affine. Apply the sheaf-stage result (D2) of [F4] to the finitely presented quasi-coherent module $\mathcal{F}$ on their limit $X$: after enlarging $T$ it descends to a finitely presented ${\mathcal{O}}_{X_T}$-module ${\mathcal{F}}_T$ with $\mathcal{F}\cong {\mathcal{F}}_T{\otimes }_{{\mathcal{O}}_{X_T}}{\mathcal{O}}_X$. On affine charts, its presentation matrices have coefficients in the chart coordinate rings; (D2) also descends their compatibility on overlaps.

Properness descends (D4). The stage morphism $f_T$ is of finite presentation, hence locally of finite type. Its pullback $f$ is proper. Apply the proper-stage result (D4) of [F4] to obtain a later stage at which $f_T$ is proper. That carrier proves eventual separatedness by descending the diagonal and universal closedness through a proper surjective Chow cover. Finite presentation persists under base change.

Flatness descends (D3). The stage morphism $X_T\to \mathrm{Spec}{A}_T$ and ${\mathcal{F}}_T$ are of finite presentation by steps 1.2 and 2.1, while their pullbacks to $A$ give the stipulated flat sheaf. Apply the flat-sheaf-stage result (D3) of [F4] to this relative pair; it gives a later stage at which the pulled-back ${\mathcal{F}}_T$ is flat over its base. Its proof reduces to finitely many affine charts and one common later stage.

Conclusion. Choose $T$ large enough for steps 1.2, 2.1, 2.2, and 3.1 simultaneously (the system is filtered, so the finitely many enlargements can be combined), put $A_i=A_T$, $X_i=X_T$, $f_i=f_T$ and ${\mathcal{F}}_i={\mathcal{F}}_T$. Then $A_i$ is a finitely generated $\mathbb{Z}$-subalgebra of $A$, $f_i$ is proper of finite presentation, ${\mathcal{F}}_i$ is finitely presented and flat over $A_i$, and the pullbacks to $A$ recover $f$ and $\mathcal{F}$ up to the stated isomorphisms.

Boundaries and choice accounting. If $A$ is finitely generated over $\mathbb{Z}$, then $A=A_T$ for a finite $T$ generating it and the descent is trivial (the identity stage, with the given $f$ and $\mathcal{F}$). If $X=\mathrm{\varnothing }$, the empty $X_T$ over a stage works; if $\mathcal{F}=0$, the zero module is finitely presented and flat over every ring. The finite enlargements combine because the index system is filtered. The stated Axiom of Choice is inherited from the four proved finite-stage carriers [F4]; combining these finitely many stages uses no further choice.