Integral quasi-coherent algebras over qcqs bases are unions of finite subalgebras

Fix a finite affine open cover $S=U_1\cup \cdots \cup U_n$ with each $U_i=\mathrm{Spec}{R}_i$ and with all intersections $U_i\cap U_j$ quasi-compact; such a cover exists by [F2]. On $U_i$ the algebra is $\mathcal{A}{\mathrm{\mid }}_{U_i}=\widetilde{A_i}$ for an integral $R_i$-algebra $A_i$ by [F3]. Keep the affine domain of each specified local section in a finite working family; on each such domain the section generates a finite-type submodule. Coefficients of monic certificates lie in the structure sheaf and belong to every ${\mathcal{O}}_S$-subalgebra. Only finitely many affine domains and choices occur.

Quasi-compact open pushforward. Let $j:U\hookrightarrow S$ be a quasi-compact open immersion and $\mathcal{G}$ a quasi-coherent module on $U$. On any affine $V=\mathrm{Spec}R\subseteq S$, the intersection $W=U\cap V$ is quasi-compact because $S$ is quasi-separated. Choose finitely many principal opens $D(f_i)$ of $V$ covering $W$. The sheaf equalizer computes $\mathrm{\Gamma }(W,\mathcal{G})$ from the finite product of $\mathrm{\Gamma }(D(f_i),\mathcal{G})$ and the finite product of $\mathrm{\Gamma }(D(f_i{f}_j),\mathcal{G})$. For $r\in R$, localization at $r$ is exact and commutes with these finite products; using quasi-coherence on each principal open, the localized equalizer is the equalizer for the cover $D(r{f}_i)$ of $W\cap D(r)$. Thus $\mathrm{\Gamma }(W,\mathcal{G}{)}_r\cong \mathrm{\Gamma }(W\cap D(r),\mathcal{G})$. This is exactly the affine-local criterion that $j_{\ast }\mathcal{G}$ is quasi-coherent.

Extension of a quasi-coherent subsheaf. Suppose $\mathcal{G}\subseteq \mathcal{F}{\mathrm{\mid }}_U$ with $U$ quasi-compact open and $\mathcal{F}$ quasi-coherent on $S$. The maps $\mathcal{F}\oplus j_{\ast }\mathcal{G}\to j_{\ast }{j}^{\ast }\mathcal{F}$, $(a,b)\mapsto a{\mathrm{\mid }}_U-b$, are maps of quasi-coherent modules by step 1.2. Their kernel $\mathcal{H}$ is quasi-coherent by [F7]. Its projection to $\mathcal{F}$ is injective, since $j_{\ast }\mathcal{G}\to j_{\ast }{j}^{\ast }\mathcal{F}$ is injective, and on $U$ its image is precisely $\mathcal{G}$. Hence $\mathcal{G}$ extends to a quasi-coherent subsheaf $\mathcal{H}\subseteq \mathcal{F}$.

Finite-type extension. Assume $\mathcal{G}$ in step 2.1 is of finite type. Add one affine $V=\mathrm{Spec}R$ to $U$ at a time from a finite affine cover of $S$. Apply step 2.1 over $U\cup V$ to obtain an extension $\mathcal{H}\subseteq \mathcal{F}$. Write $\mathcal{H}{\mathrm{\mid }}_V=\tilde{M}$. The quasi-compact intersection $U\cap V$ has a finite principal cover $D(f_i)$. Since $\mathcal{G}$ is finite type there, each $M_{f_i}$ has finitely many generators; choose numerator representatives in $M$ for all of them and let $M^{\mathrm{\prime }}\subseteq M$ be the finite submodule they generate. Then $M_{f_i}^{\mathrm{\prime }}=M_{f_i}$ for each $i$, so $\widetilde{M^{\mathrm{\prime }}}$ and $\mathcal{G}$ agree on $U\cap V$ and glue to a finite-type quasi-coherent subsheaf of $\mathcal{F}$ on $U\cup V$. Finite induction over the affine cover extends $\mathcal{G}$ to all of $S$.

Finite-type submodules and subalgebras. Any local section of a quasi-coherent module $\mathcal{F}$ over an affine open $U$ lies in the finite-type submodule generated by it on $U$; step 3.1 extends that submodule to a finite-type quasi-coherent subsheaf of $\mathcal{F}$ on $S$. Finite sums give one such subsheaf containing any finite family of local sections. For the quasi-coherent algebra $\mathcal{A}$, take the subalgebra generated by such a finite-type submodule and $1$: on every affine chart it has finitely many algebra generators, and localization commutes with forming the algebra generated by a module, so these chart subalgebras glue as a finite-type quasi-coherent subalgebra. Finite joins of these subalgebras are again finite type, so they form a filtered system whose union is $\mathcal{A}$.

Every member of the filtered system in step 4.1 is finite over ${\mathcal{O}}_S$. Indeed, on an affine open $V=\mathrm{Spec}R$ it has finitely many algebra generators, each integral over $R$ because $\mathcal{A}$ is integral. By [F6] the resulting $R$-algebra is finite as an $R$-module. Thus the system is a filtered union of finite quasi-coherent subalgebras. For the finite set of sections and coefficients in the Statement, step 4.1 supplies a common finite-type subalgebra, and the preceding argument makes it finite.

The construction works on a finite affine cover from step 1.1 and contains each specified local section and certificate by step 5.1. The empty family uses the subalgebra generated by $1$, which is finite; the zero-ring chart gives a zero algebra and causes no exception. AC enters only in selecting the finite affine covers, generators and representatives in steps 1.1–3.1 and in [F6]. No Noetherian or separated hypothesis beyond qcqs is used.