Fpqc covers are universally submersive

Suppose every affine chart map $A\to B$ is flat and fix $x\in X$ with image $s\in S$. Choose an affine neighborhood $V=\mathrm{Spec}A$ of $s$ and an affine neighborhood $U=\mathrm{Spec}B$ of $x$ contained in $f^{-1}(V)$. For the corresponding primes $\mathfrak{p}\subset A$ and $\mathfrak{q}\subset B$, the map $A_{\mathfrak{p}}\to A_{\mathfrak{p}}{\otimes }_{A}B$ is flat by [F7], and $A_{\mathfrak{p}}{\otimes }_{A}B\to B_{\mathfrak{q}}$ is a localization, hence flat by [F7]. Their composite is the local-ring map at $x$ by [F2], so $f$ is flat by [F1]. Conversely, if $f$ is flat, fix any affine chart $U=\mathrm{Spec}B\to V=\mathrm{Spec}A$ and let $\mathfrak{q}\in \mathrm{Spec}B$ contract to $\mathfrak{p}\subseteq A$. [F1] and [F2] make $A_{\mathfrak{p}}\to B_{\mathfrak{q}}$ flat. For a finitely generated ideal $I\subseteq A$, let $K=\ker (I{\otimes }_{A}B\to B)$. Exact localization and the tensor-localization isomorphisms of [F4] identify $K_{\mathfrak{q}}$ with the kernel of $I_{\mathfrak{p}}{\otimes }_{A_{\mathfrak{p}}}{B}_{\mathfrak{q}}\to B_{\mathfrak{q}}$, which is zero by flatness. This holds for every $\mathfrak{q}$. If $B=0$, $K=0$; otherwise, if $K\mathrm{\ne }0$, take $0\mathrm{\ne }k\in K$. Its annihilator in $B$ is proper, so under AC [A1] and [F6] give a maximal ideal $\mathfrak{q}$ containing it. Then $k\mathrm{/}1\mathrm{\ne }0$ in $K_{\mathfrak{q}}$ by [F5], a contradiction. Thus $I{\otimes }_{A}B\to B$ is injective for every finitely generated $I$, and [F3] makes $B$ flat. This proves both directions of the affine-chart criterion.

Since $p$ is quasi-compact and surjective, [F9] says every base change $p_T$ is quasi-compact and surjective.

If $Z\subseteq T$ is closed, then $p_T^{-1}(Z)$ is closed because a scheme morphism is continuous.

Let $p:S^{\mathrm{\prime }}\to S$ be page-local fpqc and $T\to S$ arbitrary. Choose affine opens $W=\mathrm{Spec}C\subseteq T$ and $V=\mathrm{Spec}A\subseteq S$ with $W$ mapping into $V$, and cover $p^{-1}(V)$ by affine opens $U_i=\mathrm{Spec}{B}_i$. Since $p$ is flat, step 1.1 shows every $A\to B_i$ is flat. By [F8], the affine schemes $W{\times }_V{U}_i=\mathrm{Spec}(B_i{\otimes }_{A}C)$ cover $p_T^{-1}(W)$, and each chart map $C\to B_i{\otimes }_{A}C$ is flat by [F7]. Step 1.1 then proves that $p_T$ is flat.

Assume $p_T^{-1}(Z)$ is closed. For each affine open $U=\mathrm{Spec}A\subseteq T$, the inverse image $Y=p_T^{-1}(U)$ is quasi-compact by steps 1.2, [F15], and [F16], since affine schemes are quasi-compact. Choose a finite affine open cover $Y={\bigcup }_{i=1}^n{U}_i$, where $U_i=\mathrm{Spec}{B}_i$, omitting empty members; [F10] supplies affine neighborhoods and finite subcover extraction. The finite disjoint union of these charts is $\mathrm{Spec}B$ for $B={\prod }_i{B}_i$ by [F11]. Its map to $U$ is surjective by step 1.2, and $A\to B$ is flat: step 1.1 applies to each chart map $A\to B_i$, while [F11] says the finite direct sum of these flat $A$-modules is flat. The inverse image of $Z\cap U$ in $\mathrm{Spec}B$ is closed, hence equals $V(J)$ for an ideal $J\subseteq B$ by [F12]. If $\mathfrak{p}\in Z\cap U$, surjectivity gives a prime $\mathfrak{q}\subseteq B$ above it; since $\mathfrak{q}$ lies in the inverse image $V(J)$, the quotient-prime correspondence gives a point of $\mathrm{Spec}(B\mathrm{/}J)$ above $\mathfrak{p}$. Conversely every point of $\mathrm{Spec}(B\mathrm{/}J)$ corresponds to a prime in $V(J)$ and therefore maps into $Z\cap U$. Hence $Z\cap U=\mathrm{image}(\mathrm{Spec}(B\mathrm{/}J)\to \mathrm{Spec}A)$. If $\mathfrak{p}\subseteq {\mathfrak{p}}^{\mathrm{\prime }}$ in $\mathrm{Spec}A$ and $\mathfrak{p}\in Z\cap U$, surjectivity gives ${\mathfrak{q}}^{\mathrm{\prime }}\subseteq B$ over ${\mathfrak{p}}^{\mathrm{\prime }}$. Flat going-down [F13] gives $\mathfrak{q}\subseteq {\mathfrak{q}}^{\mathrm{\prime }}$ over $\mathfrak{p}$. Since the full inverse image of $Z\cap U$ is $V(J)$, $J\subseteq \mathfrak{q}\subseteq {\mathfrak{q}}^{\mathrm{\prime }}$, so ${\mathfrak{p}}^{\mathrm{\prime }}$ belongs to the image. Thus this image is stable under specialization.

The morphism $\mathrm{Spec}(B\mathrm{/}J)\to \mathrm{Spec}A$ is quasi-compact: the inverse image of every affine open is affine by [F8], hence quasi-compact by [F16] and [F15]. Its image is $Z\cap U$ and is stable under specialization by step 3.1, so [F14] makes $Z\cap U$ closed. Since affine opens cover $T$, $Z$ is closed.

Step 1.1 proves both directions of the affine-chart flatness criterion; steps 1.2 and 2.1 prove universal quasi-compactness, surjectivity, and flatness; and steps 1.3, 3.1, and 4.1 prove both directions of the closed-subset criterion. The zero-ring chart in step 1.1 has zero kernel; the zero ideal in its flatness test gives the zero map with zero kernel, while the unit ideal gives $A{\otimes }_{A}B\cong B$. An empty target chart has empty source, and the empty source is flat vacuously. A nonempty affine target in step 3.1 has nonempty inverse image by surjectivity. AC is used in step 1.1 to find a maximal ideal containing the annihilator; it is also required by the base-change-surjectivity, going-down, and specialization-image suppliers [F9, F13, F14]. Finite affine covers use only finite subcover extraction.