Finite étale covers descend effectively along fpqc covers

For an affine faithfully flat map $A\to B$, let $N$ be the upstairs finite étale $B$-algebra and let $D$ be its invariant algebra. By [F1], $D{\otimes }_{A}B\cong N$. To descend finite generation, express finitely many $B$-module generators of $N$ as finite sums of tensors $d\otimes b$. The finitely many occurring $d$ generate a submodule $D_0$ whose base change surjects onto $N$. Thus $(D\mathrm{/}{D}_0){\otimes }_{A}B=0$, and faithful flatness gives $D=D_0$.

Choose a finite free surjection $A^r\to D$ with kernel $K$. Flatness of $B\mathrm{/}A$ identifies $K{\otimes }_{A}B$ with the kernel of $B^r\to N$. This kernel is finitely generated because $N$ is finitely presented as a $B$-module by [F2]; the assertion for an arbitrary finite free surjection follows by comparing it with a fixed finite presentation and eliminating finitely many auxiliary generators. Applying the same finite-tensor-generator argument as step 1.1 to $K$ proves that $K$ is finitely generated. Hence $D$ is finitely presented as an $A$-module. It is flat by [F2]. Since ${\mathrm{\Omega }}_{D\mathrm{/}A}{\otimes }_{A}B\cong {\mathrm{\Omega }}_{N\mathrm{/}B}=0$, faithful flatness gives ${\mathrm{\Omega }}_{D\mathrm{/}A}=0$. Therefore $D$ is finite étale by [F2]. Maps descend with their algebra structures by [F1], proving the entire assertion for affine faithfully flat maps.

Now restrict the target to an affine open $U=\mathrm{Spec}A\subseteq S$. By [F4], $S_U^{\mathrm{\prime }}$ is quasi-compact, so choose finitely many affine opens $V_j=\mathrm{Spec}{B}_j$ covering it. Their disjoint union is affine, with ring $B={\prod }_j{B}_j$. The map $A\to B$ is flat and surjective on spectra, hence faithfully flat by [F3]. Pull the upstairs cover and cocycle back to this disjoint union, and use steps 1.1 and 2.1 to construct a finite étale cover of $U$. Its pullback is canonically the given cover on each $V_j$; the isomorphisms agree on intersections because their restrictions to $V_j{\times }_U{V}_k$ are precisely the original descent datum. Thus they glue to identify its pullback over all of $S_U^{\mathrm{\prime }}$ with the original cover and datum.

On the intersections of two affine opens of $S$, cover the intersection by affine opens and repeat step 3.1. Full faithfulness in step 2.1 makes the resulting comparison isomorphisms unique after requiring compatibility upstairs; uniqueness makes them agree on further overlaps and satisfy the cocycle identity. The algebras and then their relative spectra glue by [F3]. Finiteness and étaleness are affine local, so the glued cover has both properties. A compatible upstairs morphism descends on these affine opens by [F1] and its unique local descents agree, hence glue. This proves the claimed equivalence, including effectiveness and uniqueness, over arbitrary base schemes. AC is used through the suppliers recorded in [F4].