Finite field descent is effective for schemes with affine-contained descent orbits

The datum makes $D=Y{\times }_{\mathrm{Spec}k}\mathrm{Spec}K$ into a relation on the underlying $k$-scheme $Y$: its first map is projection, and its second map uses the given isomorphism between the two $K{\otimes }_{k}K$ base changes. Both projections are finite locally free of rank $[K:k]$. The cocycle gives composition, and the diagonal and exchange of the two scalar factors give identity and inverse. The map $D\to Y{\times }_{k}Y$ is a monomorphism: the source point together with the scalar structure of the target uniquely determines the scalar point in the second factor, and the datum then uniquely determines the target. Thus [F1] gives $p:Y\to Y_0=Y\mathrm{/}D$, finite locally free, onto, with kernel pair $D$.

The map $(p,\text{structure}):Y\to Y_0{\times }_{k}K$ becomes an isomorphism after the faithfully flat cover $Y\to Y_0$: its pullback is $Y{\times }_{Y_0}Y=D$, which is exactly $Y{\times }_{k}K$, with the isomorphism supplied by the datum. Its inverse descends by [F2], so the map itself is an isomorphism. The same morphism descent gives uniqueness and descent of compatible morphisms. This uses the entire cocycle over the tensor algebras; automorphism invariance alone is insufficient for inseparable $K\mathrm{/}k$. AC is inherited from [F1]–[F2].