Affineness and finiteness of morphisms descend under fppf base change

Work over an affine open $U=\mathrm{Spec}A\subset S$. Choose finitely many affine opens covering $S_U^{\mathrm{\prime }}$; their disjoint union is affine, $V=\mathrm{Spec}B$, and is a faithfully flat affine cover of $U$. The pullback $X_V$ is affine over $V$, say $\mathrm{Spec}C$. Its two pullbacks to $V{\times }_{U}V$ have the canonical isomorphism coming from the scheme $X_U$, and satisfy the cocycle. By [F1], $C$ descends to an $A$-algebra $D$, with $D{\otimes }_{A}B\cong C$ as algebras with datum.

This algebra isomorphism gives a compatible isomorphism $X_V\cong (\mathrm{Spec}D{)}_V$. The map from $X_V$ to $\mathrm{Spec}D$ descends along $X_V\to X_U$ by the morphism part of [F1], and the inverse map descends along $(\mathrm{Spec}D{)}_V\to \mathrm{Spec}D$. The resulting two maps are inverse because their composites become identities after the faithful cover and uniqueness in [F1] detects equality. Thus $X_U\cong \mathrm{Spec}D$, proving affineness of $f$ locally on its target and hence globally.

If $f_{S^{\mathrm{\prime }}}$ is finite, $C$ is a finite $B$-module. By [F2], $D$ is a finite $A$-module: equivalently express finitely many generators of $D{\otimes }_{A}B$ using finitely many tensor coefficients in $D$, let $D_0$ be their $A$-span, and use faithfulness to deduce $D\mathrm{/}{D}_0=0$. Hence $X_U\to U$ is finite. Finiteness is affine local on the target by this module description, so $f$ is finite. AC is inherited from [F1]–[F2]; quasi-compactness supplies the finite affine subcover used in step 1.1.