Scheme morphisms satisfy fppf descent

For any faithfully flat ring map $A\to B$, $A\to B\rightrightarrows B{\otimes }_{A}B$ is an equalizer. Check exactness after the faithful flat tensor extension by $B$. The extended sequence is $0\to B\to B{\otimes }_{A}B\to B{\otimes }_{A}B{\otimes }_{A}B$, with the first arrow $b\mapsto b\otimes 1$. That arrow is split by multiplication. If $x=\sum b_i\otimes c_i$ has equal images in the triple tensor product, applying multiplication to its first two factors gives $x=(\sum b_i{c}_i)\otimes 1$, proving exactness. Flatness preserves kernels and cokernels, and their vanishing descends by [F1]; thus the original sequence is exact.

For an affine open $V\subset Z$, the open $W=(f^{\mathrm{\prime }}{)}^{-1}(V)$ is stable under the two relation projections. Any two points of $X^{\mathrm{\prime }}$ over the same point of $X$ lift to a common point of the fibre product: their residue-field tensor product over the base residue field is nonzero. Hence each fibre lies entirely in $W$ or entirely outside it. The image $U=p(W)$ is open by [F1], and $W=p^{-1}(U)$. Such $U$ cover $X$ as $V$ ranges over an affine target cover.

On an affine open $T=\mathrm{Spec}A\subset U$, choose a finite affine open cover of $p^{-1}(T)$; quasi-compactness gives finiteness. Its disjoint union is affine, say $\mathrm{Spec}B$, and maps faithfully flat to $T$, since restriction of $p$ to each open remains flat and the union is onto. The restriction of $f^{\mathrm{\prime }}$ to this cover, with affine target $V$, gives a ring map $\mathcal{O}(V)\to B$ whose two composites into $B{\otimes }_{A}B$ agree. By step 1.1 it takes values in $A$, yielding a unique morphism $T\to V$. The same equalizer shows that its pullback agrees with $f^{\mathrm{\prime }}$ on the whole $p^{-1}(T)$, by checking on these affine opens.

The descended morphisms agree on overlaps: their pullbacks agree, and the equalizer argument on affine source covers detects equality. Thus they glue uniquely on $X$. Conversely every pulled-back morphism satisfies compatibility. For a family of fppf covers, work over an affine open of $X$, take affine source opens whose open images cover it, and select finitely many by quasi-compactness. Their disjoint union is a faithfully flat affine refinement of the family to which the same argument applies. This proves the sheaf condition and uniqueness for represented functors. AC is inherited from [F1] and the scheme-affine-cover suppliers.