Effective fppf descent for separated locally quasi-finite morphisms

Affine refinement and reduction. Work on an affine open of the original target. Flat locally finitely presented maps are open by [F2], so finitely many affine source opens of the given covering have images covering this affine target. Their disjoint union gives a single affine faithfully flat finitely presented cover $X=\mathrm{Spec}B\to S=\mathrm{Spec}A$. Pull the datum back to it. Effectivity on such a refinement implies effectivity for the original datum: over each original member the two pulled-back schemes become compatibly isomorphic on its base change by $X\to S$, so [F4] descends the isomorphism and its inverse; uniqueness makes all cocycles agree. Local solutions on target affine opens likewise glue uniquely by [F4] and [F6]. It therefore suffices to treat this single affine cover. Write $V\to X$ for the scheme with its datum. If the target is empty, the unique solution is the empty scheme.

Saturated quasi-compact opens. Let $W^1\subseteq V$ be an affine open and let $\phi :V{\times }_{S}X\to X{\times }_{S}V$ be the descent transport. Set $W={\mathrm{pr}}_V(\phi (W^1{\times }_{S}X))$. It is open by [F2] and quasi-compact, since $W^1{\times }_{S}X$ is affine and its continuous image is quasi-compact. The diagonal identity gives $W^1\subseteq W$. The cocycle makes $W$ invariant under transport: applying two transports to a point has the same result as their composite. Points over a common base point may first be lifted to a common residue-field extension, since their residue-field tensor product is nonzero. This gives $\phi (W{\times }_{S}X)=X{\times }_{S}W$ and restricts the datum to $W$.

Quasi-affineness. The map $W\to X$ is separated and locally quasi-finite by [F5]. It is quasi-compact: a distinguished open in the affine $X$ pulls back to the nonvanishing locus of a global function on the quasi-compact $W$; choose a finite affine cover of $W$, where each such locus is principal affine. Thus $W\to X$ is quasi-finite. By [F3] it is an open subscheme of a finite $X$-scheme, which is affine, hence $W$ is quasi-affine. It is also separated over $S$, since $X\to S$ is affine.

Canonical affinization and flat base change. Put $C=\mathrm{\Gamma }(W,{\mathcal{O}}_W)$. The canonical map $j:W\to \mathrm{Spec}C$ is an open immersion. To verify this, embed $W$ into an affine $\mathrm{Spec}D$ and cover this quasi-compact open by finitely many principal opens $D_D(f_a)\subseteq W$. For any quasi-compact separated scheme, sections are the equalizer of the finite products of the coordinate rings of a finite affine cover and its affine pair intersections; intersections are affine because the separated diagonal is closed in the product of the affine charts. Localizing this equalizer is exact and commutes with its finite products, so $C_{f_a}=\mathrm{\Gamma }(W_{f_a},{\mathcal{O}}_W)=D_{f_a}$. Thus $j$ is an isomorphism on each $W_{f_a}$ onto $D_C(f_a)$ and is an open immersion globally. The same equalizer shows that, for any flat ring extension $B\to B^{\mathrm{\prime }}$, $C{\otimes }_B{B}^{\mathrm{\prime }}\cong \mathrm{\Gamma }(W_{B^{\mathrm{\prime }}},\mathcal{O})$: tensor preserves this finite equalizer and the affine intersection rings base change. These identifications respect restriction and composition.

Effective descent of the quasi-affine piece. The two projections $B\to B{\otimes }_{A}B$ are flat. Therefore step 3.1 turns the datum on $W$ into an algebra descent datum on $C$, satisfying its cocycle by functoriality. By [F6] it descends to an $A$-algebra $C_0$ with $B{\otimes }_A{C}_0\cong C$. The canonical open immersion $W\subseteq \mathrm{Spec}C$ is compatible with that datum. Let $p:\mathrm{Spec}C\to \mathrm{Spec}{C}_0$ be the faithfully flat finitely presented base change of $X\to S$. Its invariant open $W$ descends to the open $W_0=p(W)$: openness follows from [F2], and invariance says that any two points in one fibre either both lie in $W$ or both do not. The residue-field tensor argument of step 1.2 proves $p^{-1}(W_0)=W$. Hence $W_0{\times }_{S}X\cong W$, with its original datum. This proves quasi-affine effectivity here without an external descent lemma.

Gluing the descended pieces. The saturated opens of step 1.2 cover $V$. For two such opens, their intersection is an invariant open in each. Under the affine faithfully flat cover $W\to W_0$, invariance descends this intersection to an open of $W_0$ by the image argument of step 4.1, and likewise for the other piece. Their canonical upstairs identification descends with its inverse by [F4]. These identifications satisfy the cocycle by uniqueness of morphism descent. Glue the descended pieces by [F6], using their affine covers, to a scheme $V_0\to S$. Its pullback is the given $V$, compatibly with the datum.

Descent of separatedness. The diagonal of $V_0\to S$ becomes a closed immersion after the affine faithfully flat cover $X\to S$, because $V\to X$ is separated. Closed immersions descend here: on an affine open $T=\mathrm{Spec}R$ of the diagonal target, its pullback $T_X$ is affine and the upstairs closed subscheme is specified by an ideal $I\subseteq R{\otimes }_{A}B$ with the canonical descent datum. Module descent [F6] descends the inclusion $I\hookrightarrow R{\otimes }_{A}B$ to an ideal $J\subseteq R$; the ideal property is preserved by transport. The quotient $R\mathrm{/}J$ base changes to the upstairs quotient. By [F4] the resulting closed subscheme and the original diagonal fibre product are isomorphic: descend their compatible upstairs isomorphism and its inverse. Thus the diagonal is a closed immersion, and $V_0\to S$ is separated.

Descent of local quasi-finiteness. Restrict $V_0$ to an affine open $Z=\mathrm{Spec}D$ over the affine $S$. Its base change $Z_X$ is affine and locally quasi-finite over $X$, hence quasi-finite because it is quasi-compact. In particular $D{\otimes }_{A}B$ is finitely generated as a $B$-algebra. Finitely many tensor coefficients $d_a\in D$ of such generators generate an $A$-subalgebra $D^{\mathrm{\prime }}\subseteq D$ whose tensor with $B$ surjects onto $D{\otimes }_{A}B$; faithful flatness applied to the module $D\mathrm{/}{D}^{\mathrm{\prime }}$ gives $D=D^{\mathrm{\prime }}$. Thus $Z\to S$ is of finite type. For a point $s\in S$, choose a point of $X$ above it with residue field $L\mathrm{/}\kappa (s)$. The fibre algebra $(D{\otimes }_A\kappa (s)){\otimes }_{\kappa (s)}L$ is finite-dimensional over $L$: apply [F3] to the separated quasi-finite $Z_X\to X$, whose fibre is an open subscheme of a finite fibre. A finite-dimensional algebra is Artinian; its prime spectrum is finite and discrete, and every open subscheme is a product of some of its local factors, hence again finite-dimensional. Linear independence is preserved by field extension, so $D{\otimes }_A\kappa (s)$ is already finite-dimensional over $\kappa (s)$. Its localizations at primes are finite-dimensional, which is the pointwise quasi-finite condition of [F5]. Each $Z\to S$ is therefore quasi-finite, and $V_0\to S$ is locally quasi-finite. Undoing step 1.1 gives the entire original claim. AC is inherited from [F3], [F4], [F6] and the affine-cover choices.