Lemma 40.15.3 (Gabber). Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i\in I}$ be an fpqc covering. Let $(V_ i/X_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to S\} $, see Descent, Definition 35.34.3. If each morphism $V_ i \to X_ i$ is ind-quasi-affine, then the descent datum is effective.
Proof. Being ind-quasi-affine is a property of morphisms of schemes which is preserved under any base change, see More on Morphisms, Lemma 37.66.6. Hence Descent, Lemma 35.36.2 applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{ X \to S\} $ flat surjective morphism of affines. Say $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi )$ be a descent datum relative to $X$ over $S$ and assume that $V \to X$ is ind-quasi-affine, in other words, $V$ is ind-quasi-affine.
Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with $U = X$ and $R = X \times _ S X$ and $s$, $t$, $c$ as usual. By Groupoids, Lemma 39.21.3 the pair $(V, \varphi )$ corresponds to a cartesian morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes. Let $u' \in U'$ be any point. By Groupoids, Lemmas 39.19.2, 39.19.3, and 39.19.4 we can choose $u' \in W \subset E \subset U'$ where $W$ is open and $R'$-invariant, and $E$ is set-theoretically $R'$-invariant and an intersection of a nonempty family of quasi-compact opens.
Translating back to $(V, \varphi )$, for any $v \in V$ we can find $v \in W \subset E \subset V$ with the following properties: (a) $W$ is open and $\varphi (W \times _ S X) = X \times _ S W$ and (b) $E$ an intersection of quasi-compact opens and $\varphi (E \times _ S X) = X \times _ S E$ set-theoretically. Here we use the notation $E \times _ S X$ to mean the inverse image of $E$ in $V \times _ S X$ by the projection morphism and similarly for $X \times _ S E$. By Lemma 40.15.2 this implies that $\varphi $ defines an isomorphism
\begin{align*} \Gamma (E, \mathcal{O}_ V|_ E) \otimes _ R A & = \Gamma (E \times _ S X, \mathcal{O}_{V \times _ S X}|_{E \times _ S X}) \\ & \to \Gamma (X \times _ S E, \mathcal{O}_{X \times _ S V}|_{X \times _ S E}) \\ & = A \otimes _ R \Gamma (E, \mathcal{O}_ V|_ E) \end{align*}
of $A \otimes _ R A$-algebras which we will call $\psi $. The cocycle condition for $\varphi $ translates into the cocycle condition for $\psi $ as in Descent, Definition 35.3.1 (details omitted). By Descent, Proposition 35.3.9 we find an $R$-algebra $R'$ and an isomorphism $\chi : R' \otimes _ R A \to \Gamma (E, \mathcal{O}_ V|_ E)$ of $A$-algebras, compatible with $\psi $ and the canonical descent datum on $R' \otimes _ R A$.
By Lemma 40.15.1 we obtain a canonical “embedding”
\[ j : (E, \mathcal{O}_ V|_ E) \longrightarrow \mathop{\mathrm{Spec}}(\Gamma (E, \mathcal{O}_ V|_ E)) = \mathop{\mathrm{Spec}}(R' \otimes _ R A) \]
of locally ringed spaces. The construction of this map is canonical and we get a commutative diagram
\[ \xymatrix{ & E \times _ S X \ar[rr]_\varphi \ar[ld] \ar[rd]^{j'} & & X \times _ S E \ar[rd] \ar[ld]_{j''} \\ E \ar[rd]^ j & & \mathop{\mathrm{Spec}}(R' \otimes _ R A \otimes _ R A) \ar[ld] \ar[rd] & & E \ar[ld]_ j \\ & \mathop{\mathrm{Spec}}(R' \otimes _ R A) \ar[rd] & & \mathop{\mathrm{Spec}}(R' \otimes _ R A) \ar[ld] \\ & & \mathop{\mathrm{Spec}}(R') } \]
where $j'$ and $j''$ come from the same construction applied to $E \times _ S X \subset V \times _ S X$ and $X \times _ S E \subset X \times _ S V$ via $\chi $ and the identifications used to construct $\psi $. It follows that $j(W)$ is an open subscheme of $\mathop{\mathrm{Spec}}(R' \otimes _ R A)$ whose inverse image under the two projections $\mathop{\mathrm{Spec}}(R' \otimes _ R A \otimes _ R A) \to \mathop{\mathrm{Spec}}(R' \otimes _ R A)$ are equal. By Descent, Lemma 35.13.6 we find an open $W_0 \subset \mathop{\mathrm{Spec}}(R')$ whose base change to $\mathop{\mathrm{Spec}}(A)$ is $j(W)$. Contemplating the diagram above we see that the descent datum $(W, \varphi |_{W \times _ S X})$ is effective. By Descent, Lemma 35.35.13 we see that our descent datum is effective. $\square$
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