The Stacks project

35.38 Descending quasi-affine morphisms

In this section we show that “quasi-affine morphisms satisfy descent for fpqc-coverings”. Here is the formal statement.

Lemma 35.38.1. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i\in I}$ be an fpqc covering, see Topologies, Definition 34.9.1. Let $(V_ i/X_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to S\} $. If each morphism $V_ i \to X_ i$ is quasi-affine, then the descent datum is effective.

Proof. Being quasi-affine is a property of morphisms of schemes which is preserved under any base change, see Morphisms, Lemmas 29.13.3 and 29.13.5. Hence 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 $\pi : V \to X$ is quasi-affine.

According to Morphisms, Lemma 29.13.3 this means that

\[ V \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ X(\pi _*\mathcal{O}_ V) = W \]

is a quasi-compact open immersion of schemes over $X$. The projections $\text{pr}_ i : X \times _ S X \to X$ are flat and hence we have

\[ \text{pr}_0^*\pi _*\mathcal{O}_ V = (\pi \times \text{id}_ X)_*\mathcal{O}_{V \times _ S X}, \quad \text{pr}_1^*\pi _*\mathcal{O}_ V = (\text{id}_ X \times \pi )_*\mathcal{O}_{X \times _ S V} \]

by flat base change (Cohomology of Schemes, Lemma 30.5.2). Thus the isomorphism $\varphi : V \times _ S X \to X \times _ S V$ (which is an isomorphism over $X \times _ S X$) induces an isomorphism of quasi-coherent sheaves of algebras

\[ \varphi ^\sharp : \text{pr}_0^*\pi _*\mathcal{O}_ V \longrightarrow \text{pr}_1^*\pi _*\mathcal{O}_ V \]

on $X \times _ S X$. The cocycle condition for $\varphi $ implies the cocycle condition for $\varphi ^\sharp $. Another way to say this is that it produces a descent datum $\varphi '$ on the affine scheme $W$ relative to $X$ over $S$, which moreover has the property that the morphism $V \to W$ is a morphism of descent data. Hence by Lemma 35.37.1 (or by effectivity of descent for quasi-coherent algebras) we obtain a scheme $U' \to S$ with an isomorphism $(W, \varphi ') \cong (X \times _ S U', can)$ of descent data. We note in passing that $U'$ is affine by Lemma 35.23.18.

And now we can think of $V$ as a (quasi-compact) open $V \subset X \times _ S U'$ with the property that it is stable under the descent datum

\[ can : X \times _ S U' \times _ S X \to X \times _ S X \times _ S U', (x_0, u', x_1) \mapsto (x_0, x_1, u'). \]

In other words $(x_0, u') \in V \Rightarrow (x_1, u') \in V$ for any $x_0, x_1, u'$ mapping to the same point of $S$. Because $X \to S$ is surjective we immediately find that $V$ is the inverse image of a subset $U \subset U'$ under the morphism $X \times _ S U' \to U'$. Because $X \to S$ is quasi-compact, flat and surjective also $X \times _ S U' \to U'$ is quasi-compact flat and surjective. Hence by Morphisms, Lemma 29.25.12 this subset $U \subset U'$ is open and we win. $\square$

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