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Tag 0246

34.35. Descending quasi-affine morphisms

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

Lemma 34.35.1. Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering, see Topologies, Definition 33.9.1. 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, Lemma 28.12.5. Hence Lemma 34.33.2 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 $X = \mathop{\rm Spec}(A)$ and $S = \mathop{\rm Spec}(R)$ 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 28.12.3 this means that $$V \longrightarrow \underline{\mathop{\rm Spec}}_X(\pi_*\mathcal{O}_V) = W$$ 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 29.5.2). 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 34.34.1 (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 34.20.18.

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 28.24.11 this subset $U \subset U'$ is open and we win. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 8241–8341 (see updates for more information).

\section{Descending quasi-affine morphisms}
\label{section-quasi-affine}

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

\begin{lemma}
\label{lemma-quasi-affine}
Let $S$ be a scheme.
Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering, see
Topologies, Definition \ref{topologies-definition-fpqc-covering}.
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.
\end{lemma}

\begin{proof}
Being quasi-affine is a property of morphisms of schemes
which is preserved under any base change, see
Morphisms, Lemma \ref{morphisms-lemma-base-change-quasi-affine}.
Hence Lemma \ref{lemma-descending-types-morphisms} 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 $X = \Spec(A)$ and $S = \Spec(R)$ 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.

\medskip\noindent
According to Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine}
this means that
$$V \longrightarrow \underline{\Spec}_X(\pi_*\mathcal{O}_V) = W$$
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 \ref{coherent-lemma-flat-base-change-cohomology}).
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 \ref{lemma-affine}
(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 \ref{lemma-descending-property-affine}.

\medskip\noindent
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 \ref{morphisms-lemma-fpqc-quotient-topology}
this subset $U \subset U'$ is open and we win.
\end{proof}

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