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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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