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Tag 02L5

Chapter 34: Descent > Section 34.20: Properties of morphisms local in the fpqc topology on the target

Lemma 34.20.16. The property $\mathcal{P}(f) =$''$f$ is affine'' is fpqc local on the base.

Proof. A base change of an affine morphism is affine, see Morphisms, Lemma 28.11.8. Being affine is Zariski local on the base, see Morphisms, Lemma 28.11.3. Finally, let $g : S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is affine. In other words, $X'$ is affine, say $X' = \mathop{\rm Spec}(A')$. Also write $S = \mathop{\rm Spec}(R)$ and $S' = \mathop{\rm Spec}(R')$. We have to show that $X$ is affine.

By Lemmas 34.20.1 and 34.20.5 we see that $X \to S$ is separated and quasi-compact. Thus $f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras, see Schemes, Lemma 25.24.1. Hence $f_*\mathcal{O}_X = \widetilde{A}$ for some $R$-algebra $A$. In fact $A = \Gamma(X, \mathcal{O}_X)$ of course. Also, by flat base change (see for example Cohomology of Schemes, Lemma 29.5.2) we have $g^*f_*\mathcal{O}_X = f'_*\mathcal{O}_{X'}$. In other words, we have $A' = R' \otimes_R A$. Consider the canonical morphism $$ X \longrightarrow \mathop{\rm Spec}(A) $$ over $S$ from Schemes, Lemma 25.6.4. By the above the base change of this morphism to $S'$ is an isomorphism. Hence it is an isomorphism by Lemma 34.20.15. Therefore Lemma 34.19.4 applies and we win. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 4906–4910 (see updates for more information).

    \begin{lemma}
    \label{lemma-descending-property-affine}
    The property $\mathcal{P}(f) =$``$f$ is affine''
    is fpqc local on the base.
    \end{lemma}
    
    \begin{proof}
    A base change of an affine morphism is affine, see
    Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}.
    Being affine is Zariski local on the base, see
    Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}.
    Finally, let
    $g : S' \to S$ be a flat surjective morphism of affine schemes,
    and let $f : X \to S$ be a morphism. Assume that the base change
    $f' : X' \to S'$ is affine. In other words, $X'$ is affine, say
    $X' = \Spec(A')$. Also write $S = \Spec(R)$
    and $S' = \Spec(R')$. We have to show that $X$ is affine.
    
    \medskip\noindent
    By Lemmas \ref{lemma-descending-property-quasi-compact}
    and \ref{lemma-descending-property-separated} we see that
    $X \to S$ is separated and quasi-compact. Thus
    $f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras,
    see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
    Hence $f_*\mathcal{O}_X = \widetilde{A}$ for some $R$-algebra $A$.
    In fact $A = \Gamma(X, \mathcal{O}_X)$ of course.
    Also, by flat base change
    (see for example
    Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology})
    we have $g^*f_*\mathcal{O}_X = f'_*\mathcal{O}_{X'}$.
    In other words, we have $A' = R' \otimes_R A$.
    Consider the canonical morphism
    $$
    X \longrightarrow \Spec(A)
    $$
    over $S$ from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}.
    By the above the base change of this morphism to $S'$ is an isomorphism.
    Hence it is an isomorphism by
    Lemma \ref{lemma-descending-property-isomorphism}.
    Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
    \end{proof}

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