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Tag 0AP7

Chapter 36: More on Morphisms > Section 36.55: Ind-quasi-affine morphisms

Lemma 36.55.2. The property of being ind-quasi-affine is stable under base change.

Proof. Let $f : X \to Y$ be an ind-quasi-affine morphism. Let $Z$ be an affine scheme and let $Z \to Y$ be a morphism. To show: $Z \times_Y X$ is ind-quasi-affine. Let $W \subset Z \times_Y X$ be a quasi-compact open. We can find finitely many affine opens $V_1, \ldots, V_n$ of $Y$ and finitely many quasi-compact opens $U_i \subset f^{-1}(V_i)$ such that $Z$ maps into $\bigcup V_i$ and $W$ maps into $\bigcup U_i$. Then we may replace $Y$ by $\bigcup V_i$ and $X$ by $\bigcup W_i$. In this case $f^{-1}(V_i)$ is quasi-compact open (details omitted; use that $f$ is separated) and hence quasi-affine. Thus now $X \to Y$ is a quasi-affine morphism (Morphisms, Lemma 28.12.3) and the result follows from the fact that the base change of a quasi-affine morphism is quasi-affine (Morphisms, Lemma 28.12.5). $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 16479–16482 (see updates for more information).

    \begin{lemma}
    \label{lemma-base-change-ind-quasi-affine}
    The property of being ind-quasi-affine is stable under base change.
    \end{lemma}
    
    \begin{proof}
    Let $f : X \to Y$ be an ind-quasi-affine morphism. Let $Z$ be an affine
    scheme and let $Z \to Y$ be a morphism. To show: $Z \times_Y X$ is
    ind-quasi-affine. Let $W \subset Z \times_Y X$ be a quasi-compact open.
    We can find finitely many affine opens $V_1, \ldots, V_n$ of $Y$
    and finitely many quasi-compact opens $U_i \subset f^{-1}(V_i)$
    such that $Z$ maps into $\bigcup V_i$ and $W$ maps into $\bigcup U_i$.
    Then we may replace $Y$ by $\bigcup V_i$ and $X$ by $\bigcup W_i$.
    In this case $f^{-1}(V_i)$ is quasi-compact open (details omitted; use
    that $f$ is separated) and hence quasi-affine. Thus now $X \to Y$
    is a quasi-affine morphism (Morphisms, Lemma
    \ref{morphisms-lemma-characterize-quasi-affine}) and the result
    follows from the fact that the base change of a quasi-affine morphism
    is quasi-affine (Morphisms, Lemma
    \ref{morphisms-lemma-base-change-quasi-affine}).
    \end{proof}

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