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Tag 081H

Chapter 28: Morphisms of Schemes > Section 28.24: Flat morphisms

Lemma 28.24.14. Let $f : X \to Y$ be a flat morphism of schemes. Let $V \subset Y$ be a retrocompact open which is scheme theoretically dense. Then $f^{-1}V$ is scheme theoretically dense in $X$.

Proof. We will use the characterization of Lemma 28.7.5. We have to show that for any open $U \subset X$ the map $\mathcal{O}_X(U) \to \mathcal{O}_X(U \cap f^{-1}V)$ is injective. It suffices to prove this when $U$ is an affine open which maps into an affine open $W \subset Y$. Say $W = \mathop{\rm Spec}(A)$ and $U = \mathop{\rm Spec}(B)$. Then $V \cap W = D(f_1) \cup \ldots \cup D(f_n)$ for some $f_i \in A$, see Algebra, Lemma 10.28.1. Thus we have to show that $B \to B_{f_1} \times \ldots \times B_{f_n}$ is injective. We are given that $A \to A_{f_1} \times \ldots \times A_{f_n}$ is injective and that $A \to B$ is flat. Since $B_{f_i} = A_{f_i} \otimes_A B$ we win. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4477–4482 (see updates for more information).

    \begin{lemma}
    \label{lemma-flat-morphism-scheme-theoretically-dense-open}
    Let $f : X \to Y$ be a flat morphism of schemes. Let $V \subset Y$ be
    a retrocompact open which is scheme theoretically dense. Then $f^{-1}V$
    is scheme theoretically dense in $X$.
    \end{lemma}
    
    \begin{proof}
    We will use the characterization of
    Lemma \ref{lemma-characterize-scheme-theoretically-dense}.
    We have to show that for any open $U \subset X$ the map
    $\mathcal{O}_X(U) \to \mathcal{O}_X(U \cap f^{-1}V)$ is injective.
    It suffices to prove this when $U$ is an affine open which maps into
    an affine open $W \subset Y$. Say $W = \Spec(A)$ and $U = \Spec(B)$.
    Then $V \cap W = D(f_1) \cup \ldots \cup D(f_n)$ for some
    $f_i \in A$, see
    Algebra, Lemma \ref{algebra-lemma-qc-open}.
    Thus we have to show that
    $B \to B_{f_1} \times \ldots \times B_{f_n}$ is injective.
    We are given that $A \to A_{f_1} \times \ldots \times A_{f_n}$ is injective
    and that $A \to B$ is flat. Since $B_{f_i} = A_{f_i} \otimes_A B$ we win.
    \end{proof}

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