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Tag 02JZ

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

Lemma 28.24.12. Let $h : X \to Y$ be a morphism of schemes over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Let $x \in X$ with $y = h(x) \in Y$. If $h$ is flat at $x$, then $$ \mathcal{G}\text{ flat over }S\text{ at }y \Leftrightarrow h^*\mathcal{G}\text{ flat over }S\text{ at }x. $$ In particular: If $h$ is surjective and flat, then $\mathcal{G}$ is flat over $S$, if and only if $h^*\mathcal{G}$ is flat over $S$. If $h$ is surjective and flat, and $X$ is flat over $S$, then $Y$ is flat over $S$.

Proof. You can prove this by applying Algebra, Lemma 10.38.9. Here is a direct proof. Let $s \in S$ be the image of $y$. Consider the local ring maps $\mathcal{O}_{S, s} \to \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$. By assumption the ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is faithfully flat, see Algebra, Lemma 10.38.17. Let $N = \mathcal{G}_y$. Note that $h^*\mathcal{G}_x = N \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x}$, see Sheaves, Lemma 6.26.4. Let $M' \to M$ be an injection of $\mathcal{O}_{S, s}$-modules. By the faithful flatness mentioned above we have \begin{align*} \mathop{\rm Ker}( M' \otimes_{\mathcal{O}_{S, s}} N \to M \otimes_{\mathcal{O}_{S, s}} N) \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \\ = \mathop{\rm Ker}( M' \otimes_{\mathcal{O}_{S, s}} N \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \to M \otimes_{\mathcal{O}_{S, s}} N \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x}) \end{align*} Hence the equivalence of the lemma follows from the second characterization of flatness in Algebra, Lemma 10.38.5. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4413–4427 (see updates for more information).

    \begin{lemma}
    \label{lemma-flat-permanence}
    Let $h : X \to Y$ be a morphism of schemes over $S$.
    Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$.
    Let $x \in X$ with $y = h(x) \in Y$. If $h$ is flat at $x$, then
    $$
    \mathcal{G}\text{ flat over }S\text{ at }y
    \Leftrightarrow
    h^*\mathcal{G}\text{ flat over }S\text{ at }x.
    $$
    In particular: If $h$ is surjective and flat, then
    $\mathcal{G}$ is flat over $S$, if and only if
    $h^*\mathcal{G}$ is flat over $S$. If $h$ is surjective and
    flat, and $X$ is flat over $S$, then $Y$ is flat over $S$.
    \end{lemma}
    
    \begin{proof}
    You can prove this by applying
    Algebra, Lemma \ref{algebra-lemma-flatness-descends-more-general}.
    Here is a direct proof. Let $s \in S$ be the image of $y$.
    Consider the local ring maps
    $\mathcal{O}_{S, s} \to \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$.
    By assumption the ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$
    is faithfully flat, see
    Algebra, Lemma \ref{algebra-lemma-local-flat-ff}.
    Let $N = \mathcal{G}_y$. Note that
    $h^*\mathcal{G}_x = N \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x}$, see
    Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback-modules}.
    Let $M' \to M$ be an injection of $\mathcal{O}_{S, s}$-modules.
    By the faithful flatness mentioned above we have
    \begin{align*}
    \Ker(
    M' \otimes_{\mathcal{O}_{S, s}} N \to M \otimes_{\mathcal{O}_{S, s}} N)
    \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \\
    =
    \Ker(
    M' \otimes_{\mathcal{O}_{S, s}} N
    \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x}
    \to
    M \otimes_{\mathcal{O}_{S, s}} N
    \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x})
    \end{align*}
    Hence the equivalence of the lemma follows from the second characterization
    of flatness in
    Algebra, Lemma \ref{algebra-lemma-flat}.
    \end{proof}

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