The Stacks project

Lemma 29.24.13. 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{\mathrm{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{\mathrm{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. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 29.24: Flat morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02JZ. Beware of the difference between the letter 'O' and the digit '0'.