The Stacks project

Lemma 37.16.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume

  1. $X$ is locally of finite presentation over $S$,

  2. $\mathcal{F}$ an $\mathcal{O}_ X$-module of finite presentation,

  3. $\mathcal{F}$ is flat over $S$, and

  4. $Y$ is locally of finite type over $S$.

Then the set

\[ U = \{ x \in X \mid \mathcal{F} \text{ flat at }x \text{ over }Y\} . \]

is open in $X$ and its formation commutes with arbitrary base change: If $S' \to S$ is a morphism of schemes, and $U'$ is the set of points of $X' = X \times _ S S'$ where $\mathcal{F}' = \mathcal{F} \times _ S S'$ is flat over $Y' = Y \times _ S S'$, then $U' = U \times _ S S'$.

Proof. By Morphisms, Lemma 29.21.11 the morphism $f$ is locally of finite presentation. Hence $U$ is open by Theorem 37.15.1. Because we have assumed that $\mathcal{F}$ is flat over $S$ we see that Theorem 37.16.2 implies

\[ U = \{ x \in X \mid \mathcal{F}_ s \text{ flat at }x \text{ over }Y_ s\} . \]

where $s$ always denotes the image of $x$ in $S$. (This description also works trivially when $\mathcal{F}_ x = 0$.) Moreover, the assumptions of the lemma remain in force for the morphism $f' : X' \to Y'$ and the sheaf $\mathcal{F}'$. Hence $U'$ has a similar description. In other words, it suffices to prove that given $s' \in S'$ mapping to $s \in S$ we have

\[ \{ x' \in X'_{s'} \mid \mathcal{F}'_{s'} \text{ flat at }x' \text{ over }Y'_{s'}\} \]

is the inverse image of the corresponding locus in $X_ s$. This is true by Lemma 37.15.2 because in the cartesian diagram

\[ \xymatrix{ X'_{s'} \ar[d] \ar[r] & X_ s \ar[d] \\ Y'_{s'} \ar[r] & Y_ s } \]

the horizontal morphisms are flat as they are base changes by the flat morphism $\mathop{\mathrm{Spec}}(\kappa (s')) \to \mathop{\mathrm{Spec}}(\kappa (s))$. $\square$


Comments (0)


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Your email address will not be published.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 05VJ. Beware of the difference between the letter 'O' and the digit '0'.