The Stacks project

37.15 Openness of the flat locus

This result takes some work to prove, and (perhaps) deserves its own section. Here it is.

reference

Theorem 37.15.1. Let $S$ be a scheme. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module which is locally of finite presentation. Then

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

is open in $X$.

Proof. We may test for openness locally on $X$ hence we may assume that $f$ is a morphism of affine schemes. In this case the theorem is exactly Algebra, Theorem 10.129.4. $\square$

Lemma 37.15.2. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

be a cartesian diagram of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x' \in X'$ with images $x = g'(x')$ and $s' = g'(x')$.

  1. If $\mathcal{F}$ is flat over $S$ at $x$, then $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$.

  2. If $g$ is flat at $s'$ and $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$, then $\mathcal{F}$ is flat over $S$ at $x$.

In particular, if $g$ is flat, $f$ is locally of finite presentation, and $\mathcal{F}$ is locally of finite presentation, then formation of the open subset of Theorem 37.15.1 commutes with base change.

Proof. Consider the commutative diagram of local rings

\[ \xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{S', s'} \ar[u] & \mathcal{O}_{S, s} \ar[l] \ar[u] } \]

Note that $\mathcal{O}_{X', x'}$ is a localization of $\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S', s'}$, and that $((g')^*\mathcal{F})_{x'}$ is equal to $\mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \mathcal{O}_{X', x'}$. Hence the lemma follows from Algebra, Lemma 10.100.1. $\square$


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Your email address will not be published.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 0398. Beware of the difference between the letter 'O' and the digit '0'.