The Stacks project

Lemma 76.24.3. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $X$ be an algebraic space locally of finite presentation over $S = \mathop{\mathrm{Spec}}(A)$. For $n \geq 1$ set $S_ n = \mathop{\mathrm{Spec}}(A/I^ n)$ and $X_ n = S_ n \times _ S X$. Let $\mathcal{F}$ be coherent $\mathcal{O}_ X$-module. If for every $n \geq 1$ the pullback $\mathcal{F}_ n$ of $\mathcal{F}$ to $X$ is flat over $S_ n$, then the (open) locus where $\mathcal{F}$ is flat over $X$ contains the inverse image of $V(I)$ under $X \to S$.

Proof. The locus where $\mathcal{F}$ is flat over $S$ is open in $|X|$ by Theorem 76.22.1. The statement is insensitive to replacing $X$ by the members of an ├ętale covering, hence we may assume $X$ is an affine scheme. In this case the result follows immediately from Algebra, Lemma 10.99.11. Some details omitted. $\square$

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Your email address will not be published.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 08VP. Beware of the difference between the letter 'O' and the digit '0'.