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 étale 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.
$\square$

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