Lemma 15.19.5. In Situation 15.19.1. Let $I \subset R$ be an ideal. Assume

1. $R$ is a Noetherian ring,

2. $S$ is a Noetherian ring,

3. $M$ is a finite $S$-module, and

4. for each $n \geq 1$ and any prime $\mathfrak q \in V(J + IS)$ the module $(M/I^ n M)_{\mathfrak q}$ is flat over $R/I^ n$.

Then (15.19.1.1) holds for $(R, I)$, i.e., for every prime $\mathfrak q \in V(J + IS)$ the localization $M_{\mathfrak q}$ is flat over $R$.

Proof. Let $\mathfrak q \in V(J + IS)$. Then Algebra, Lemma 10.99.11 applied to $R \to S_{\mathfrak q}$ and $M_{\mathfrak q}$ implies that $M_{\mathfrak q}$ is flat over $R$. $\square$

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