Lemma 10.128.10 (Critère de platitude par fibres: locally nilpotent case). Let
\xymatrix{ S \ar[rr] & & S' \\ & R \ar[lu] \ar[ru] }
be a commutative diagram in the category of rings. Let I \subset R be a locally nilpotent ideal and M an S'-module. Assume
R \to S is of finite type,
R \to S' is of finite presentation,
M is a finitely presented S'-module,
M/IM is flat as a S/IS-module, and
M is flat as an R-module.
Then M is a flat S-module and S_\mathfrak q is flat and essentially of finite presentation over R for every \mathfrak q \subset S such that M \otimes _ S \kappa (\mathfrak q) is nonzero.
Proof.
If M \otimes _ S \kappa (\mathfrak q) is nonzero, then S' \otimes _ S \kappa (\mathfrak q) is nonzero and hence there exists a prime \mathfrak q' \subset S' lying over \mathfrak q (Lemma 10.18.6). Let \mathfrak p \subset R be the image of \mathfrak q in \mathop{\mathrm{Spec}}(R). Then I \subset \mathfrak p as I is locally nilpotent hence M/\mathfrak p M is flat over S/\mathfrak pS. Hence we may apply Lemma 10.128.9 to R_\mathfrak p \to S_\mathfrak q \to S'_{\mathfrak q'} and M_{\mathfrak q'}. We conclude that M_{\mathfrak q'} is flat over S and S_\mathfrak q is flat and essentially of finite presentation over R. Since \mathfrak q' was an arbitrary prime of S' we also see that M is flat over S (Lemma 10.39.18).
\square
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