The Stacks project

Lemma 10.101.8 (Critère de platitude par fibres: 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 nilpotent ideal and $M$ an $S'$-module. Assume

  1. The module $M/IM$ is a flat $S/IS$-module.

  2. The module $M$ is a flat $R$-module.

Then $M$ is a flat $S$-module and $S_{\mathfrak q}$ is flat over $R$ for every $\mathfrak q \subset S$ such that $M \otimes _ S \kappa (\mathfrak q)$ is nonzero.

Proof. As $M$ is flat over $R$ tensoring with the short exact sequence $0 \to I \to R \to R/I \to 0$ gives a short exact sequence

\[ 0 \to I \otimes _ R M \to M \to M/IM \to 0. \]

Note that $I \otimes _ R M \to IS \otimes _ S M$ is surjective. Combined with the above this means both maps in

\[ I \otimes _ R M \to IS \otimes _ S M \to M \]

are injective. Hence $\text{Tor}_1^ S(IS, M) = 0$ (see Remark 10.75.9) and we conclude that $M$ is a flat $S$-module by Lemma 10.99.8. To finish we need to show that $S_{\mathfrak q}$ is flat over $R$ for any prime $\mathfrak q \subset S$ such that $M \otimes _ S \kappa (\mathfrak q)$ is nonzero. This follows from Lemma 10.39.15 and 10.39.10. $\square$

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