The Stacks project

Lemma 38.25.12. Let $A$ be a local ring. Let $I \subset A$ be an ideal. Let $U \subset \mathop{\mathrm{Spec}}(A)$ be quasi-compact open. Let $M$ be an $A$-module. Assume that

  1. $M/IM$ is flat over $A/I$,

  2. $M$ is flat over $U$,

Then $M/I_2M$ is flat over $A/I_2$ where $I_2 = \mathop{\mathrm{Ker}}(I \to \Gamma (U, I/I^2))$.

Proof. It suffices to show that $M \otimes _ A I/I_2 \to IM/I_2M$ is injective, see Algebra, Lemma 10.99.9. This is true over $U$ by assumption (2). Thus it suffices to show that $M \otimes _ A I/I_2$ injects into its sections over $U$. We have $M \otimes _ A I/I_2 = M/IM \otimes _ A I/I_2$ and $M/IM$ is a filtered colimit of finite free $A/I$-modules (Algebra, Theorem 10.81.4). Hence it suffices to show that $I/I_2$ injects into its sections over $U$, which follows from the construction of $I_2$. $\square$

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