The Stacks project

Lemma 38.25.5. Let $A \to B$ be a local ring homomorphism of local rings which is essentially of finite type. Let $u : N \to M$ be a $B$-module map. If $N$ is a finite $B$-module, $M$ is flat over $A$, and $\overline{u} : N/\mathfrak m_ A N \to M/\mathfrak m_ A M$ is injective, then $u$ is $A$-universally injective, $N$ is of finite presentation over $B$, and $N$ is flat over $A$.

Proof. Let $A \to A^ h$ be the henselization of $A$. Let $B'$ be the localization of $B \otimes _ A A^ h$ at the maximal ideal $\mathfrak m_ B \otimes A^ h + B \otimes \mathfrak m_{A^ h}$. Since $B \to B'$ is flat (hence faithfully flat, see Algebra, Lemma 10.39.17), we may replace $A \to B$ with $A^ h \to B'$, the module $M$ by $M \otimes _ B B'$, the module $N$ by $N \otimes _ B B'$, and $u$ by $u \otimes \text{id}_{B'}$, see Algebra, Lemmas 10.83.2 and 10.39.9. Thus we may assume that $A$ is a henselian local ring. In this case our lemma follows from the more general Lemma 38.25.4. $\square$

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