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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)