Lemma 10.156.5. Let $A \to B$ and $A \to C$ be local homomorphisms of local rings. If $A \to C$ is integral and either $\kappa (\mathfrak m_ C)/\kappa (\mathfrak m_ A)$ or $\kappa (\mathfrak m_ B)/\kappa (\mathfrak m_ A)$ is purely inseparable, then $D = B \otimes _ A C$ is a local ring and $B \to D$ and $C \to D$ are local.

**Proof.**
Any maximal ideal of $D$ lies over the maximal ideal of $B$ by going up for the integral ring map $B \to D$ (Lemma 10.36.22). Now $D/\mathfrak m_ B D = \kappa (\mathfrak m_ B) \otimes _ A C = \kappa (\mathfrak m_ B) \otimes _{\kappa (\mathfrak m_ A)} C/\mathfrak m_ A C$. The spectrum of $C/\mathfrak m_ A C$ consists of a single point, namely $\mathfrak m_ C$. Thus the spectrum of $D/\mathfrak m_ B D$ is the same as the spectrum of $\kappa (\mathfrak m_ B) \otimes _{\kappa (\mathfrak m_ A)} \kappa (\mathfrak m_ C)$ which is a single point by our assumption that either $\kappa (\mathfrak m_ C)/\kappa (\mathfrak m_ A)$ or $\kappa (\mathfrak m_ B)/\kappa (\mathfrak m_ A)$ is purely inseparable. This proves that $D$ is local and that the ring maps $B \to D$ and $C \to D$ are local.
$\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)