The Stacks project

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. $\square$

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