The Stacks project

Lemma 10.46.1. Let $\varphi : R \to S$ be a surjective map with locally nilpotent kernel. Then $\varphi $ induces a homeomorphism of spectra and isomorphisms on residue fields. For any ring map $R \to R'$ the ring map $R' \to R' \otimes _ R S$ is surjective with locally nilpotent kernel.

Proof. By Lemma 10.17.7 the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is a homeomorphism onto the closed subset $V(\mathop{\mathrm{Ker}}(\varphi ))$. Of course $V(\mathop{\mathrm{Ker}}(\varphi )) = \mathop{\mathrm{Spec}}(R)$ because every prime ideal of $R$ contains every nilpotent element of $R$. This also implies the statement on residue fields. By right exactness of tensor product we see that $\mathop{\mathrm{Ker}}(\varphi )R'$ is the kernel of the surjective map $R' \to R' \otimes _ R S$. Hence the final statement by Lemma 10.32.3. $\square$

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