Lemma 10.46.10. Let $\varphi : R \to S$ be a ring map. Assume

1. $\varphi$ is integral,

2. $\varphi$ induces an bijective map of spectra,

3. $\varphi$ induces purely inseparable residue field extensions.

Then $\varphi$ induces a homeomorphism on spectra and for any ring map $R \to R'$ properties (1), (2), (3) are true for $R' \to R' \otimes _ R S$.

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