The Stacks project

Lemma 10.46.9. Let $\varphi : R \to S$ be a ring map. Assume

  1. $\varphi $ is integral,

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

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

Then $\varphi $ induces a homeomorphism from $\mathop{\mathrm{Spec}}(S)$ onto a closed subset of $\mathop{\mathrm{Spec}}(R)$ and for any ring map $R \to R'$ properties (1), (2), (3) are true for $R' \to R' \otimes _ R S$.

Proof. The map on spectra is closed by Lemmas 10.41.6 and 10.36.22. The properties are preserved under base change by Lemmas 10.46.8 and 10.36.13. $\square$

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