Lemma 115.4.2. Let $\varphi : R \to S$ be a ring map. If

for any $x \in S$ there exists $n > 0$ such that $x^ n$ is in the image of $\varphi $, and

for any $x \in \mathop{\mathrm{Ker}}(\varphi )$ there exists $n > 0$ such that $x^ n = 0$,

then $\varphi $ induces a homeomorphism on spectra. Given a prime number $p$ such that

$S$ is generated as an $R$-algebra by elements $x$ such that there exists an $n > 0$ with $x^{p^ n} \in \varphi (R)$ and $p^ nx \in \varphi (R)$, and

the kernel of $\varphi $ is generated by nilpotent elements,

then (1) and (2) hold, and for any ring map $R \to R'$ the ring map $R' \to R' \otimes _ R S$ also satisfies (a), (b), (1), and (2) and in particular induces a homeomorphism on spectra.

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