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

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

2. 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

1. $S$ 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

2. 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.

Proof. This is a combination of Algebra, Lemmas 10.46.3 and 10.46.7. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00I5. Beware of the difference between the letter 'O' and the digit '0'.