The Stacks project

Lemma 10.30.5. Let $R \subset S$ be an injective ring map. Then $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ hits all the minimal primes.

Proof. Let $\mathfrak p \subset R$ be a minimal prime. In this case $R_{\mathfrak p}$ has a unique prime ideal. Hence it suffices to show that $S_{\mathfrak p}$ is not zero. And this follows from the fact that localization is exact, see Proposition 10.9.12. $\square$


Comments (6)

Comment #4211 by Aaron on

I think it should be: hits all the minimal primes.

Comment #6009 by Ivan on

could you please explain why it suffices to show that is not zero and we get a such that ? in additon, here is a ring?

Comment #6010 by Fan on

See the section on localisation (00CM)

Comment #6012 by Ivan on

but 00CM doesn't mention about it. even if is exact and is non-zero, is it possible that ? why is non-zero and then there exist such that ?

Comment #6014 by Ivan on

I think I got it, thanks


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Your email address will not be published.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 00FK. Beware of the difference between the letter 'O' and the digit '0'.