The Stacks project

Lemma 10.30.6. Let $R \to S$ be a ring map. The following are equivalent:

  1. The kernel of $R \to S$ consists of nilpotent elements.

  2. The minimal primes of $R$ are in the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$.

  3. The image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is dense in $\mathop{\mathrm{Spec}}(R)$.

Proof. Let $I = \mathop{\mathrm{Ker}}(R \to S)$. Note that $\sqrt{(0)} = \bigcap _{\mathfrak q \subset S} \mathfrak q$, see Lemma 10.17.2. Hence $\sqrt{I} = \bigcap _{\mathfrak q \subset S} R \cap \mathfrak q$. Thus $V(I) = V(\sqrt{I})$ is the closure of the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$. This shows that (1) is equivalent to (3). It is clear that (2) implies (3). Finally, assume (1). We may replace $R$ by $R/I$ and $S$ by $S/IS$ without affecting the topology of the spectra and the map. Hence the implication (1) $\Rightarrow $ (2) follows from Lemma 10.30.5. $\square$


Comments (2)

Comment #9517 by Goodluckthere on

The equality is better to be written as "the preimage of the nilradical of equals ."

Comment #9518 by Goodluckthere on

Also in the penultimate line, it's strange to take since is not inside and its image in is zero. I think what it means is that the map is an injective ring map, therefore by lemma \ref{https://stacks.math.columbia.edu/tag/00FK}, the corresponding spectra map hits all minimal prime ideals of . A minimal prime ideal of will contain the nilpotent elements of , therefore also by (1) thus corresponding to a minimal prime ideal of which in turn will have a preimage in .


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Your email address will not be published.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