The Stacks project

Lemma 10.30.3. Let $\varphi : R \to S$ be a ring map. The following are equivalent:

  1. The map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is surjective.

  2. For any ideal $I \subset R$ the inverse image of $\sqrt{IS}$ in $R$ is equal to $\sqrt{I}$.

  3. For any radical ideal $I \subset R$ the inverse image of $IS$ in $R$ is equal to $I$.

  4. For every prime $\mathfrak p$ of $R$ the inverse image of $\mathfrak p S$ in $R$ is $\mathfrak p$.

In this case the same is true after any base change: Given a ring map $R \to R'$ the ring map $R' \to R' \otimes _ R S$ has the equivalent properties (1), (2), (3) as well.

Proof. If $J \subset S$ is an ideal, then $\sqrt{\varphi ^{-1}(J)} = \varphi ^{-1}(\sqrt{J})$. This shows that (2) and (3) are equivalent. The implication (3) $\Rightarrow $ (4) is immediate. If $I \subset R$ is a radical ideal, then Lemma 10.17.2 guarantees that $I = \bigcap _{I \subset \mathfrak p} \mathfrak p$. Hence (4) $\Rightarrow $ (2). By Lemma 10.17.9 we have $\mathfrak p = \varphi ^{-1}(\mathfrak p S)$ if and only if $\mathfrak p$ is in the image. Hence (1) $\Leftrightarrow $ (4). Thus (1), (2), (3), and (4) are equivalent.

Assume (1) holds. Let $R \to R'$ be a ring map. Let $\mathfrak p' \subset R'$ be a prime ideal lying over the prime $\mathfrak p$ of $R$. To see that $\mathfrak p'$ is in the image of $\mathop{\mathrm{Spec}}(R' \otimes _ R S) \to \mathop{\mathrm{Spec}}(R')$ we have to show that $(R' \otimes _ R S) \otimes _{R'} \kappa (\mathfrak p')$ is not zero, see Lemma 10.17.9. But we have

\[ (R' \otimes _ R S) \otimes _{R'} \kappa (\mathfrak p') = S \otimes _ R \kappa (\mathfrak p) \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak p') \]

which is not zero as $S \otimes _ R \kappa (\mathfrak p)$ is not zero by assumption and $\kappa (\mathfrak p) \to \kappa (\mathfrak p')$ is an extension of fields. $\square$

Comments (2)

Comment #4244 by Kazuki Masugi on

"Hence (3) (2)" should be "Hence (4) (2)"

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Your email address will not be published.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