Lemma 10.18.6. Let $\varphi : R \to S$ be a ring map. Let $\mathfrak p$ be a prime of $R$. The following are equivalent
$\mathfrak p$ is in the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$,
$S \otimes _ R \kappa (\mathfrak p) \not= 0$,
$S_{\mathfrak p}/\mathfrak p S_{\mathfrak p} \not= 0$,
$(S/\mathfrak pS)_{\mathfrak p} \not= 0$, and
$\mathfrak p = \varphi ^{-1}(\mathfrak pS)$.
Proof. We have already seen the equivalence of the first two in Remark 10.18.5. The others are just reformulations of this. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: