The Stacks project

Lemma 10.36.20. Suppose $R \to S$ is integral. Let $\mathfrak q, \mathfrak q' \in \mathop{\mathrm{Spec}}(S)$ be distinct primes having the same image in $\mathop{\mathrm{Spec}}(R)$. Then neither $\mathfrak q \subset \mathfrak q'$ nor $\mathfrak q' \subset \mathfrak q$.

Proof. Let $\mathfrak p \subset R$ be the image. By Remark 10.18.5 the primes $\mathfrak q, \mathfrak q'$ correspond to ideals in $S \otimes _ R \kappa (\mathfrak p)$. Thus the lemma follows from Lemma 10.36.19. $\square$

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