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
Comments (0)
There are also: