Lemma 38.14.1. Let R be a local ring with maximal ideal \mathfrak m. Let R \to S be a ring map. Let N be an S-module. Assume
N is projective as an R-module, and
S/\mathfrak mS is Noetherian and N/\mathfrak mN is a finite S/\mathfrak mS-module.
Then for any prime \mathfrak q \subset S which is an associated prime of N \otimes _ R \kappa (\mathfrak p) where \mathfrak p = R \cap \mathfrak q we have \mathfrak q + \mathfrak m S \not= S.
Proof. Note that the hypotheses of Lemmas 38.7.1 and 38.7.6 are satisfied. We will use the conclusions of these lemmas without further mention. Let \Sigma \subset S be the multiplicative set of elements which are not zerodivisors on N/\mathfrak mN. The map N \to \Sigma ^{-1}N is R-universally injective. Hence we see that any \mathfrak q \subset S which is an associated prime of N \otimes _ R \kappa (\mathfrak p) is also an associated prime of \Sigma ^{-1}N \otimes _ R \kappa (\mathfrak p). Clearly this implies that \mathfrak q corresponds to a prime of \Sigma ^{-1}S. Thus \mathfrak q \subset \mathfrak q' where \mathfrak q' corresponds to an associated prime of N/\mathfrak mN and we win. \square
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