Lemma 38.8.4. Let R be a ring. Let I \subset R be an ideal. Let R \to S be a ring map, and N an S-module. Assume
R is a Noetherian ring,
S is a Noetherian ring,
N is a finite S-module,
N is flat over R, and
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 IS + \mathfrak q \not= S.
Then the map N \to N^\wedge of N into the I-adic completion of N is universally injective as a map of R-modules.
Proof. This follows from Lemma 38.8.3 because Algebra, Lemma 10.65.5 and Remark 10.65.6 guarantee that the set of associated primes of tensor products N \otimes _ R Q are contained in the set of associated primes of the modules N \otimes _ R \kappa (\mathfrak p). \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Your email address will not be published.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)