Lemma 10.51.4 (Krull's intersection theorem). Let R be a Noetherian local ring. Let I \subset R be a proper ideal. Let M be a finite R-module. Then \bigcap _{n \geq 0} I^ nM = 0.
Proof. Let N = \bigcap _{n \geq 0} I^ nM. Then N = I^ nM \cap N for all n \geq 0. By the Artin-Rees Lemma 10.51.2 we see that N = I^ nM \cap N \subset IN for some suitably large n. By Nakayama's Lemma 10.20.1 we see that N = 0. \square
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