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Tag 00IP

Chapter 10: Commutative Algebra > Section 10.50: More Noetherian rings

Lemma 10.50.4 (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 10.50.2 we see that $N = I^nM \cap N \subset IN$ for some suitably large $n$. By Nakayama's Lemma 10.19.1 we see that $N = 0$. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 11631–11636 (see updates for more information).

    \begin{lemma}[Krull's intersection theorem]
    \label{lemma-intersect-powers-ideal-module-zero}
    Let $R$ be a Noetherian local ring. Let $I \subset R$ be
    a proper ideal. Let $M$ be a finite $R$-module.
    Then $\bigcap_{n \geq 0} I^nM = 0$.
    \end{lemma}
    
    \begin{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 \ref{lemma-Artin-Rees}
    we see that $N = I^nM \cap N \subset IN$ for
    some suitably large $n$. By Nakayama's Lemma \ref{lemma-NAK}
    we see that $N = 0$.
    \end{proof}

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