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I am currently engaged in independent study of algebraic geometry, using Dan Bump's book. One of the exercises in it outlines a proof of the Krull Intersection Theorem, which [here] is the following:

Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$, and let $M$ be the intersection of all of the $\mathfrak{m}^n$. Then $M = 0$.

The hints direct me to use the Artin-Rees lemma to show that $\mathfrak{m} M = M$, then use Nakayama's lemma to show that $M = 0$ (this second step is easy). I showed this to a professor and he accused the book of using big machinery for no reason, arguing that

$$\mathfrak{m} M = \mathfrak{m} \bigcap_{n \ge 0} \mathfrak{m}^n = \bigcap_{n \ge 1} \mathfrak{m}^n = M.$$

Does this argument work? Does Bump apply Artin-Rees because that argument works in some broader context where the above argument fails?

  • 4
    Don't believe everything that professors tell you. How did he show that $m\bigcap m^n=\bigcap m^{n+1}$?2010-11-04
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    @user3120: no problem. It's clear that m times the intersection of the m^n is contained in the intersection of the m^{n+1}, but there is no reason to expect the reverse inclusion in general. I don't know a counterexample off the top of my head, though.2010-11-04
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    Yeah, I don't recall the argument he gave. Will hunt for a counterexample.2010-11-04
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    Incidenally, the Artin--Rees Lemma is not particlarly heavy machinery: it is a direct application of the Hilbert Basis Theorem (although not always explained this way), which makes it a pretty basic and fundamental fact about Noetherian rings.2010-11-05
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    $(x^2) \cap (x) = (x^2) \neq (x^3)$ which is the product of the ideals inside $\mathbb{Z}[x]$2010-12-05

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