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I an not 100% sure that this question has an answer, but still I would like to ask it.

There is a short an simple proof of Krull intersection theorem (for example page 12 in http://www.jmilne.org/math/xnotes/CA.pdf or page 154 http://math.uga.edu/~pete/integral.pdf ). As far as I got this short proof is very recent (at least in Pete Clark's notes there is a reference to an article of 2004).

I would like to know if this proof can be stated in a more "geometric" language. Even though I seem to understand the proof it looks to me a bit as a trick...

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There is another result, called the Artin-Rees lemma, which implies Krull's intersection theorem. The proofs I know are more complicated than the one you mention, but the Artin-Rees lemma is also related to Krull's Hauptidealsatz which has clearly a more geometric flavour. You could try to look at the following article : D. Rees, Two classical theorems of ideal theory. Proc. Cambridge Philos. Soc. 52 (1956), 155–157. –  François Brunault Oct 8 '12 at 19:55
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EDIT: This is a geometric interpretation of the statement of the Krull Intersection Theorem, and not the proof, as the OP asked for. Here it is anyways.

Ok, so let's work in the setting of a Noetherian local ring $(R, \mathfrak{m})$. Then the Krull intersection says that $\bigcap_{n \geq 1} \mathfrak{m}^n = (0)$. (You can remove the local hypothesis if you assume that $R$ is a domain, and then $\mathfrak{m}$ can be any ideal).

The easiest geometric interpretation of this statement that I can think of is something like the following.

There is no hypersurface passing through a point (or subvariety) of a variety $X$ with infinite order of vanishing through that point/subvariety.

Or,

The only function which vanishes to arbitrarily high order at a point is the zero function.

The interpretation should be pretty easy. Given an $f \in R$, we can measure the order of vanishing of $f$ by asking what's the biggest power $n$ of $\mathfrak{m}$ such that $f \in \mathfrak{m}^n$ but such that $f \notin \mathfrak{m}^{n+1}$. Given now a scheme $X$, go look at a stalk of some (possibly non-closed) point.

Obviously, this result shows up a lot when studying completion, so it has geometric applications as well.

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Karl, thanks a lot for the answer! In fact for me indeed this is a geometric interpretation of the statement of "Krull intersection". But I wonder still (and this is my question): do you think that one can cast the proof of "Krull intersection" theorem given in Milne's or Clark's notes in this geometric language? Do you think this is possible/useful thing try to achieve? –  aglearner Oct 8 '12 at 19:50
    
Ah, you are right, I misread the question. I'll try to edit my answer. –  Karl Schwede Oct 8 '12 at 20:07
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