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.pdfhttp://alpha.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...