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What is the importance of the Krull's principal ideal theorem in later study of commutative algebra and algebraic geometry?

Can any one tell me the geometric picture of this theorem?

Thank you very much !

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1 Answer 1

It is very closely linked to dimension theory in algebraic geometry, as a proper ideal with only one generator in a polynomial ring in $n$ variables over an algebraically closed field have a zero set of dimension $n-1$ by this theorem. The generalization by induction say that for each new generator, the dimension of the zero set goes down in dimension by at most 1.

As for a more concrete geometric picture, most irreducible real polynomials (I say "most", as the reals are not algebraically closed) in 3 variables will define a surface in $\mathbb{R}^3$. Of course, it might have folds, singularities and self-intersections. Two polynomials usually define one or more curves (but might define a surface if one polynomial's zero set is entirely contained in the other's), while three usually defines single points.

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Thank you very much Arthur. I pointed my question out on my own when I continued reading Commutative Algebra book. –  Arsenaler Mar 13 '12 at 17:41

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