Krull's height theorem

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Statement

Let R be a Noetherian commutative unital ring and x_1, x_2, \ldots, x_c be elements in R. Let P be minimal among primes containing all the x_is. Then, the codimension of P is at most c.

There is also a converse of this statement viz converse of Krull's height theorem.

Proof

Starting assumptions

Replacing R by R_P if necessary, we may assume that R is a local ring with unique maximal ideal P.

In particular, we see that the ring R/(x_1,x_2,\ldots,x_c) is a local Artinian ring with unique maximal ideal P/(x_1,x_2,\ldots,x_c), hence P is nilpotent modulo (x_1,x_2,\ldots,x_c).

Main proof

Supose P_1 is a prime contained in P, with no primes between. Then, it suffices to show, inductively, that P_1 is minimal over an ideal generated by c-1 elements.