Krull's height theorem
From Commalg
Statement
Let be a Noetherian commutative unital ring and
be elements in
. Let
be minimal among primes containing all the
s. Then, the codimension of
is at most
.
There is also a converse of this statement viz converse of Krull's height theorem.
Proof
Starting assumptions
Replacing by
if necessary, we may assume that
is a local ring with unique maximal ideal
.
In particular, we see that the ring is a local Artinian ring with unique maximal ideal
, hence
is nilpotent modulo
.
Main proof
Supose is a prime contained in
, with no primes between. Then, it suffices to show, inductively, that
is minimal over an ideal generated by
elements.