In commutative algebra, the Krull–Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring,[1] K its total ring of fractions.
Suppose L is a finite extension of K.[2] If
and B is reduced, then B is a noetherian ring of dimension at most one.
Furthermore, for every nonzero ideal
[3][4] Note that the theorem does not say that B is finite over A.
The theorem does not extend to higher dimension.
One important consequence of the theorem is that the integral closure of a Dedekind domain A in a finite extension of the field of fractions of A is again a Dedekind domain.
This consequence does generalize to a higher dimension: the Mori–Nagata theorem states that the integral closure of a noetherian domain is a Krull domain.
and KB is a finite extension of K, so we may assume without loss of generality that
is integral over K, there exists
Then C is a one-dimensional noetherian ring, and
denotes the total ring of fractions of C. Thus we can substitute C for A and reduce to the case
be minimal prime ideals of A; there are finitely many of them.
be the field of fractions of
the kernel of the natural map
Then we have: Now, if the theorem holds when A is a domain, then this implies that B is a one-dimensional noetherian domain since each
Hence, we reduced the proof to the case A is a domain.
be an ideal and let a be a nonzero element in the nonzero ideal
is a zero-dim noetherian ring; thus, artinian, there is an
We claim Since it suffices to establish the inclusion locally, we may assume A is a local ring with the maximal ideal
Let x be a nonzero element in B.
Thus, Now, assume n is a minimum integer such that
and the last inclusion holds.
But then the above inclusion holds for
and this establishes the claim.
has finite length as A-module.
there is finitely generated and so
is finitely generated.
Finally, the exact sequence
of A-modules shows that