Krull–Akizuki theorem


In algebra, the Krull–Akizuki theorem states the following: let A be a at most one-dimensional reduced noetherian ring, K its total ring of fractions. If B is a subring of a finite extension L of K containing A
then B is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal I of B, is finite over A.
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.

Proof

Here, we give a proof when. Let be minimal prime ideals of A; there are finitely many of them. Let be the field of fractions of and 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 is and since. Hence, we reduced the proof to the case A is a domain. Let be an ideal and let a be a nonzero element in the nonzero ideal . Set. Since is a zero-dim noetherian ring; thus, artinian, there is an l such that for all. 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. Then, since A is noetherian, there is an n such that and so. Thus,
Now, assume n is a minimum integer such that and the last inclusion holds. If, then we easily see that. But then the above inclusion holds for, contradiction. Hence, we have and this establishes the claim. It now follows:
Hence, has finite length as A-module. In particular, the image of I there is finitely generated and so I is finitely generated. Finally, the above shows that has zero dimension and so B has dimension one.