Codazzi tensor

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In algebra, the Krull–Akizuki theorem states the following: let A be a one-dimensional reduced noetherian ring,[1] K its total ring of fractions. If B is a subring of a finite extension L of K containing A and is not a field, then B is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal I of B, B/I is finite over A.[2]

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 L=K. Let pi be minimal prime ideals of A; there are finitely many of them. Let Ki be the field of fractions of A/pi and Ii the kernel of the natural map BKKi. Then we have:

A/piB/IiKi.

Now, if the theorem holds when A is a domain, then this implies that B is a one-dimensional noetherian domain since each B/Ii is and since B=B/Ii. Hence, we reduced the proof to the case A is a domain. Let 0IB be an ideal and let a be a nonzero element in the nonzero ideal IA. Set In=anBA+aA. Since A/aA is a zero-dim noetherian ring; thus, artinian, there is an l such that In=Il for all nl. We claim

alBal+1B+A.

Since it suffices to establish the inclusion locally, we may assume A is a local ring with the maximal ideal m. Let x be a nonzero element in B. Then, since A is noetherian, there is an n such that mn+1x1A and so an+1xan+1BAIn+2. Thus,

anxan+1BA+A.

Now, assume n is a minimum integer such that nl and the last inclusion holds. If n>l, then we easily see that anxIn+1. But then the above inclusion holds for n1, contradiction. Hence, we have n=l and this establishes the claim. It now follows:

B/aBalB/al+1B(al+1B+A)/al+1BA/al+1BA.

Hence, B/aB 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 B/aB has zero dimension and so B has dimension one.

References

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  1. In this article, a ring is commutative and has unity.
  2. Template:Harvnb