Krull–Akizuki theorem

Short description: About extensions of one-dimensional Noetherian rings (commutative algebra)

In commutative algebra, the Krull–Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring, K its total ring of fractions. Suppose L is a finite extension of K. If $A\subset B\subset L$ and B is reduced, then B is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal $I$ of B, $B/I$ 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

First observe that $A\subset B\subset KB$ and KB is a finite extension of K, so we may assume without loss of generality that $L=KB$. Then $L=K{x}_1+\cdots +K{x}_n$ for some $x_1,\dots ,x_n\in B$. Since each $x_i$ is integral over K, there exists $a_i\in A$ such that $a_i{x}_i$ is integral over A. Let $C=A[a_1{x}_1,\dots ,a_n{x}_n]$. Then C is a one-dimensional noetherian ring, and $C\subset B\subset Q(C)$, where $Q(C)$ denotes the total ring of fractions of C. Thus we can substitute C for A and reduce to the case $L=K$.

Let ${\mathfrak{𝔭}}_i$ be minimal prime ideals of A; there are finitely many of them. Let $K_i$ be the field of fractions of $A/{\mathfrak{𝔭}}_i$ and $I_i$ the kernel of the natural map $B\to K\to K_i$. Then we have:

$A/{\mathfrak{𝔭}}_i\subset B/I_i\subset K_i$ and $K\simeq \prod K_i$.

Now, if the theorem holds when A is a domain, then this implies that B is a one-dimensional noetherian domain since each $B/I_i$ is and since $B\simeq \prod B/I_i$. Hence, we reduced the proof to the case A is a domain. Let $0\ne I\subset B$ be an ideal and let a be a nonzero element in the nonzero ideal $I\cap A$. Set $I_n=a^{n}B\cap A+aA$. Since $A/aA$ is a zero-dim noetherian ring; thus, artinian, there is an $l$ such that $I_n=I_l$ for all $n\ge l$. We claim

$a^{l}B\subset a^{l+1}B+A.$

Since it suffices to establish the inclusion locally, we may assume A is a local ring with the maximal ideal $\mathfrak{𝔪}$. Let x be a nonzero element in B. Then, since A is noetherian, there is an n such that ${\mathfrak{𝔪}}^{n+1}\subset x^{-1}A$ and so $a^{n+1}x\in a^{n+1}B\cap A\subset I_{n+2}$. Thus,

$a^{n}x\in a^{n+1}B\cap A+A.$

Now, assume n is a minimum integer such that $n\ge l$ and the last inclusion holds. If $n>l$, then we easily see that $a^{n}x\in I_{n+1}$. But then the above inclusion holds for $n-1$, contradiction. Hence, we have $n=l$ and this establishes the claim. It now follows:

$B/aB\simeq a^{l}B/a^{l+1}B\subset (a^{l+1}B+A)/a^{l+1}B\simeq A/(a^{l+1}B\cap A).$

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. The above shows that $B/aB$ has dimension zero and so B has dimension one. Finally, the exact sequence $B/aB\to B/I\to (0)$ of A-modules shows that $B/I$ is finite over A. $◻$

References

• Bourbaki, Nicolas (1989). Commutative algebra. Berlin Heidelberg: Springer. ISBN 978-3-540-64239-8.