In algebra, the Krull–Akizuki theorem states the following: let ''A'' be a
one-dimensional reduced noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
, ''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 In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' i ...
of a
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
''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 In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which ...
.
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.
References
*
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...
, ''Commutative algebra''
{{DEFAULTSORT:Krull-Akizuki theorem
Theorems in algebra