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 are exactly the Krull domains of dimension at most 1.

In this article, a ring is commutative and has unity.

Formal definition

be an integral domain be the set of all prime ideals of height is a Krull ring if

1. is a discrete valuation ring,

2. ),

3. is contained in only a finite number of height 1 prime ideals.

It is also possible to characterize Krull rings by mean of valuations only:

such that:

1. ,

2. .

are called essential valuations.

., but it must is the minimal set of normalized valuations satisfying the equivalent definition.

Properties

its quotient field.

• ..

• . into the free Abelian group.

• are pairwise independent. As a consequence, there holds the so-called weak approximation theorem, an homologue of the Chinese remainder theorem: .

• is noetherian if and only if by height-1 primes are noetherian.

• are coprime.

• .

• Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain.

• is a Krull domain.

• is a multiplicatively closed set.

• is a Krull domain.

Examples

1. Any unique factorization domain is a Krull domain. Conversely, a Krull domain is a unique factorization domain if (and only if) every prime ideal of height one is principal.

2. Every integrally closed noetherian domain is a Krull domain. In particular, Dedekind domains are Krull domains. Conversely, Krull domains are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.

3. is a Krull domain then so is the polynomial ring and the formal power series ring.

4. in infinitely many variables over a unique factorization domain is a Krull domain which is not noetherian.

5. be a Noetherian domain with quotient field be a finite algebraic extension. Then the integral closure is a Krull domain (Mori–Nagata theorem).

6. be a Zariski ring is a Krull domain (Mori).

7. is a Krull domain (Nagata).

The divisor class group of a Krull ring

is its quotient field. A prime divisor in the sequel. A (Weil) divisor is a formal integral linear combination). The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group.

lies above.

by

. This leads to several results. For example, the following generalizes a theorem of Gauss:

.

The divisor class group of a Krull rings are also used to set up powerful descent methods, and in particular the Galoisian descent.

Cartier divisor

A Cartier divisor of a Krull ring is a locally principal (Weil) divisor. The Cartier divisors form a subgroup of the group of divisors containing the principal divisors. The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the Picard group of invertible sheaves on Spec(A).

Example: in the ring k[x,y,z]/(xy–z2) the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.

References

• Hideyuki Matsumura, Commutative Ring Theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. xiv+320 pp.