Krull ring

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 (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

Let $A$ be an integral domain and let $P$ be the set of all prime ideals of $A$ of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then $A$ is a Krull ring if

1. $A_{\mathfrak{p}}$ is a discrete valuation ring for all $\mathfrak{p} \in P$,
2. $A$ is the intersection of these discrete valuation rings (considered as subrings of the quotient field of $A$).
3. Any nonzero element of $A$ is contained in only a finite number of height 1 prime ideals.

Properties

A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.

Let A be a Zariski ring (e.g., a local noetherian ring). If the completion $\widehat{A}$ is a Krull domain, then A is a Krull domain.

Examples

1. Every integrally closed noetherian domain is a Krull ring. In particular, Dedekind domains are Krull rings. Conversely Krull rings are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
2. If $A$ is a Krull ring then so is the polynomial ring $A[x]$ and the formal power series ring $A[[x]]$.
3. The polynomial ring $R[x_1, x_2, x_3, \ldots]$ in infinitely many variables over a unique factorization domain $R$ is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring.
4. Let $A$ be a Noetherian domain with quotient field $K$, and $L$ be a finite algebraic extension of $K$. Then the integral closure of $A$ in $L$ is a Krull ring (Mori–Nagata theorem).

The divisor class group of a Krull ring

A (Weil) divisor of a Krull ring A is a formal integral linear combination of the height 1 prime ideals, and these form a group D(A). A divisor of the form div(x) for some non-zero x in A is called a principal divisor, and the principal divisors form a subgroup of the group of divisors. The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A.

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 Algebra. Second Edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
• 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. ISBN 0-521-25916-9

• Bourbaki, 7.1, no 10, Proposition 16.