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.
Contents
In this article, a ring is commutative and has unity.
Formal definition
Let
-
A p p ∈ P , -
A is the intersection of these discrete valuation rings (considered as subrings of the quotient field ofA ). - 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
Examples
- 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.
- If
A is a Krull ring then so is the polynomial ringA [ x ] and the formal power series ringA [ [ x ] ] . - The polynomial ring
R [ x 1 , x 2 , x 3 , … ] in infinitely many variables over a unique factorization domainR is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring. - Let
A be a Noetherian domain with quotient fieldK , andL be a finite algebraic extension ofK . Then the integral closure ofA inL 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.