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Krull ring

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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 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 p is a discrete valuation ring for all p ∈ 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 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 , … ] 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

Krull ring Wikipedia
(Text) CC BY-SA


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