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.

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

Formal definition[edit]

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[edit]

A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.[1]

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.[2]

Examples[edit]

  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).[3]

The divisor class group of a Krull ring[edit]

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]/(xyz2] the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.

References[edit]