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

  1. is a discrete valuation ring for all ,
  2. is the intersection of these discrete valuation rings (considered as subrings of the quotient field of ).
  3. Any nonzero element of 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 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 is a Krull ring then so is the polynomial ring and the formal power series ring .
  3. The polynomial ring in infinitely many variables over a unique factorization domain is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring.
  4. Let be a Noetherian domain with quotient field , and be a finite algebraic extension of . Then the integral closure of in 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]

  1. ^ "Krull ring - Encyclopedia of Mathematics". eom.springer.de. Retrieved 2016-04-14. 
  2. ^ Bourbaki, 7.1, no 10, Proposition 16.
  3. ^ Huneke, Craig; Swanson, Irena (2006-10-12). Integral Closure of Ideals, Rings, and Modules. Cambridge University Press. ISBN 9780521688604.