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 in 1931.[1] 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

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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.

It is also possible to characterize Krull rings by mean of valuations only:[2]

An integral domain   is a Krull ring if there exists a family   of discrete valuations on the field of fractions   of   such that:

  1. for any   and all  , except possibly a finite number of them,  ,
  2. for any  ,   belongs to   if and only if   for all  .

The valuations   are called essential valuations of  .

The link between the two definitions is as follows: for every  , one can associate a unique normalized valuation   of   whose valuation ring is  .[3] Then the set   satisfies the conditions of the equivalent definition. Conversely, if the set   is as above, and the   have been normalized, then   may be bigger than  , but it must contain  . In other words,   is the minimal set of normalized valuations satisfying the equivalent definition.

Properties

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With the notations above, let   denote the normalized valuation corresponding to the valuation ring  ,   denote the set of units of  , and   its quotient field.

  • An element   belongs to   if, and only if,   for every  . Indeed, in this case,   for every  , hence  ; by the intersection property,  . Conversely, if   and   are in  , then  , hence  , since both numbers must be  .
  • An element   is uniquely determined, up to a unit of  , by the values  ,  . Indeed, if   for every  , then  , hence   by the above property (q.e.d). This shows that the application   is well defined, and since   for only finitely many  , it is an embedding of   into the free Abelian group generated by the elements of  . Thus, using the multiplicative notation " " for the later group, there holds, for every  ,  , where the   are the elements of   containing  , and  .
  • The valuations   are pairwise independent.[4] As a consequence, there holds the so-called weak approximation theorem,[5] an homologue of the Chinese remainder theorem: if   are distinct elements of  ,   belong to   (resp.  ), and   are   natural numbers, then there exist   (resp.  ) such that   for every  .
  • A consequence of the weak approximation theorem is a characterization of when Krull rings are noetherian; namely, a Krull ring   is noetherian if and only if all of its quotients   by height-1 primes are noetherian.
  • Two elements   and   of   are coprime if   and   are not both   for every  . The basic properties of valuations imply that a good theory of coprimality holds in  .
  • Every prime ideal of   contains an element of  .[6]
  • Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain.[7]
  • If   is a subfield of  , then   is a Krull domain.[8]
  • If   is a multiplicatively closed set not containing 0, the ring of quotients   is again a Krull domain. In fact, the essential valuations of   are those valuation   (of  ) for which  .[9]
  • If   is a finite algebraic extension of  , and   is the integral closure of   in  , then   is a Krull domain.[10]

Examples

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  1. Any unique factorization domain is a Krull domain. Conversely, a Krull domain is a unique factorization domain if (and only if) every prime ideal of height one is principal.[11][12]
  2. Every integrally closed noetherian domain is a Krull domain.[13] In particular, Dedekind domains are Krull domains. Conversely, Krull domains are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
  3. If   is a Krull domain then so is the polynomial ring   and the formal power series ring  .[14]
  4. The polynomial ring   in infinitely many variables over a unique factorization domain   is a Krull domain which is not noetherian.
  5. Let   be a Noetherian domain with quotient field  , and   be a finite algebraic extension of  . Then the integral closure of   in   is a Krull domain (Mori–Nagata theorem).[15]
  6. Let   be a Zariski ring (e.g., a local noetherian ring). If the completion   is a Krull domain, then   is a Krull domain (Mori).[16][17]
  7. Let   be a Krull domain, and   be the multiplicatively closed set consisting in the powers of a prime element  . Then   is a Krull domain (Nagata).[18]

The divisor class group of a Krull ring

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Assume that   is a Krull domain and   is its quotient field. A prime divisor of   is a height 1 prime ideal of  . The set of prime divisors of   will be denoted   in the sequel. A (Weil) divisor of   is a formal integral linear combination of prime divisors. They form an Abelian group, noted  . A divisor of the form  , for some non-zero   in  , is called a principal divisor. The principal divisors of   form a subgroup of the group of divisors (it has been shown above that this group is isomorphic to  , where   is the group of unities of  ). The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of  ; it is usually denoted  .

Assume that   is a Krull domain containing  . As usual, we say that a prime ideal   of   lies above a prime ideal   of   if  ; this is abbreviated in  .

Denote the ramification index of   over   by  , and by   the set of prime divisors of  . Define the application   by

 

(the above sum is finite since every   is contained in at most finitely many elements of  ). Let extend the application   by linearity to a linear application  . One can now ask in what cases   induces a morphism  . This leads to several results.[19] For example, the following generalizes a theorem of Gauss:

The application   is bijective. In particular, if   is a unique factorization domain, then so is  .[20]

The divisor class group of a Krull rings are also used to set up powerful descent methods, and in particular the Galoisian descent.[21]

Cartier divisor

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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.[22]

References

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  1. ^ Wolfgang Krull (1931).
  2. ^ P. Samuel, Lectures on Unique Factorization Domain, Theorem 3.5.
  3. ^ A discrete valuation   is said to be normalized if  , where   is the valuation ring of  . So, every class of equivalent discrete valuations contains a unique normalized valuation.
  4. ^ If   and  were both finer than a common valuation   of  , the ideals   and   of their corresponding valuation rings would contain properly the prime ideal   hence   and   would contain the prime ideal   of  , which is forbidden by definition.
  5. ^ See Moshe Jarden, Intersections of local algebraic extensions of a Hilbertian field , in A. Barlotti et al., Generators and Relations in Groups and Geometries, Dordrecht, Kluwer, coll., NATO ASI Series C (no 333), 1991, p. 343-405. Read online: archive, p. 17, Prop. 4.4, 4.5 and Rmk 4.6.
  6. ^ P. Samuel, Lectures on Unique Factorization Domains, Lemma 3.3.
  7. ^ Idem, Prop 4.1 and Corollary (a).
  8. ^ Idem, Prop 4.1 and Corollary (b).
  9. ^ Idem, Prop. 4.2.
  10. ^ Idem, Prop 4.5.
  11. ^ P. Samuel, Lectures on Factorial Rings, Thm. 5.3.
  12. ^ "Krull ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994], retrieved 2016-04-14
  13. ^ P. Samuel, Lectures on Unique Factorization Domains, Theorem 3.2.
  14. ^ Idem, Proposition 4.3 and 4.4.
  15. ^ Huneke, Craig; Swanson, Irena (2006-10-12). Integral Closure of Ideals, Rings, and Modules. Cambridge University Press. ISBN 9780521688604.
  16. ^ Bourbaki, 7.1, no 10, Proposition 16.
  17. ^ P. Samuel, Lectures on Unique Factorization Domains, Thm. 6.5.
  18. ^ P. Samuel, Lectures on Unique Factorization Domains, Thm. 6.3.
  19. ^ P. Samuel, Lectures on Unique Factorization Domains, p. 14-25.
  20. ^ Idem, Thm. 6.4.
  21. ^ See P. Samuel, Lectures on Unique Factorization Domains, P. 45-64.
  22. ^ Hartshorne, GTM52, Example 6.5.2, p.133 and Example 6.11.3, p.142.