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 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[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.

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.

There are other ways to introduce and define Krull rings. In fact, the theory of Krull ring is best exposed in synergy with the theory of divisorial ideals. One of the best reference on the subject is Lecture on Unique Factorization Domains by P. Samuel.

Properties[edit]

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 .
  • 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[edit]

  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] This follows easily from numero 2 above.
  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[edit]

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 setup powerful descent methods, and in particular the Galoisian descent.[21]

Cartier divisor[edit]

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[edit]

  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.