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.
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
is the intersection of these discrete valuation rings (considered as subrings of the quotient field of ).
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:
for any and all , except possibly a finite number of them, ;
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. The theory of Krull rings can be exposed in synergy with the theory of divisorial ideals. One of the best[according to whom?] references on the subject is Lecture on Unique Factorization Domains by P. Samuel.
PropertiesEdit
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 .
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]
ExamplesEdit
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]
Every integrally closednoetheriandomain 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.
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]
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 ringEdit
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 divisorEdit
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.[22]
^P. Samuel, Lectures on Unique Factorization Domain, Theorem 3.5.
^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.
^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.
^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.
^P. Samuel, Lectures on Unique Factorization Domains, Lemma 3.3.
Hideyuki Matsumura, Commutative Ring Theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. xiv+320 pp. ISBN0-521-25916-9
Samuel, Pierre (1964), Murthy, M. Pavman (ed.), Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 30, Bombay: Tata Institute of Fundamental Research, MR 0214579
August 28, 2023
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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 Contents 1 Formal definition 2 Properties 3 Examples 4 The divisor class group of a Krull ring 5 Cartier divisor 6 ReferencesFormal definition EditLet A displaystyle A be an integral domain and let P displaystyle P be the set of all prime ideals of A displaystyle A of height one that is the set of all prime ideals properly containing no nonzero prime ideal Then A displaystyle A is a Krull ring if A p displaystyle A mathfrak p is a discrete valuation ring for all p P displaystyle mathfrak p in P A displaystyle A is the intersection of these discrete valuation rings considered as subrings of the quotient field of A displaystyle A Any nonzero element of A displaystyle A 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 A displaystyle A is a Krull ring if there exists a family v i i I displaystyle v i i in I of discrete valuations on the field of fractions K displaystyle K of A displaystyle A such that for any x K 0 displaystyle x in K setminus 0 and all i displaystyle i except possibly a finite number of them v i x 0 displaystyle v i x 0 for any x K 0 displaystyle x in K setminus 0 x displaystyle x belongs to A displaystyle A if and only if v i x 0 displaystyle v i x geq 0 for all i I displaystyle i in I The valuations v i displaystyle v i are called essential valuations of A displaystyle A The link between the two definitions is as follows for every p P displaystyle mathfrak p in P one can associate a unique normalized valuation v p displaystyle v mathfrak p of K displaystyle K whose valuation ring is A p displaystyle A mathfrak p 3 Then the set V v p displaystyle mathcal V v mathfrak p satisfies the conditions of the equivalent definition Conversely if the set V v i displaystyle mathcal V v i is as above and the v i displaystyle v i have been normalized then V displaystyle mathcal V may be bigger than V displaystyle mathcal V but it must contain V displaystyle mathcal V In other words V displaystyle mathcal V is the minimal set of normalized valuations satisfying the equivalent definition There are other ways to introduce and define Krull rings The theory of Krull rings can be exposed in synergy with the theory of divisorial ideals One of the best according to whom references on the subject is Lecture on Unique Factorization Domains by P Samuel Properties EditWith the notations above let v p displaystyle v mathfrak p denote the normalized valuation corresponding to the valuation ring A p displaystyle A mathfrak p U displaystyle U denote the set of units of A displaystyle A and K displaystyle K its quotient field An element x K displaystyle x in K belongs to U displaystyle U if and only if v p x 0 displaystyle v mathfrak p x 0 for every p P displaystyle mathfrak p in P Indeed in this case x A p p displaystyle x not in A mathfrak p mathfrak p for every p P displaystyle mathfrak p in P hence x 1 A p displaystyle x 1 in A mathfrak p by the intersection property x 1 A displaystyle x 1 in A Conversely if x displaystyle x and x 1 displaystyle x 1 are in A displaystyle A then v p x x 1 v p 1 0 v p x v p x 1 displaystyle v mathfrak p xx 1 v mathfrak p 1 0 v mathfrak p x v mathfrak p x 1 hence v p x v p x 1 0 displaystyle v mathfrak p x v mathfrak p x 1 0 since both numbers must be 0 displaystyle geq 0 An element x A displaystyle x in A is uniquely determined up to a unit of A displaystyle A by the values v p x displaystyle v mathfrak p x p P displaystyle mathfrak p in P Indeed if v p x v p y displaystyle v mathfrak p x v mathfrak p y for every p P displaystyle mathfrak p in P then v p x y 1 0 displaystyle v mathfrak p xy 1 0 hence x y 1 U displaystyle xy 1 in U by the above property q e d This shows that the application x m o d U v p x p P displaystyle x rm mod U mapsto left v mathfrak p x right mathfrak p in P is well defined and since v p x 0 displaystyle v mathfrak p x not 0 for only finitely many p displaystyle mathfrak p it is an embedding of A U displaystyle A times U into the free Abelian group generated by the elements of P displaystyle P Thus using the multiplicative notation displaystyle cdot for the later group there holds for every x A displaystyle x in A times x 1 p 1 a 1 p 2 a 2 p n a n m o d U displaystyle x 1 cdot mathfrak p 1 alpha 1 cdot mathfrak p 2 alpha 2 cdots mathfrak p n alpha n rm mod U where the p i displaystyle mathfrak p i are the elements of P displaystyle P containing x displaystyle x and a i v p i x displaystyle alpha i v mathfrak p i x The valuations v p displaystyle v mathfrak p are pairwise independent 4 As a consequence there holds the so called weak approximation theorem 5 an homologue of the Chinese remainder theorem if p 1 p n displaystyle mathfrak p 1 ldots mathfrak p n are distinct elements of P displaystyle P x 1 x n displaystyle x 1 ldots x n belong to K displaystyle K resp A p displaystyle A mathfrak p and a 1 a n displaystyle a 1 ldots a n are n displaystyle n natural numbers then there exist x K displaystyle x in K resp x A p displaystyle x in A mathfrak p such that v p i x x i n i displaystyle v mathfrak p i x x i n i for every i displaystyle i Two elements x displaystyle x and y displaystyle y of A displaystyle A are coprime if v p x displaystyle v mathfrak p x and v p y displaystyle v mathfrak p y are not both gt 0 displaystyle gt 0 for every p P displaystyle mathfrak p in P The basic properties of valuations imply that a good theory of coprimality holds in A displaystyle A Every prime ideal of A displaystyle A contains an element of P displaystyle P 6 Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain 7 If L displaystyle L is a subfield of K displaystyle K then A L displaystyle A cap L is a Krull domain 8 If S A displaystyle S subset A is a multiplicatively closed set not containing 0 the ring of quotients S 1 A displaystyle S 1 A is again a Krull domain In fact the essential valuations of S 1 A displaystyle S 1 A are those valuation v p displaystyle v mathfrak p of K displaystyle K for which p S displaystyle mathfrak p cap S emptyset 9 If L displaystyle L is a finite algebraic extension of K displaystyle K and B displaystyle B is the integral closure of A displaystyle A in L displaystyle L then B displaystyle B is a Krull domain 10 Examples EditAny 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 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 If A displaystyle A is a Krull domain then so is the polynomial ring A x displaystyle A x and the formal power series ring A x displaystyle A x 14 The polynomial ring R x 1 x 2 x 3 displaystyle R x 1 x 2 x 3 ldots in infinitely many variables over a unique factorization domain R displaystyle R is a Krull domain which is not noetherian Let A displaystyle A be a Noetherian domain with quotient field K displaystyle K and L displaystyle L be a finite algebraic extension of K displaystyle K Then the integral closure of A displaystyle A in L displaystyle L is a Krull domain Mori Nagata theorem 15 This follows easily from numero 2 above Let A displaystyle A be a Zariski ring e g a local noetherian ring If the completion A displaystyle widehat A is a Krull domain then A displaystyle A is a Krull domain Mori 16 17 Let A displaystyle A be a Krull domain and V displaystyle V be the multiplicatively closed set consisting in the powers of a prime element p A displaystyle p in A Then S 1 A displaystyle S 1 A is a Krull domain Nagata 18 The divisor class group of a Krull ring EditAssume that A displaystyle A is a Krull domain and K displaystyle K is its quotient field A prime divisor of A displaystyle A is a height 1 prime ideal of A displaystyle A The set of prime divisors of A displaystyle A will be denoted P A displaystyle P A in the sequel A Weil divisor of A displaystyle A is a formal integral linear combination of prime divisors They form an Abelian group noted D A displaystyle D A A divisor of the form d i v x p P v p x p displaystyle div x sum p in P v p x cdot p for some non zero x displaystyle x in K displaystyle K is called a principal divisor The principal divisors of A displaystyle A form a subgroup of the group of divisors it has been shown above that this group is isomorphic to A U displaystyle A times U where U displaystyle U is the group of unities of A displaystyle A The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A displaystyle A it is usually denoted C A displaystyle C A Assume that B displaystyle B is a Krull domain containing A displaystyle A As usual we say that a prime ideal P displaystyle mathfrak P of B displaystyle B lies above a prime ideal p displaystyle mathfrak p of A displaystyle A if P A p displaystyle mathfrak P cap A mathfrak p this is abbreviated in P p displaystyle mathfrak P mathfrak p Denote the ramification index of v P displaystyle v mathfrak P over v p displaystyle v mathfrak p by e P p displaystyle e mathfrak P mathfrak p and by P B displaystyle P B the set of prime divisors of B displaystyle B Define the application P A D B displaystyle P A to D B by j p P p P P B e P p P displaystyle j mathfrak p sum mathfrak P mathfrak p mathfrak P in P B e mathfrak P mathfrak p mathfrak P the above sum is finite since every x p displaystyle x in mathfrak p is contained in at most finitely many elements of P B displaystyle P B Let extend the application j displaystyle j by linearity to a linear application D A D B displaystyle D A to D B One can now ask in what cases j displaystyle j induces a morphism j C A C B displaystyle bar j C A to C B This leads to several results 19 For example the following generalizes a theorem of Gauss The application j C A C A X displaystyle bar j C A to C A X is bijective In particular if A displaystyle A is a unique factorization domain then so is A X displaystyle A X 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 EditA 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 22 References Edit Wolfgang Krull 1931 P Samuel Lectures on Unique Factorization Domain Theorem 3 5 A discrete valuation v displaystyle v is said to be normalized if v O v N displaystyle v O v mathbb N where O v displaystyle O v is the valuation ring of v displaystyle v So every class of equivalent discrete valuations contains a unique normalized valuation If v p 1 displaystyle v mathfrak p 1 and v p 2 displaystyle v mathfrak p 2 were both finer than a common valuation w displaystyle w of K displaystyle K the ideals A p 1 p 1 displaystyle A mathfrak p 1 mathfrak p 1 and A p 2 p 2 displaystyle A mathfrak p 2 mathfrak p 2 of their corresponding valuation rings would contain properly the prime ideal p w x K w x gt 0 displaystyle mathfrak p w x in K w x gt 0 hence p 1 displaystyle mathfrak p 1 and p 2 displaystyle mathfrak p 2 would contain the prime ideal p w A displaystyle mathfrak p w cap A of A displaystyle A which is forbidden by definition 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 P Samuel Lectures on Unique Factorization Domains Lemma 3 3 Idem Prop 4 1 and Corollary a Idem Prop 4 1 and Corollary b Idem Prop 4 2 Idem Prop 4 5 P Samuel Lectures on Factorial Rings Thm 5 3 Krull ring Encyclopedia of Mathematics EMS Press 2001 1994 retrieved 2016 04 14 P Samuel Lectures on Unique Factorization Domains Theorem 3 2 Idem Proposition 4 3 and 4 4 Huneke Craig Swanson Irena 2006 10 12 Integral Closure of Ideals Rings and Modules Cambridge University Press ISBN 9780521688604 Bourbaki 7 1 no 10 Proposition 16 P Samuel Lectures on Unique Factorization Domains Thm 6 5 P Samuel Lectures on Unique Factorization Domains Thm 6 3 P Samuel Lectures on Unique Factorization Domains p 14 25 Idem Thm 6 4 See P Samuel Lectures on Unique Factorization Domains P 45 64 Hartshorne GTM52 Example 6 5 2 p 133 and Example 6 11 3 p 142 N Bourbaki Commutative algebra Krull ring Encyclopedia of Mathematics EMS Press 2001 1994 Krull Wolfgang 1931 Allgemeine Bewertungstheorie J Reine Angew Math 167 160 196 archived from the original on January 6 2013 Hideyuki Matsumura Commutative Algebra Second Edition Mathematics Lecture Note Series 56 Benjamin Cummings Publishing Co Inc Reading Mass 1980 xv 313 pp ISBN 0 8053 7026 9 Hideyuki Matsumura Commutative Ring Theory Translated from the Japanese by M Reid Cambridge Studies in Advanced Mathematics 8 Cambridge University Press Cambridge 1986 xiv 320 pp ISBN 0 521 25916 9 Samuel Pierre 1964 Murthy M Pavman ed Lectures on unique factorization domains Tata Institute of Fundamental Research Lectures on Mathematics vol 30 Bombay Tata Institute of Fundamental Research MR 0214579 Retrieved from https en wikipedia org w index php title Krull ring amp oldid 1156780739, wikipedia, wiki, book, books, library,