{{Short description|Commutative ring with a well behaved theory of prime factorization}} 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.<ref>{{harvs|txt|authorlink=Wolfgang Krull|first=Wolfgang |last=Krull|year=1931}}.</ref> 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== Let <math> A </math> be an integral domain and let <math> P </math> be the set of all prime ideals of <math> A </math> of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then <math> A </math> is a '''Krull ring''' if # <math> A_{\mathfrak{p}} </math> is a discrete valuation ring for all <math> \mathfrak{p} \in P </math>, #<math> A </math> is the intersection of these discrete valuation rings (considered as subrings of the quotient field of <math> A </math>), #any nonzero element of <math> A </math> 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:<ref>P. Samuel, ''Lectures on Unique Factorization Domain'', Theorem 3.5.</ref>
An integral domain <math>A</math> is a Krull ring if there exists a family <math> \{ v _ {i} \} _ {i \in I } </math> of discrete valuations on the field of fractions <math>K</math> of <math>A</math> such that: # for any <math> x \in K \setminus \{ 0 \} </math> and all <math>i</math>, except possibly a finite number of them, <math> v _ {i} ( x) = 0 </math>, # for any <math> x \in K \setminus \{ 0 \}</math>, <math> x </math> belongs to <math>A</math> if and only if <math> v _ {i} ( x) \geq 0 </math> for all <math>i \in I </math>.
The valuations <math>v_i</math> are called '''essential valuations''' of <math>A</math>.
The link between the two definitions is as follows: for every <math>\mathfrak p\in P</math>, one can associate a unique normalized valuation <math>v_{\mathfrak p}</math> of <math>K</math> whose valuation ring is <math>A_{\mathfrak p}</math>.<ref>A discrete valuation <math>v</math> is said to be ''normalized'' if <math>v(O_v) = \mathbb N</math>, where <math>O_v</math> is the valuation ring of <math>v</math>. So, every class of equivalent discrete valuations contains a unique normalized valuation.</ref> Then the set <math>\mathcal V = \{v_{\mathfrak p}\}</math> satisfies the conditions of the equivalent definition. Conversely, if the set <math>\mathcal V' = \{v_i\}</math> is as above, and the <math>v_i</math> have been normalized, then <math>\mathcal V'</math> may be bigger than <math>\mathcal V</math>, but it ''must'' contain <math>\mathcal V</math>. In other words, <math>\mathcal V </math> is the minimal set of normalized valuations satisfying the equivalent definition.
== Properties ==
With the notations above, let <math>v_{\mathfrak p}</math> denote the normalized valuation corresponding to the valuation ring <math>A_{\mathfrak p}</math>, <math>U</math> denote the set of units of <math>A</math>, and <math>K</math> its quotient field.
* ''An element <math>x \in K</math> belongs to <math>U</math> if, and only if, <math>v_{\mathfrak p} (x) = 0</math> for every <math>\mathfrak p \in P</math>.'' Indeed, in this case, <math>x \not\in A_{\mathfrak p}\mathfrak p</math> for every <math>\mathfrak p\in P</math>, hence <math>x^{-1} \in A_{\mathfrak p}</math>; by the intersection property, <math>x^{-1}\in A</math>. Conversely, if <math>x</math> and <math>x^{-1}</math> are in <math>A</math>, then <math>v_{\mathfrak p} (xx^{-1}) = v_{\mathfrak p} (1) = 0 = v_{\mathfrak p} (x) + v_{\mathfrak p} (x^{-1})</math>, hence <math>v_{\mathfrak p} (x) = v_{\mathfrak p} (x^{-1}) = 0</math>, since both numbers must be <math>\geq 0</math>. * ''An element <math>x \in A</math> is uniquely determined, up to a unit of <math>A</math>, by the values <math>v_{\mathfrak p} (x)</math>, <math>\mathfrak p \in P</math>.'' Indeed, if <math>v_{\mathfrak p} (x) = v_{\mathfrak p} (y)</math> for every <math>\mathfrak p \in P</math>, then <math>v_{\mathfrak p} (xy^{-1}) = 0</math>, hence <math>xy^{-1}\in U</math> by the above property (q.e.d). This shows that the application <math>x\ {\rm mod}\ U\mapsto \left(v_{\mathfrak p}(x) \right)_{\mathfrak p \in P}</math> is well defined, and since <math>v_{\mathfrak p}(x)\not = 0</math> for only finitely many <math>\mathfrak p</math>, it is an embedding of <math>A^{\times}/U</math> into the free Abelian group generated by the elements of <math>P</math>. Thus, using the multiplicative notation "<math>\cdot</math>" for the later group, there holds, for every <math>x\in A^\times</math>, <math>x = 1\cdot \mathfrak p_1^{\alpha_1}\cdot\mathfrak p_2^{\alpha_2}\cdots \mathfrak p_n^{\alpha_n}\ {\rm mod}\ U</math>, where the <math>\mathfrak p_i</math> are the elements of <math>P</math> containing <math>x</math>, and <math>\alpha_i = v_{\mathfrak p_i} (x)</math>. * The valuations <math>v_{\mathfrak p} </math> are pairwise independent.<ref>If <math>v_{\mathfrak p_1} </math> and <math>v_{\mathfrak p_2} </math>were both finer than a common valuation <math>w</math> of <math>K</math>, the ideals <math>A_{\mathfrak p_1}\mathfrak p_1</math> and <math>A_{\mathfrak p_2}\mathfrak p_2</math> of their corresponding valuation rings would contain properly the prime ideal <math>\mathfrak p_w= \{x\in K:\ w(x) > 0\},</math> hence <math>\mathfrak p_1</math> and <math>\mathfrak p_2</math> would contain the prime ideal <math>\mathfrak p_w\cap A</math> of <math>A</math>, which is forbidden by definition.</ref> As a consequence, there holds the so-called ''weak approximation theorem'',<ref>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: [https://archive.wikiwix.com/cache/20211123084502/http://www.math.tau.ac.il/~jarden/Articles/paper56.pdf archive], p. 17, Prop. 4.4, 4.5 and Rmk 4.6.</ref> an homologue of the Chinese remainder theorem: ''if <math>\mathfrak p_1, \ldots \mathfrak p_n</math> are distinct elements of <math>P</math>, <math> x_1,\ldots x_n</math> belong to <math>K</math> (resp. <math>A_{\mathfrak p}</math>), and <math>a_1, \ldots a_n</math> are <math>n</math> natural numbers, then there exist <math>x\in K</math> (resp. <math>x\in A_{\mathfrak p}</math>) such that <math>v_{\mathfrak p_i} (x - x_i) = n_i</math> for every <math>i</math>.'' * A consequence of the weak approximation theorem is a characterization of when Krull rings are noetherian; namely, a Krull ring <math>A</math> is noetherian if and only if all of its quotients <math>A/{\mathfrak p}</math> by height-1 primes are noetherian. * Two elements <math>x</math> and <math>y</math> of <math>A</math> are ''coprime'' if <math>v_{\mathfrak p} (x) </math> and <math>v_{\mathfrak p} (y)</math> are not both <math>> 0</math> for every <math>\mathfrak p\in P</math>. The basic properties of valuations imply that a good theory of coprimality holds in <math>A</math>. * Every prime ideal of <math>A</math> contains an element of <math>P</math>.<ref>P. Samuel, ''Lectures on Unique Factorization Domains'', Lemma 3.3.</ref> * Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain.<ref>Idem, Prop 4.1 and Corollary (a).</ref> * If <math>L</math> is a subfield of <math>K</math>, then <math>A\cap L</math> is a Krull domain.<ref>Idem, Prop 4.1 and Corollary (b).</ref> * If <math>S\subset A</math> is a multiplicatively closed set not containing 0, the ring of quotients <math>S^{-1}A</math> is again a Krull domain. In fact, the essential valuations of <math>S^{-1}A</math> are those valuation <math>v_{\mathfrak p}</math> (of <math>K</math>) for which <math>\mathfrak p \cap S = \emptyset</math>.<ref>Idem, Prop. 4.2.</ref> * If <math>L</math> is a finite algebraic extension of <math>K</math>, and <math>B</math> is the integral closure of <math>A</math> in <math>L</math>, then <math>B</math> is a Krull domain.<ref>Idem, Prop 4.5.</ref>
==Examples== #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.<ref>P. Samuel, ''Lectures on Factorial Rings'', Thm. 5.3.</ref><ref>{{SpringerEOM|title = Krull ring |access-date = 2016-04-14}}</ref> # Every integrally closed noetherian domain is a Krull domain.<ref>P. Samuel, ''Lectures on Unique Factorization Domains'', Theorem 3.2.</ref> 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 <math> A </math> is a Krull domain then so is the polynomial ring <math> A[x] </math> and the formal power series ring <math> Ax </math>.<ref>Idem, Proposition 4.3 and 4.4.</ref> # The polynomial ring <math>R[x_1, x_2, x_3, \ldots]</math> in infinitely many variables over a unique factorization domain <math> R </math> is a Krull domain which is not noetherian. # Let <math> A </math> be a Noetherian domain with quotient field <math> K </math>, and <math> L </math> be a finite algebraic extension of <math> K </math>. Then the integral closure of <math> A </math> in <math> L </math> is a Krull domain (Mori–Nagata theorem).<ref>{{Cite book|url = https://books.google.com/books?id=APPtnn84FMIC|title = Integral Closure of Ideals, Rings, and Modules|last = Huneke|first = Craig|last2 = Swanson|first2 = Irena|author2-link= Irena Swanson |date = 2006-10-12|publisher = Cambridge University Press|isbn = 9780521688604|language = en}}</ref> #Let <math>A</math> be a Zariski ring (e.g., a local noetherian ring). If the completion <math>\widehat{A}</math> is a Krull domain, then <math>A</math> is a Krull domain (Mori).<ref>Bourbaki, 7.1, no 10, Proposition 16.</ref><ref>P. Samuel, ''Lectures on Unique Factorization Domains'', Thm. 6.5.</ref> #Let <math>A</math> be a Krull domain, and <math>V</math> be the multiplicatively closed set consisting in the powers of a prime element <math>p\in A</math>. Then <math>S^{-1}A</math> is a Krull domain (Nagata).<ref>P. Samuel, ''Lectures on Unique Factorization Domains'', Thm. 6.3.</ref>
==The divisor class group of a Krull ring==
Assume that <math>A</math> is a Krull domain and <math>K</math> is its quotient field. A '''prime divisor''' of <math>A</math> is a height 1 prime ideal of <math>A</math>. The set of prime divisors of <math>A</math> will be denoted <math>P(A)</math> in the sequel. A (Weil) '''divisor''' of <math>A</math> is a formal integral linear combination of prime divisors. They form an Abelian group, noted <math>D(A)</math>. A divisor of the form <math>div(x)=\sum_{p\in P}v_p(x)\cdot p</math>, for some non-zero <math>x</math> in <math>K</math>, is called a principal divisor. The principal divisors of <math>A</math> form a subgroup of the group of divisors (it has been shown above that this group is isomorphic to <math>A^\times /U</math>, where <math>U</math> is the group of unities of <math>A</math>). The quotient of the group of divisors by the subgroup of principal divisors is called the '''divisor class group''' of <math>A</math>; it is usually denoted <math>C(A)</math>.
Assume that <math>B</math> is a Krull domain containing <math>A</math>. As usual, we say that a prime ideal <math>\mathfrak P</math> of <math>B</math> ''lies above'' a prime ideal <math>\mathfrak p</math> of <math>A</math> if <math>\mathfrak P\cap A = \mathfrak p</math>; this is abbreviated in <math>\mathfrak P|\mathfrak p</math>.
Denote the ramification index of <math>v_{\mathfrak P}</math> over <math>v_{\mathfrak p}</math> by <math>e(\mathfrak P,\mathfrak p)</math>, and by <math>P(B)</math> the set of prime divisors of <math>B</math>. Define the application <math>P(A)\to D(B)</math> by :<math> j(\mathfrak p) = \sum_{\mathfrak P|\mathfrak p,\ \mathfrak P\in P(B)} e(\mathfrak P, \mathfrak p) \mathfrak P</math> (the above sum is finite since every <math>x\in \mathfrak p</math> is contained in at most finitely many elements of <math>P(B)</math>). Let extend the application <math>j</math> by linearity to a linear application <math>D(A)\to D(B)</math>. One can now ask in what cases <math>j</math> induces a morphism <math>\bar j:C(A)\to C(B)</math>. This leads to several results.<ref>P. Samuel, ''Lectures on Unique Factorization Domains'', p. 14-25.</ref> For example, the following generalizes a theorem of Gauss:
''The application <math>\bar j:C(A)\to C(A[X])</math> is bijective. In particular, if <math>A</math> is a unique factorization domain, then so is <math>A[X]</math>.''<ref>Idem, Thm. 6.4.</ref>
The divisor class group of a Krull rings are also used to set up powerful ''descent methods'', and in particular the Galoisian descent.<ref>See P. Samuel, ''Lectures on Unique Factorization Domains'', P. 45-64.</ref>
==Cartier divisor== 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''–''z''<sup>2</sup>) the divisor class group has order 2, generated by the divisor ''y''=''z'', but the Picard subgroup is the trivial group.<ref>Hartshorne, GTM52, Example 6.5.2, p.133 and Example 6.11.3, p.142.</ref>
==References== {{reflist}} *{{cite book |first=Nicolas |last=Bourbaki |author-link=Nicolas Bourbaki |title=Commutative algebra|url=https://archive.org/details/commutativealgeb0000bour |url-access=registration }} * {{springer|title=Krull ring|id=p/k055930}} *{{Citation | last1=Krull | first1=Wolfgang | author1-link=Wolfgang Krull | title=Allgemeine Bewertungstheorie | url=http://gdz.sub.uni-goettingen.de/en/dms/load/img/?IDDOC=260807 | archive-url=https://archive.today/20130106080253/http://gdz.sub.uni-goettingen.de/en/dms/load/img/?IDDOC=260807 | url-status=dead | archive-date=January 6, 2013 | year=1931 | journal=J. Reine Angew. Math. | volume=167 | pages=160–196 }} * {{Matsumura CA}} * 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}} *{{Citation | last1=Samuel | first1=Pierre | author1-link=Pierre Samuel | editor1-last=Murthy | editor1-first=M. Pavman | title=Lectures on unique factorization domains | url=http://www.math.tifr.res.in/~publ/ln/ | publisher=Tata Institute of Fundamental Research | location=Bombay | series=Tata Institute of Fundamental Research Lectures on Mathematics |mr=0214579 | year=1964 | volume=30}}
Category:Ring theory Category:Commutative algebra