Krull ring
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In commutative algebra, a Krull ring, or Krull domain, is a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
with a well behaved theory of prime factorization. They were introduced by
Wolfgang Krull Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject. Krull was born and went to school in Baden-Baden. H ...
in 1931. They are a higher-dimensional generalization of
Dedekind domain In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...
s, which are exactly the Krull domains of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
at most 1. In this article, a ring is commutative and has unity.


Formal definition

Let A be an
integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
and let P be the set of all
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
s of A of
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For an example of vertical extent, "This basketball player is 7 foot 1 inches in height." For an e ...
one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then A is a Krull ring if # A_ is a
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' that satisfies any and all of the following equivalent conditions: # '' ...
for all \mathfrak \in P , # A is the intersection of these discrete valuation rings (considered as subrings of the quotient field of A ), #any nonzero element of 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: An integral domain A is a Krull ring if there exists a family \ _ of discrete valuations on the field of fractions K of A such that: # for any x \in K \setminus \ and all i, except possibly a finite number of them, v _ ( x) = 0 , # for any x \in K \setminus \, x belongs to A if and only if v _ ( x) \geq 0 for all i \in I . The valuations v_i are called essential valuations of A. The link between the two definitions is as follows: for every \mathfrak p\in P, one can associate a unique normalized valuation v_ of K whose valuation ring is A_. Then the set \mathcal V = \ satisfies the conditions of the equivalent definition. Conversely, if the set \mathcal V' = \ is as above, and the v_i have been normalized, then \mathcal V' may be bigger than \mathcal V, but it ''must'' contain \mathcal V. In other words, \mathcal V is the minimal set of normalized valuations satisfying the equivalent definition.


Properties

With the notations above, let v_ denote the normalized valuation corresponding to the valuation ring A_, U denote the set of units of A, and K its quotient field. * ''An element x \in K belongs to U if, and only if, v_ (x) = 0 for every \mathfrak p \in P.'' Indeed, in this case, x \not\in A_\mathfrak p for every \mathfrak p\in P, hence x^ \in A_; by the intersection property, x^\in A. Conversely, if x and x^ are in A, then v_ (xx^) = v_ (1) = 0 = v_ (x) + v_ (x^), hence v_ (x) = v_ (x^) = 0, since both numbers must be \geq 0. * ''An element x \in A is uniquely determined, up to a unit of A, by the values v_ (x), \mathfrak p \in P.'' Indeed, if v_ (x) = v_ (y) for every \mathfrak p \in P, then v_ (xy^) = 0, hence xy^\in U by the above property (q.e.d). This shows that the application x\ \ U\mapsto \left(v_(x) \right)_ is well defined, and since v_(x)\not = 0 for only finitely many \mathfrak p, it is an embedding of A^/U into the
free Abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
generated by the elements of P. Thus, using the multiplicative notation "\cdot" for the later group, there holds, for every x\in A^\times, x = 1\cdot \mathfrak p_1^\cdot\mathfrak p_2^\cdots \mathfrak p_n^\ \ U, where the \mathfrak p_i are the elements of P containing x, and \alpha_i = v_ (x). * The valuations v_ are pairwise independent. As a consequence, there holds the so-called ''weak approximation theorem'', an homologue of the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
: ''if \mathfrak p_1, \ldots \mathfrak p_n are distinct elements of P, x_1,\ldots x_n belong to K (resp. A_), and a_1, \ldots a_n are n natural numbers, then there exist x\in K (resp. x\in A_) such that v_ (x - x_i) = n_i for every i.'' * A consequence of the weak approximation theorem is a characterization of when Krull rings are noetherian; namely, a Krull ring A is noetherian
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
all of its quotients A/ by height-1 primes are noetherian. * Two elements x and y of A are ''coprime'' if v_ (x) and v_ (y) are not both > 0 for every \mathfrak p\in P. The basic properties of valuations imply that a good theory of coprimality holds in A. * Every prime ideal of A contains an element of P. * Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain. * If L is a subfield of K, then A\cap L is a Krull domain. * If S\subset A is a multiplicatively closed set not containing 0, the ring of quotients S^A is again a Krull domain. In fact, the essential valuations of S^A are those valuation v_ (of K) for which \mathfrak p \cap S = \emptyset. * If L is a finite algebraic extension of K, and B is the integral closure of A in L, then B is a Krull domain.


Examples

#Any unique factorization domain is a Krull domain. Conversely, a Krull domain is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
if (and only if) every prime ideal of height one is principal. # Every integrally closed
noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
domain is a Krull domain. In particular,
Dedekind domain In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...
s 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 is a Krull domain then so is the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
A and the formal power series ring A x . # The polynomial ring R _1, x_2, x_3, \ldots/math> in infinitely many variables over a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
R is a Krull domain which is not noetherian. # Let A be a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
domain with
quotient field In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fiel ...
K , and L be a finite algebraic extension of K . Then the
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of some monic polynomial over ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and ...
of A in L is a Krull domain ( Mori–Nagata theorem). #Let A be a Zariski ring (e.g., a local noetherian ring). If the completion \widehat is a Krull domain, then A is a Krull domain (Mori). #Let A be a Krull domain, and V be the multiplicatively closed set consisting in the powers of a prime element p\in A. Then S^A is a Krull domain (Nagata).


The divisor class group of a Krull ring

Assume that A is a Krull domain and K is its quotient field. A prime divisor of A is a height 1 prime ideal of A. The set of prime divisors of A will be denoted P(A) in the sequel. A (Weil) divisor of A is a formal integral linear combination of prime divisors. They form an Abelian group, noted D(A). A divisor of the form div(x)=\sum_v_p(x)\cdot p, for some non-zero x in K, is called a principal divisor. The principal divisors of A form a subgroup of the group of divisors (it has been shown above that this group is isomorphic to A^\times /U, where U is the group of unities of A). The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A; it is usually denoted C(A). Assume that B is a Krull domain containing A. As usual, we say that a prime ideal \mathfrak P of B ''lies above'' a prime ideal \mathfrak p of A if \mathfrak P\cap A = \mathfrak p; this is abbreviated in \mathfrak P, \mathfrak p. Denote the ramification index of v_ over v_ by e(\mathfrak P,\mathfrak p), and by P(B) the set of prime divisors of B. Define the application P(A)\to D(B) by : j(\mathfrak p) = \sum_ e(\mathfrak P, \mathfrak p) \mathfrak P (the above sum is finite since every x\in \mathfrak p is contained in at most finitely many elements of P(B)). Let extend the application j by linearity to a linear application D(A)\to D(B). One can now ask in what cases j induces a morphism \bar j:C(A)\to C(B). This leads to several results. For example, the following generalizes a theorem of Gauss: ''The application \bar j:C(A)\to C(A is bijective. In particular, if A is a unique factorization domain, then so is A /math>.'' The divisor class group of a Krull rings are also used to set up powerful ''descent methods'', and in particular the Galoisian descent.


Cartier divisor

A
Cartier divisor In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumf ...
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 In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ver ...
of invertible sheaves on Spec(''A''). Example: in the ring ''k'' 'x'',''y'',''z''(''xy''–''z''2) the divisor class group has order 2, generated by the divisor ''y''=''z'', but the Picard subgroup is the trivial group.Hartshorne, GTM52, Example 6.5.2, p.133 and Example 6.11.3, p.142.


References

* * * * * 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. *{{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 Ring theory Commutative algebra