Krull ring

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. 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]\displaystyle{ A }[/math] be an integral domain and let [math]\displaystyle{ P }[/math] be the set of all prime ideals of [math]\displaystyle{ A }[/math] of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then [math]\displaystyle{ A }[/math] is a Krull ring if

1. [math]\displaystyle{ A_{\mathfrak{p}} }[/math] is a discrete valuation ring for all [math]\displaystyle{ \mathfrak{p} \in P }[/math],
2. [math]\displaystyle{ A }[/math] is the intersection of these discrete valuation rings (considered as subrings of the quotient field of [math]\displaystyle{ A }[/math]).
3. Any nonzero element of [math]\displaystyle{ 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:

An integral domain [math]\displaystyle{ A }[/math] is a Krull ring if there exists a family [math]\displaystyle{ \{ v _ {i} \} _ {i \in I } }[/math] of discrete valuations on the field of fractions [math]\displaystyle{ K }[/math] of [math]\displaystyle{ A }[/math] such that:

1. for any [math]\displaystyle{ x \in K \setminus \{ 0 \} }[/math] and all [math]\displaystyle{ i }[/math], except possibly a finite number of them, [math]\displaystyle{ v _ {i} ( x) = 0 }[/math];
2. for any [math]\displaystyle{ x \in K \setminus \{ 0 \} }[/math], [math]\displaystyle{ x }[/math] belongs to [math]\displaystyle{ A }[/math] if and only if [math]\displaystyle{ v _ {i} ( x) \geq 0 }[/math] for all [math]\displaystyle{ i \in I }[/math].

The valuations [math]\displaystyle{ v_i }[/math] are called essential valuations of [math]\displaystyle{ A }[/math].

The link between the two definitions is as follows: for every [math]\displaystyle{ \mathfrak p\in P }[/math], one can associate a unique normalized valuation [math]\displaystyle{ v_{\mathfrak p} }[/math] of [math]\displaystyle{ K }[/math] whose valuation ring is [math]\displaystyle{ A_{\mathfrak p} }[/math]. Then the set [math]\displaystyle{ \mathcal V = \{v_{\mathfrak p}\} }[/math] satisfies the conditions of the equivalent definition. Conversely, if the set [math]\displaystyle{ \mathcal V' = \{v_i\} }[/math] is as above, and the [math]\displaystyle{ v_i }[/math] have been normalized, then [math]\displaystyle{ \mathcal V' }[/math] may be bigger than [math]\displaystyle{ \mathcal V }[/math], but it must contain [math]\displaystyle{ \mathcal V }[/math]. In other words, [math]\displaystyle{ \mathcal V }[/math] 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 references on the subject is Lecture on Unique Factorization Domains by P. Samuel.

Properties

With the notations above, let [math]\displaystyle{ v_{\mathfrak p} }[/math] denote the normalized valuation corresponding to the valuation ring [math]\displaystyle{ A_{\mathfrak p} }[/math], [math]\displaystyle{ U }[/math] denote the set of units of [math]\displaystyle{ A }[/math], and [math]\displaystyle{ K }[/math] its quotient field.

• An element [math]\displaystyle{ x \in K }[/math] belongs to [math]\displaystyle{ U }[/math] if, and only if, [math]\displaystyle{ v_{\mathfrak p} (x) = 0 }[/math] for every [math]\displaystyle{ \mathfrak p \in P }[/math]. Indeed, in this case, [math]\displaystyle{ x \not\in A_{\mathfrak p}\mathfrak p }[/math] for every [math]\displaystyle{ \mathfrak p\in P }[/math], hence [math]\displaystyle{ x^{-1} \in A_{\mathfrak p} }[/math]; by the intersection property, [math]\displaystyle{ x^{-1}\in A }[/math]. Conversely, if [math]\displaystyle{ x }[/math] and [math]\displaystyle{ x^{-1} }[/math] are in [math]\displaystyle{ A }[/math], then [math]\displaystyle{ v_{\mathfrak p} (xx^{-1}) = v_{\mathfrak p} (1) = 0 = v_{\mathfrak p} (x) + v_{\mathfrak p} (x^{-1}) }[/math], hence [math]\displaystyle{ v_{\mathfrak p} (x) = v_{\mathfrak p} (x^{-1}) = 0 }[/math], since both numbers must be [math]\displaystyle{ \geq 0 }[/math].
• An element [math]\displaystyle{ x \in A }[/math] is uniquely determined, up to a unit of [math]\displaystyle{ A }[/math], by the values [math]\displaystyle{ v_{\mathfrak p} (x) }[/math], [math]\displaystyle{ \mathfrak p \in P }[/math]. Indeed, if [math]\displaystyle{ v_{\mathfrak p} (x) = v_{\mathfrak p} (y) }[/math] for every [math]\displaystyle{ \mathfrak p \in P }[/math], then [math]\displaystyle{ v_{\mathfrak p} (xy^{-1}) = 0 }[/math], hence [math]\displaystyle{ xy^{-1}\in U }[/math] by the above property (q.e.d). This shows that the application [math]\displaystyle{ x\ {\rm mod}\ U\mapsto \left(v_{\mathfrak p}(x) \right)_{\mathfrak p \in P} }[/math] is well defined, and since [math]\displaystyle{ v_{\mathfrak p}(x)\not = 0 }[/math] for only finitely many [math]\displaystyle{ \mathfrak p }[/math], it is an embedding of [math]\displaystyle{ A^{\times}/U }[/math] into the free Abelian group generated by the elements of [math]\displaystyle{ P }[/math]. Thus, using the multiplicative notation "[math]\displaystyle{ \cdot }[/math]" for the later group, there holds, for every [math]\displaystyle{ x\in A^\times }[/math], [math]\displaystyle{ 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]\displaystyle{ \mathfrak p_i }[/math] are the elements of [math]\displaystyle{ P }[/math] containing [math]\displaystyle{ x }[/math], and [math]\displaystyle{ \alpha_i = v_{\mathfrak p_i} (x) }[/math].
• The valuations [math]\displaystyle{ v_{\mathfrak p} }[/math] are pairwise independent. As a consequence, there holds the so-called weak approximation theorem, an homologue of the Chinese remainder theorem: if [math]\displaystyle{ \mathfrak p_1, \ldots \mathfrak p_n }[/math] are distinct elements of [math]\displaystyle{ P }[/math], [math]\displaystyle{ x_1,\ldots x_n }[/math] belong to [math]\displaystyle{ K }[/math] (resp. [math]\displaystyle{ A_{\mathfrak p} }[/math]), and [math]\displaystyle{ a_1, \ldots a_n }[/math] are [math]\displaystyle{ n }[/math] natural numbers, then there exist [math]\displaystyle{ x\in K }[/math] (resp. [math]\displaystyle{ x\in A_{\mathfrak p} }[/math]) such that [math]\displaystyle{ v_{\mathfrak p_i} (x - x_i) = n_i }[/math] for every [math]\displaystyle{ i }[/math].
• A consequence of the weak approximation theorem is a characterization of when Krull rings are noetherian; namely, a Krull ring [math]\displaystyle{ A }[/math] is noetherian if and only if all of its quotients [math]\displaystyle{ A/{\mathfrak p} }[/math] by height-1 primes are noetherian.
• Two elements [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] of [math]\displaystyle{ A }[/math] are coprime if [math]\displaystyle{ v_{\mathfrak p} (x) }[/math] and [math]\displaystyle{ v_{\mathfrak p} (y) }[/math] are not both [math]\displaystyle{ \gt 0 }[/math] for every [math]\displaystyle{ \mathfrak p\in P }[/math]. The basic properties of valuations imply that a good theory of coprimality holds in [math]\displaystyle{ A }[/math].
• Every prime ideal of [math]\displaystyle{ A }[/math] contains an element of [math]\displaystyle{ P }[/math].
• Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain.
• If [math]\displaystyle{ L }[/math] is a subfield of [math]\displaystyle{ K }[/math], then [math]\displaystyle{ A\cap L }[/math] is a Krull domain.
• If [math]\displaystyle{ S\subset A }[/math] is a multiplicatively closed set not containing 0, the ring of quotients [math]\displaystyle{ S^{-1}A }[/math] is again a Krull domain. In fact, the essential valuations of [math]\displaystyle{ S^{-1}A }[/math] are those valuation [math]\displaystyle{ v_{\mathfrak p} }[/math] (of [math]\displaystyle{ K }[/math]) for which [math]\displaystyle{ \mathfrak p \cap S = \emptyset }[/math].
• If [math]\displaystyle{ L }[/math] is a finite algebraic extension of [math]\displaystyle{ K }[/math], and [math]\displaystyle{ B }[/math] is the integral closure of [math]\displaystyle{ A }[/math] in [math]\displaystyle{ L }[/math], then [math]\displaystyle{ B }[/math] is a Krull domain.

Examples

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.
2. Every integrally closed noetherian domain is a Krull domain. 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 [math]\displaystyle{ A }[/math] is a Krull domain then so is the polynomial ring [math]\displaystyle{ A[x] }[/math] and the formal power series ring [math]\displaystyle{ Ax }[/math].
4. The polynomial ring [math]\displaystyle{ R[x_1, x_2, x_3, \ldots] }[/math] in infinitely many variables over a unique factorization domain [math]\displaystyle{ R }[/math] is a Krull domain which is not noetherian.
5. Let [math]\displaystyle{ A }[/math] be a Noetherian domain with quotient field [math]\displaystyle{ K }[/math], and [math]\displaystyle{ L }[/math] be a finite algebraic extension of [math]\displaystyle{ K }[/math]. Then the integral closure of [math]\displaystyle{ A }[/math] in [math]\displaystyle{ L }[/math] is a Krull domain (Mori–Nagata theorem).
6. Let [math]\displaystyle{ A }[/math] be a Zariski ring (e.g., a local noetherian ring). If the completion [math]\displaystyle{ \widehat{A} }[/math] is a Krull domain, then [math]\displaystyle{ A }[/math] is a Krull domain (Mori).
7. Let [math]\displaystyle{ A }[/math] be a Krull domain, and [math]\displaystyle{ V }[/math] be the multiplicatively closed set consisting in the powers of a prime element [math]\displaystyle{ p\in A }[/math]. Then [math]\displaystyle{ S^{-1}A }[/math] is a Krull domain (Nagata).

The divisor class group of a Krull ring

Assume that [math]\displaystyle{ A }[/math] is a Krull domain and [math]\displaystyle{ K }[/math] is its quotient field. A prime divisor of [math]\displaystyle{ A }[/math] is a height 1 prime ideal of [math]\displaystyle{ A }[/math]. The set of prime divisors of [math]\displaystyle{ A }[/math] will be denoted [math]\displaystyle{ P(A) }[/math] in the sequel. A (Weil) divisor of [math]\displaystyle{ A }[/math] is a formal integral linear combination of prime divisors. They form an Abelian group, noted [math]\displaystyle{ D(A) }[/math]. A divisor of the form [math]\displaystyle{ div(x)=\sum_{p\in P}v_p(x)\cdot p }[/math], for some non-zero [math]\displaystyle{ x }[/math] in [math]\displaystyle{ K }[/math], is called a principal divisor. The principal divisors of [math]\displaystyle{ A }[/math] form a subgroup of the group of divisors (it has been shown above that this group is isomorphic to [math]\displaystyle{ A^\times /U }[/math], where [math]\displaystyle{ U }[/math] is the group of unities of [math]\displaystyle{ A }[/math]). The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of [math]\displaystyle{ A }[/math]; it is usually denoted [math]\displaystyle{ C(A) }[/math].

Assume that [math]\displaystyle{ B }[/math] is a Krull domain containing [math]\displaystyle{ A }[/math]. As usual, we say that a prime ideal [math]\displaystyle{ \mathfrak P }[/math] of [math]\displaystyle{ B }[/math] lies above a prime ideal [math]\displaystyle{ \mathfrak p }[/math] of [math]\displaystyle{ A }[/math] if [math]\displaystyle{ \mathfrak P\cap A = \mathfrak p }[/math]; this is abbreviated in [math]\displaystyle{ \mathfrak P|\mathfrak p }[/math].

Denote the ramification index of [math]\displaystyle{ v_{\mathfrak P} }[/math] over [math]\displaystyle{ v_{\mathfrak p} }[/math] by [math]\displaystyle{ e(\mathfrak P,\mathfrak p) }[/math], and by [math]\displaystyle{ P(B) }[/math] the set of prime divisors of [math]\displaystyle{ B }[/math]. Define the application [math]\displaystyle{ P(A)\to D(B) }[/math] by

[math]\displaystyle{ 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]\displaystyle{ x\in \mathfrak p }[/math] is contained in at most finitely many elements of [math]\displaystyle{ P(B) }[/math]). Let extend the application [math]\displaystyle{ j }[/math] by linearity to a linear application [math]\displaystyle{ D(A)\to D(B) }[/math]. One can now ask in what cases [math]\displaystyle{ j }[/math] induces a morphism [math]\displaystyle{ \bar j:C(A)\to C(B) }[/math]. This leads to several results. For example, the following generalizes a theorem of Gauss:

The application [math]\displaystyle{ \bar j:C(A)\to C(A[X]) }[/math] is bijective. In particular, if [math]\displaystyle{ A }[/math] is a unique factorization domain, then so is [math]\displaystyle{ A[X] }[/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 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.

References

• N. Bourbaki. Commutative algebra. https://archive.org/details/commutativealgeb0000bour. 
• Hazewinkel, Michiel, ed. (2001), "Krull ring", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/k055930 
• Krull, Wolfgang (1931), "Allgemeine Bewertungstheorie", J. Reine Angew. Math. 167: 160–196, http://gdz.sub.uni-goettingen.de/en/dms/load/img/?IDDOC=260807 
• 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, 30, Bombay: Tata Institute of Fundamental Research, http://www.math.tifr.res.in/~publ/ln/