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.[1] They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.

Formal definition

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

1. ${\displaystyle A_{\mathfrak {p}}}$ is a discrete valuation ring for all ${\displaystyle {\mathfrak {p}}\in P}$,
2. ${\displaystyle A}$ is the intersection of these discrete valuation rings (considered as subrings of the quotient field of ${\displaystyle A}$).
3. Any nonzero element of ${\displaystyle A}$ 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 ${\displaystyle A}$ is a Krull ring if there exists a family ${\displaystyle \{v_{i}\}_{i\in I}}$ of discrete valuations on the field of fractions ${\displaystyle K}$ of ${\displaystyle A}$ such that:

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

The valuations ${\displaystyle v_{i}}$ are called essential valuations of ${\displaystyle A}$.

The link between the two definitions is as follows: for every ${\displaystyle {\mathfrak {p}}\in P}$, one can associate a unique normalized valuation ${\displaystyle v_{\mathfrak {p}}}$ of ${\displaystyle K}$ whose valuation ring is ${\displaystyle A_{\mathfrak {p}}}$.[3] Then the set ${\displaystyle {\mathcal {V}}=\{v_{\mathfrak {p}}\}}$ satisfies the conditions of the equivalent definition. Conversely, if the set ${\displaystyle {\mathcal {V}}'=\{v_{i}\}}$ is as above, and the ${\displaystyle v_{i}}$ have been normalized, then ${\displaystyle {\mathcal {V}}'}$ may be bigger than ${\displaystyle {\mathcal {V}}}$, but it must contain ${\displaystyle {\mathcal {V}}}$. In other words, ${\displaystyle {\mathcal {V}}}$ 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

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

• An element ${\displaystyle x\in K}$ belongs to ${\displaystyle U}$ if, and only if, ${\displaystyle v_{\mathfrak {p}}(x)=0}$ for every ${\displaystyle {\mathfrak {p}}\in P}$. Indeed, in this case, ${\displaystyle x\not \in A_{\mathfrak {p}}{\mathfrak {p}}}$ for every ${\displaystyle {\mathfrak {p}}\in P}$, hence ${\displaystyle x^{-1}\in A_{\mathfrak {p}}}$; by the intersection property, ${\displaystyle x^{-1}\in A}$. Conversely, if ${\displaystyle x}$ and ${\displaystyle x^{-1}}$ are in ${\displaystyle A}$, then ${\displaystyle v_{\mathfrak {p}}(xx^{-1})=v_{\mathfrak {p}}(1)=0=v_{\mathfrak {p}}(x)+v_{\mathfrak {p}}(x^{-1})}$, hence ${\displaystyle v_{\mathfrak {p}}(x)=v_{\mathfrak {p}}(x^{-1})=0}$, since both numbers must be ${\displaystyle \geq 0}$.
• An element ${\displaystyle x\in A}$ is uniquely determined, up to a unit of ${\displaystyle A}$, by the values ${\displaystyle v_{\mathfrak {p}}(x)}$, ${\displaystyle {\mathfrak {p}}\in P}$. Indeed, if ${\displaystyle v_{\mathfrak {p}}(x)=v_{\mathfrak {p}}(y)}$ for every ${\displaystyle {\mathfrak {p}}\in P}$, then ${\displaystyle v_{\mathfrak {p}}(xy^{-1})=0}$, hence ${\displaystyle xy^{-1}\in U}$ by the above property (q.e.d). This shows that the application ${\displaystyle x\ {\rm {mod}}\ U\mapsto \left(v_{\mathfrak {p}}(x)\right)_{{\mathfrak {p}}\in P}}$ is well defined, and since ${\displaystyle v_{\mathfrak {p}}(x)\not =0}$ for only finitely many ${\displaystyle {\mathfrak {p}}}$, it is an embedding of ${\displaystyle A^{\times }/U}$ into the free Abelian group generated by the elements of ${\displaystyle P}$. Thus, using the multiplicative notation "${\displaystyle \cdot }$" for the later group, there holds, for every ${\displaystyle x\in A^{\times }}$, ${\displaystyle x=1\cdot {\mathfrak {p}}_{1}^{\alpha _{1}}\cdot {\mathfrak {p}}_{2}^{\alpha _{2}}\cdots {\mathfrak {p}}_{n}^{\alpha _{n}}\ {\rm {mod}}\ U}$, where the ${\displaystyle {\mathfrak {p}}_{i}}$ are the elements of ${\displaystyle P}$ containing ${\displaystyle x}$, and ${\displaystyle \alpha _{i}=v_{{\mathfrak {p}}_{i}}(x)}$.
• The valuations ${\displaystyle v_{\mathfrak {p}}}$ are pairwise independent.[4] As a consequence, there holds the so-called weak approximation theorem,[5] an homologue of the Chinese remainder theorem: if ${\displaystyle {\mathfrak {p}}_{1},\ldots {\mathfrak {p}}_{n}}$ are distinct elements of ${\displaystyle P}$, ${\displaystyle x_{1},\ldots x_{n}}$ belong to ${\displaystyle K}$ (resp. ${\displaystyle A_{\mathfrak {p}}}$), and ${\displaystyle a_{1},\ldots a_{n}}$ are ${\displaystyle n}$ natural numbers, then there exist ${\displaystyle x\in K}$ (resp. ${\displaystyle x\in A_{\mathfrak {p}}}$) such that ${\displaystyle v_{{\mathfrak {p}}_{i}}(x-x_{i})=n_{i}}$ for every ${\displaystyle i}$.
• Two elements ${\displaystyle x}$ and ${\displaystyle y}$ of ${\displaystyle A}$ are coprime if ${\displaystyle v_{\mathfrak {p}}(x)}$ and ${\displaystyle v_{\mathfrak {p}}(y)}$ are not both ${\displaystyle >0}$ for every ${\displaystyle {\mathfrak {p}}\in P}$. The basic properties of valuations imply that a good theory of coprimality holds in ${\displaystyle A}$.
• Every prime ideal of ${\displaystyle A}$ contains an element of ${\displaystyle P}$.[6]
• Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain.[7]
• If ${\displaystyle L}$ is a subfield of ${\displaystyle K}$, then ${\displaystyle A\cap L}$ is a Krull domain.[8]
• If ${\displaystyle S\subset A}$ is a multiplicatively closed set not containing 0, the ring of quotients ${\displaystyle S^{-1}A}$ is again a Krull domain. In fact, the essential valuations of ${\displaystyle S^{-1}A}$ are those valuation ${\displaystyle v_{\mathfrak {p}}}$ (of ${\displaystyle K}$) for which ${\displaystyle {\mathfrak {p}}\cap S=\emptyset }$.[9]
• If ${\displaystyle L}$ is a finite algebraic extension of ${\displaystyle K}$, and ${\displaystyle B}$ is the integral closure of ${\displaystyle A}$ in ${\displaystyle L}$, then ${\displaystyle B}$ is a Krull domain.[10]

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.[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 ${\displaystyle A}$ is a Krull domain then so is the polynomial ring ${\displaystyle A[x]}$ and the formal power series ring ${\displaystyle A[[x]]}$.[14]
4. The polynomial ring ${\displaystyle R[x_{1},x_{2},x_{3},\ldots ]}$ in infinitely many variables over a unique factorization domain ${\displaystyle R}$ is a Krull domain which is not noetherian.
5. Let ${\displaystyle A}$ be a Noetherian domain with quotient field ${\displaystyle K}$, and ${\displaystyle L}$ be a finite algebraic extension of ${\displaystyle K}$. Then the integral closure of ${\displaystyle A}$ in ${\displaystyle L}$ is a Krull domain (Mori–Nagata theorem).[15] This follows easily from numero 2 above.
6. Let ${\displaystyle A}$ be a Zariski ring (e.g., a local noetherian ring). If the completion ${\displaystyle {\widehat {A}}}$ is a Krull domain, then ${\displaystyle A}$ is a Krull domain (Mori).[16][17]
7. Let ${\displaystyle A}$ be a Krull domain, and ${\displaystyle V}$ be the multiplicatively closed set consisting in the powers of a prime element ${\displaystyle p\in A}$. Then ${\displaystyle S^{-1}A}$ is a Krull domain (Nagata).[18]

The divisor class group of a Krull ring

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

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

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

${\displaystyle j({\mathfrak {p}})=\sum _{{\mathfrak {P}}|{\mathfrak {p}},\ {\mathfrak {P}}\in P(B)}e({\mathfrak {P}},{\mathfrak {p}}){\mathfrak {P}}}$

(the above sum is finite since every ${\displaystyle x\in {\mathfrak {p}}}$ is contained in at most finitely many elements of ${\displaystyle P(B)}$). Let extend the application ${\displaystyle j}$ by linearity to a linear application ${\displaystyle D(A)\to D(B)}$. One can now ask in what cases ${\displaystyle j}$ induces a morphism ${\displaystyle {\bar {j}}:C(A)\to C(B)}$. This leads to several results.[19] For example, the following generalizes a theorem of Gauss:

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

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

1. ^
2. ^ P. Samuel, Lectures on Unique Factorization Domain, Theorem 3.5.
3. ^ A discrete valuation ${\displaystyle v}$ is said to be normalized if ${\displaystyle v(O_{v})=\mathbb {N} }$, where ${\displaystyle O_{v}}$ is the valuation ring of ${\displaystyle v}$. So, every class of equivalent discrete valuations contains a unique normalized valuation.
4. ^ If ${\displaystyle v_{{\mathfrak {p}}_{1}}}$ and ${\displaystyle v_{{\mathfrak {p}}_{2}}}$were both finer than a common valuation ${\displaystyle w}$ of ${\displaystyle K}$, the ideals ${\displaystyle A_{{\mathfrak {p}}_{1}}{\mathfrak {p}}_{1}}$ and ${\displaystyle A_{{\mathfrak {p}}_{2}}{\mathfrak {p}}_{2}}$ of their corresponding valuation rings would contain properly the prime ideal ${\displaystyle {\mathfrak {p}}_{w}=\{x\in K:\ w(x)>0\},}$ hence ${\displaystyle {\mathfrak {p}}_{1}}$ and ${\displaystyle {\mathfrak {p}}_{2}}$ would contain the prime ideal ${\displaystyle {\mathfrak {p}}_{w}\cap A}$ of ${\displaystyle A}$, 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.