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

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

1. for any $x\in K\setminus \{0\}$ and all $i$, except possibly a finite number of them, $v_i(x)=0$,
2. for any $x\in K\setminus \{0\}$, $x$ belongs to $A$ if and only if $v_i(x)\ge 0$ 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 $v_{\mathfrak{𝔭}}$ of $K$ whose valuation ring is $A_{\mathfrak{𝔭}}$. Then the set ${𝒱}=\{v_{\mathfrak{𝔭}}\}$ satisfies the conditions of the equivalent definition. Conversely, if the set ${{𝒱}}^{\prime }=\{v_i\}$ is as above, and the $v_i$ have been normalized, then ${{𝒱}}^{\prime }$ may be bigger than ${𝒱}$, but it must contain ${𝒱}$. In other words, ${𝒱}$ is the minimal set of normalized valuations satisfying the equivalent definition.

Properties

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

• An element $x\in K$ belongs to $U$ if, and only if, $v_{\mathfrak{𝔭}}(x)=0$ for every $\mathfrak{𝔭}\in P$. Indeed, in this case, $x\in ̸A_{\mathfrak{𝔭}}\mathfrak{𝔭}$ for every $\mathfrak{𝔭}\in P$, hence $x^{-1}\in A_{\mathfrak{𝔭}}$; by the intersection property, $x^{-1}\in A$. Conversely, if $x$ and $x^{-1}$ are in $A$, then $v_{\mathfrak{𝔭}}(x{x}^{-1})=v_{\mathfrak{𝔭}}(1)=0=v_{\mathfrak{𝔭}}(x)+v_{\mathfrak{𝔭}}(x^{-1})$, hence $v_{\mathfrak{𝔭}}(x)=v_{\mathfrak{𝔭}}(x^{-1})=0$, since both numbers must be $\ge 0$.
• An element $x\in A$ is uniquely determined, up to a unit of $A$, by the values $v_{\mathfrak{𝔭}}(x)$, $\mathfrak{𝔭}\in P$. Indeed, if $v_{\mathfrak{𝔭}}(x)=v_{\mathfrak{𝔭}}(y)$ for every $\mathfrak{𝔭}\in P$, then $v_{\mathfrak{𝔭}}(x{y}^{-1})=0$, hence $x{y}^{-1}\in U$ by the above property (q.e.d). This shows that the application $$\begin{gathered} x \\ \mathrm{m}\mathrm{o}\mathrm{d} \\ U\mapsto {(v_{\mathfrak{𝔭}}(x))}_{\mathfrak{𝔭}\in P} \end{gathered}$$ is well defined, and since $v_{\mathfrak{𝔭}}(x)=0$ for only finitely many $\mathfrak{𝔭}$, it is an embedding of $A^{\times }/U$ into the free Abelian group generated by the elements of $P$. Thus, using the multiplicative notation "$\cdot$" for the later group, there holds, for every $x\in A^{\times }$, $$\begin{gathered} x=1\cdot {\mathfrak{𝔭}}_1^{{\alpha }_1}\cdot {\mathfrak{𝔭}}_2^{{\alpha }_2}\cdots {\mathfrak{𝔭}}_n^{{\alpha }_n} \\ \mathrm{m}\mathrm{o}\mathrm{d} \\ U \end{gathered}$$, where the ${\mathfrak{𝔭}}_i$ are the elements of $P$ containing $x$, and ${\alpha }_i=v_{{\mathfrak{𝔭}}_i}(x)$.
• The valuations $v_{\mathfrak{𝔭}}$ are pairwise independent. As a consequence, there holds the so-called weak approximation theorem, an homologue of the Chinese remainder theorem: if ${\mathfrak{𝔭}}_1,\dots {\mathfrak{𝔭}}_n$ are distinct elements of $P$, $x_1,\dots x_n$ belong to $K$ (resp. $A_{\mathfrak{𝔭}}$), and $a_1,\dots a_n$ are $n$ natural numbers, then there exist $x\in K$ (resp. $x\in A_{\mathfrak{𝔭}}$) such that $v_{{\mathfrak{𝔭}}_i}(x-x_i)=n_i$ 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 all of its quotients $A/\mathfrak{𝔭}$ by height-1 primes are noetherian.
• Two elements $x$ and $y$ of $A$ are coprime if $v_{\mathfrak{𝔭}}(x)$ and $v_{\mathfrak{𝔭}}(y)$ 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 $S^{-1}A$ is again a Krull domain. In fact, the essential valuations of $S^{-1}A$ are those valuation $v_{\mathfrak{𝔭}}$ (of $K$) for which $\mathfrak{𝔭}\cap S=\mathrm{\varnothing }$.
• 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

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 $A$ is a Krull domain then so is the polynomial ring $A[x]$ and the formal power series ring $A[[x]]$.
4. The polynomial ring $R[x_1,x_2,x_3,\dots ]$ in infinitely many variables over a unique factorization domain $R$ is a Krull domain which is not noetherian.
5. Let $A$ be a Noetherian domain with quotient field $K$, and $L$ be a finite algebraic extension of $K$. Then the integral closure of $A$ in $L$ is a Krull domain (Mori–Nagata theorem).
6. Let $A$ be a Zariski ring (e.g., a local noetherian ring). If the completion $\hat{A}$ is a Krull domain, then $A$ is a Krull domain (Mori).
7. 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 $S^{-1}A$ 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 $div(x)=\sum _{p\in P}{v}_p(x)\cdot p$, 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 $v_{\mathfrak{𝔓}}$ over $v_{\mathfrak{𝔭}}$ 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

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

(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 $\overline{j}:C(A)\to C(B)$. This leads to several results. For example, the following generalizes a theorem of Gauss: The application $\overline{j}:C(A)\to C(A[X])$ is bijective. In particular, if $A$ is a unique factorization domain, then so is $A[X]$. 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.
• "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