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  $$  v _ {i} ( x) \geq  0 $$ 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 $$v_{\mathfrak p}$$ of $$K$$ whose valuation ring is $$A_{\mathfrak p}$$. Then the set $$\mathcal V = \{v_{\mathfrak p}\}$$ satisfies the conditions of the equivalent definition. Conversely, if the set $$\mathcal V' = \{v_i\}$$ 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 $$v_{\mathfrak p}$$ 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, $$x \not\in A_{\mathfrak p}\mathfrak p$$ 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 $$v_{\mathfrak p} (xx^{-1}) = v_{\mathfrak p} (1) = 0 = v_{\mathfrak p} (x) + v_{\mathfrak p} (x^{-1})$$, hence $$v_{\mathfrak p} (x) = v_{\mathfrak p} (x^{-1}) = 0$$, since both numbers must be $$\geq 0$$.
 * 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 $$v_{\mathfrak p} (xy^{-1}) = 0$$, hence $$xy^{-1}\in U$$ by the above property (q.e.d). This shows that the application $$x\ {\rm mod}\ U\mapsto \left(v_{\mathfrak p}(x) \right)_{\mathfrak p \in P}$$ is well defined, and since $$v_{\mathfrak p}(x)\not = 0$$ 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 $$x\in A^\times$$, $$x = 1\cdot \mathfrak p_1^{\alpha_1}\cdot\mathfrak p_2^{\alpha_2}\cdots \mathfrak p_n^{\alpha_n}\ {\rm mod}\ U$$, where the $$\mathfrak p_i$$ 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 $$\mathfrak p_1, \ldots \mathfrak p_n$$ are distinct elements of $$P$$, $$ x_1,\ldots x_n$$ belong to $$K$$ (resp. $$A_{\mathfrak p}$$), and $$a_1, \ldots a_n$$ 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 $$\mathfrak p \cap S = \emptyset$$.
 * 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

 * 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 $$ Ax $$.
 * 4) The polynomial ring $$R[x_1, x_2, x_3, \ldots]$$ 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 $$\widehat{A}$$ 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
 * $$ j(\mathfrak p) = \sum_{\mathfrak P|\mathfrak p,\ \mathfrak P\in P(B)} e(\mathfrak P, \mathfrak p) \mathfrak P$$

(the above sum is finite since every $$x\in \mathfrak p$$ 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 $$\bar j:C(A)\to C(A[X])$$ 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.