Divisor class group

The quotient group of the group $ D ( A) $ of divisorial ideals (cf. Divisorial ideal) of a Krull ring $ A $ by the subgroup $ F ( A) $ consisting of the principal ideals. The divisor class group is Abelian and is usually denoted by $ C ( A) $. The group $ C ( A) $ is generated by the classes of the prime ideals of height 1 in $ A $( cf. Height of an ideal).

In a sense, the divisor class group measures the divergence from uniqueness of the factorization of elements of $ A $ into irreducible factors. Thus, a factorial ring has trivial divisor class group.

Let $ \phi : A \rightarrow B $ be a homomorphism of Krull rings; then, under certain additional assumptions (for instance, when $ B $ is an integral or flat extension of $ A $), there is a canonical homomorphism $ \phi ^ {*} : C ( A) \rightarrow C ( B) $ of divisor class groups. If $ B $ is the localization of $ A $ with respect to a multiplicative system $ S $( cf. Localization in a commutative algebra), then $ \phi ^ {*} $ is surjective and the kernel of $ \phi ^ {*} $ is generated by the divisorial prime ideals of $ A $ that meet $ S $( Nagata's theorem). If $ B $ is the ring of polynomials over $ A $, then the canonical homomorphism $ \phi ^ {*} $ is bijective (this is a generalization of Gauss' theorem stating that the ring of polynomials over a field is factorial). In the more general case where $ B $ is the symmetric Noetherian algebra of an $ A $- module $ M $, the canonical homomorphism $ \phi ^ {*} $ is bijective provided that all symmetric powers $ S ^ {i} ( M) $ are reflexive. If $ B $ is the ring of formal power series over $ A $, then $ \phi ^ {*} $ is injective (and even left invertible), but not bijective, in general.

The subgroup of $ C ( A) $ generated by the invertible ideals is isomorphic to the Picard group $ \mathop{\rm Pic} ( A) $ of $ A $, and the functorial properties of $ \mathop{\rm Pic} ( A) $ and $ C ( A) $ are compatible. Thus, if $ B $ is a faithfully flat extension of a ring $ A $ and $ \phi ^ {*} : \mathop{\rm Pic} ( A) \rightarrow \mathop{\rm Pic} ( B) $ is injective, then $ \phi ^ {*} : C ( A) \rightarrow C ( B) $ is also injective. In particular, if the completion $ \widehat{A} $ of a local ring $ A $ is factorial, then $ A $ is also factorial (Mori's theorem).

Let $ A $ be a normal Noetherian ring. The group $ \mathop{\rm Pic} ( A) $ coincides with $ C ( A) $ if and only if $ A $ is locally a factorial ring, that is, if all the local rings $ A _ {\mathfrak m } $ are factorial (for instance, when $ A $ is regular). More exactly, if $ F = \{ {\mathfrak p \in \mathop{\rm Spec} ( A) } : {A _ {\mathfrak p } \textrm{ is factorial } } \} $, then $ C ( A) = \lim\limits _ \rightarrow \mathop{\rm Pic} ( U) $, where $ U $ runs over the system of open subschemes of $ \mathop{\rm Spec} ( A) $ containing $ F $. This allows one to define the divisor class group of a normal scheme [5] — the Weil divisor class group (see Divisor).

Divisor class groups were first studied for rings of algebraic numbers, and the earliest results concerning the finiteness of these groups were obtained by E. Kummer. There is a close connection between the properties of the divisor class group and number-theoretical problems, for instance, Fermat's theorem. Tables of orders of divisor class groups of certain rings of algebraic numbers are provided in [1].

In full generality, the theory of divisor class groups was obtained by W. Krull; P. Samuel studied the functorial character of divisor class groups and proposed some methods for computing them (for example, the method of descent). Other approaches to the study of the divisor class group are based on comparison with the Picard group, and cohomological and algebraic-geometrical methods are applied as well.

Every Abelian group occurs as a divisor class group.

References

[1]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[3]	P. Samuel, "Classes de diviseurs et dérivées logarithmiques" Topology , 3 (1964) pp. 81–96
[4]	R.M. Fossum, "The divisor class group of a Krull domain" , Springer (1973)
[5]	A. Grothendieck, J. Dieudonné, "Eléments de géometry algébrique IV" Publ. Math. IHES , 32 (1967)

Comments

Cf. also Class field theory for the relation between the divisor class group of a ring of algebraic integers and Abelian field extensions.