Dedekind Domain - Finitely Generated Modules Over A Dedekind Domain

Finitely Generated Modules Over A Dedekind Domain

In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.

Let us briefly recall the structure theory in the case of a finitely generated module over a PID . We define the torsion submodule to be the set of elements of such that for some nonzero in . Then:

(M1) can be decomposed into a direct sum of cyclic torsion modules, each of the form for some nonzero ideal of . By the Chinese Remainder Theorem, each can further be decomposed into a direct sum of submodules of the form, where is a power of a prime ideal. This decomposition need not be unique, but any two decompositions

differ only in the order of the factors.

(M2) The torsion submodule is a direct summand: i.e., there exists a complementary submodule of such that .

(M3PID) isomorphic to for a uniquely determined non-negative integer . In particular, a finitely generated free module.

Now let be a finitely generated module over an arbitrary Dedekind domain . Then (M1) and (M2) hold verbatim. However, it follows from (M3PID) that a finitely generated torsionfree module over a PID is free. In particular, it asserts that all fractional ideals are principal, a statement which is false whenever is not a PID. In other words, the nontriviality of the class group Cl(R) causes (M3PID) to fail. Remarkably, the additional structure in torsionfree finitely generated modules over an arbitrary Dedekind domain is precisely controlled by the class group, as we now explain. Over an arbitrary Dedekind domain one has

(M3DD) is isomorphic to a direct sum of rank one projective modules: . Moreover, for any rank one projective modules, one has

if and only if

and

Rank one projective modules can be identified with fractional ideals, and the last condition can be rephrased as

Thus a finitely generated torsionfree module of rank can be expressed as, where is a rank one projective module. The class of in Cl(R) is uniquely determined. A consequence of this is:

Theorem: Let R be a Dedekind domain. Then, where K₀(R) is the Grothendieck group of the commutative monoid of finitely generated projective R modules.

These results were established by Ernst Steinitz in 1912.