Editing Dedekind domain (section)

== 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 module]]s over a Dedekind domain.

Let us briefly recall the structure theory in the case of a finitely generated module <math>M</math> over a PID <math>R</math>.  We define the [[torsion submodule]] <math>T</math> to be the set of elements <math>m</math> of <math>M</math> such that <math>rm = 0</math> for some nonzero <math>r</math> in <math>R</math>.  Then:

(M1) <math>T</math> can be decomposed into a [[direct sum of modules|direct sum]] of [[cyclic module|cyclic]] torsion modules, each of the form <math>R/I</math> for some nonzero ideal <math>I</math> of <math>R</math>.  By the [[Chinese Remainder Theorem]], each <math>R/I</math> can further be decomposed into a direct sum of submodules of the form <math>R/P^i</math>, where <math>P^i</math> is a power of a prime ideal.  This decomposition need not be unique, but any two decompositions

: <math>T \cong R/P_1^{a_1} \oplus \cdots \oplus R/P_r^{a_r} \cong R/Q_1^{b_1} \oplus \cdots \oplus R/Q_s^{b_s} </math>

differ only in the order of the factors.

(M2) The torsion submodule is a direct summand. That is, there exists a complementary submodule <math>P</math> of <math>M</math> such that <math>M = T \oplus P</math>.

(M3PID) <math>P</math> isomorphic to <math>R^n</math> for a uniquely determined non-negative integer <math>n</math>.  In particular, <math>P</math> is a finitely generated free module.

Now let <math>M</math> be a finitely generated module over an arbitrary Dedekind domain <math>R</math>.  Then (M1) and (M2) hold verbatim.  However, it follows from (M3PID) that a finitely generated torsionfree module <math>P</math> over a PID is free.  In particular, it asserts that all fractional ideals are principal, a statement that is false whenever <math>R</math> is not a PID.  In other words, the nontriviality of the class group <math>Cl(R)</math> 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) <math>P</math> is isomorphic to a direct sum of rank one projective modules: <math>P \cong I_1 \oplus \cdots \oplus I_r</math>.  Moreover, for any rank one projective modules <math>I_1,\ldots,I_r,J_1,\ldots,J_s</math>, one has

: <math> I_1 \oplus \cdots \oplus I_r \cong J_1 \oplus \cdots \oplus J_s</math>

if and only if

: <math>r = s</math>

and

: <math>I_1 \otimes \cdots \otimes I_r \cong J_1 \otimes  \cdots \otimes J_s.\,</math>

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

: <math> [I_1 \cdots I_r] = [J_1 \cdots J_s] \in Cl(R). </math>

Thus a finitely generated torsionfree module of rank <math>n > 0</math> can be expressed as <math>R^{n-1} \oplus I</math>, where <math>I</math> is a rank one projective module.  The '''Steinitz class''' for <math>P</math> over <math>R</math> is the class <math>[I]</math> of <math>I</math> in <math>Cl(R)</math>: it is uniquely determined.<ref name=FT95>Fröhlich & Taylor (1991) p.95</ref>  A consequence of this is:

Theorem: Let <math>R</math> be a Dedekind domain.  Then <math>K_0(R) \cong \mathbb{Z} \oplus Cl(R)</math>, where <math>K_0(R)</math> is the [[Grothendieck group]] of the commutative monoid of finitely generated projective <math>R</math> modules.

These results were established by [[Ernst Steinitz]] in 1912.

An additional consequence of this structure, which is not implicit in the preceding theorem, is that if the two projective modules over a Dedekind domain have the same class in the Grothendieck group, then they are in fact abstractly isomorphic.