Dedekind Domain - Fractional Ideals and The Class Group

Fractional Ideals and The Class Group

Let R be an integral domain with fraction field K. A fractional ideal is a nonzero R-submodule I of K for which there exists a nonzero x in K such that

(We remark that this is not exactly the same as the definition given on the page describing fractional ideals: the definition given there is that a fractional ideal is a nonzero finitely generated R-submodule of K. The two definitions are equivalent if and only if R is Noetherian. Otherwise our definition is strictly weaker, being permissive enough to make all nonzero R-submodules of R — i.e., integral ideals — fractional ideals.)

Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums : the product IJ is again a fractional ideal. The set Frac(R) of all fractional ideals endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal R.

For any fractional ideal I, one may define the fractional ideal

One then tautologically has . In fact one has equality if and only if I, as an element of the monoid of Frac(R), is invertible. In other words, if I has any inverse, then the inverse must be .

A principal fractional ideal is one of the form for some nonzero x in K. Note that each principal fractional ideal is invertible, the inverse of being simply . We denote the subgroup of principal fractional ideals by Prin(R).

A domain R is a PID if and only if every fractional ideal is principal. In this case, we have Frac(R) = Prin(R) =, since two principal fractional ideals and are equal iff is a unit in R.

For a general domain R, it is meaningful to take the quotient of the monoid Frac(R) of all fractional ideals by the submonoid Prin(R) of principal fractional ideals. However this quotient itself is generally only a monoid. In fact it is easy to see that the class of a fractional ideal I in Frac(R)/Prin(R) is invertible if and only if I itself is invertible.

Now we can appreciate (DD3): in a Dedekind domain—and only in a Dedekind domain! -- is every fractional ideal invertible. Thus these are precisely the class of domains for which Frac(R)/Prin(R) forms a group, the ideal class group Cl(R) of R. This group is trivial if and only if R is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID.

We note that for an arbitrary domain one may define the Picard group Pic(R) as the group of invertible fractional ideals Inv(R) modulo the subgroup of principal fractional ideals. For a Dedekind domain this is of course the same as the ideal class group. However, on a more general class of domains—including Noetherian domains and Krull domains -- the ideal class group is constructed in a different way, and there is a canonical homomorphism

Pic(R) Cl(R)

which is however generally neither injective nor surjective. This is an affine analogue of the distinction between Cartier divisors and Weil divisors on a singular algebraic variety.

A remarkable theorem of L. Claborn (Claborn 1966) asserts that for any abelian group G whatsoever, there exists a Dedekind domain R whose ideal class group is isomorphic to G. Later, C.R. Leedham-Green showed that such an R may constructed as the integral closure of a PID in a quadratic field extension (Leedham-Green 1972). In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain which is a subring of the rational function field of an elliptic curve, and conjectured that such an "elliptic" construction should be possible for a general abelian group (Rosen 1976). Rosen's conjecture was proven in 2008 by P.L. Clark (Clark 2009).

In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.