6.5 Groups of invertible fractional ideals

The above definitions depended on a choice of ring R $\subset$ R(M), but the following definition does not. As before, let $\check{M}$ denote the collection of all $\alpha$ in K for which $\langle$$\alpha$,$\beta$$\rangle$ is an integer for every $\beta$ in M. Now, the non-degeneracy of the pairing $\langle$,$\rangle$ means that for every additive map from K to $\mathbb {Q}$ there is an $\alpha$ in K so that the additive map is given by $\beta$ $\mapsto$ $\langle$$\alpha$,$\beta$$\rangle$. It follows that $\check{M}$ can also be identified with the collection of all additive maps from M to $\mathbb {Z}$. By the usual double-duality result it follows that M = ($\check{M}$$\check{)}$. In particular, we see that $\check{M}$ is also a lattice and R($\check{M}$) = R(M).

Let [M : R] denote the collection of all $\alpha$ in K so that $\alpha$ ^. M is contained in R. Clearly, d ^. R is contained in [M : R]. On the other hand $\check{M}$ = [M : $\check{R}$] was shown above to contain all $\alpha$ that send M into $\check{R}$. The latter contains R so we see that [M : R] is contained in $\check{M}$. Thus [M : R] is also a lattice. Specifically, we define C_R as [$\check{R}$ : R].

Definition 3   Let M be a lattice in K and R $\subset$ R(M). Then we say that M is projective over R if M ^. [M : R] = R. When R = R(M) and we have [M : R] = C_R ^. $\check{M}$ then we say that M is a Gorenstein R module. Here the product of lattices L₁ ^. L₂ is the collection of all linear combinations of products $\alpha$$\beta$ with $\alpha$ in L₁ and $\beta$ in L₂.

(Further details are in the first appendix).

Armed with this result, we now consider the collection of all lattices M with the property that R(M) = R for a fixed Gorenstein order R. This collection of lattices includes R, $\check{R}$ and C_R. For any such M, the above lemma says that M ^. [M : R] = R. If we define the product of M and N as M ^. N, then this shows that we have a group with R playing the role of identity. It is further clear that M and $\alpha$ ^. M are naturally isomorphic for any non-zero $\alpha$ in K. We may further consider lattices modulo such isomorphisms. This gives us the class group of invertible fractional ideals modulo isomorphism which is denoted by Cl(R). We noted above that there could be ideals (and fractional ideals) M for R such that R is a proper subring of R(M). In this case we do not necessarily have M[M : R] = R; we do not include such M in the class group. However, since R(M) is an order as well, this situation cannot arise if R is the maximal order $\mathcal {O}$_K. The corresponding class group is sometimes loosely referred to as the class group of K and denoted Cl(K).