6.4 Lattices and ideals

To generalise one step further, we can consider any finitely-generated subgroup M of K which contains a basis of K over $\mathbb {Q}$; such an M is called a lattice. Standard arguments then show that M is of the form $\mathbb {Z}$ ^. m₁ + ... + $\mathbb {Z}$ ^. m_n for some basis m_i of K. For any fixed column vector v, let g be the invertible matrix that makes m_i ^. v the standard basis of the space of column vectors. Then after applying g, we see that M ^. v becomes the standard lattice of column vectors with integer entries. The collection R(M) of all matrices in K that take M to itself, is thus identified with the ring which we denoted as R_g above. In the following paragraphs we assume that we have made this change of co-ordinates (i. e. that g is the identity matrix). In that case R = R(M) is precisely the order consisting of integer matrices. Moreover, there is a non-zero vector v so that M is precisely the collection of all $\alpha$ so that $\alpha$ ^. v is a vector with integer entries.

By collecting the denominators of the generators of M we can find a non-zero integer d so that d ^. M is contained in R. Since this is a subgroup of R that is closed under multiplication by R, it is an ideal I in R. Thus M = d^-1I is a fractional ideal for R. It is clear that R(d ^. M) = R(M) = R. More generally, for any non-zero $\alpha$ in K, we have R($\alpha$ ^. M) = R. Moreover, $\alpha$ ^. M is obtained by replacing the v in the previous paragraph by $\alpha^{-1}_{}$v, which is just another non-zero vector.

Conversely, let I be a non-zero ideal in the ring R. Let $\alpha$ be a non-zero element of I. Then $\alpha^{-1}_{}$ is in K and by collecting the denominators we find a non-zero integer d so that d ^. $\alpha^{-1}_{}$ has integer coefficients so is in R. But then d = d$\alpha^{-1}_{}$ ^. $\alpha$ is in I; thus I contains d ^. R. In particular, I contains a basis of K and is a free group of rank n; in other words I is a lattice. Clearly R is contained in R(I) but in general the latter could be bigger.

Now, for any non-zero ideal in R we have the restriction I $\cap$ $\mathbb {Z}$ = a$\mathbb {Z}$. By the above discussion this is a non-zero ideal in $\mathbb {Z}$. We also see that R/I is a quotient of the finite group R/aR; the latter group has order aⁿ. The order of R/I is called the norm of the ideal and denoted as Nm(I). The norm of an element $\alpha$ is det($\alpha$); these two definitions are related since Nm($\alpha$ ^. R) = | det($\alpha$)| (Exercise).

Now, we noted above that Nm(d ^. R) = dⁿ for any positive integer d so we can extend the above definition by defining for M = d^-1I, Nm(M) = d^-nNm(I). Similarly, the restriction of d ^. R is clearly d, so we define the restriction of M to be d^-1(I $\cap$ $\mathbb {Z}$). When M is contained in (i. e. M is an ideal) R, the two definitions are consistent.