Proposition 4

— *Let A be a principal ideal domain, let L be a free A-module with finite basis $(u_j)$ ($1 \leq j \leq k$), let M be a submodule of L, let $(x_i)$ be a system of generators of M and let $A \alpha_i$ ($1 \leq i \leq n$) be the invariant factors of M with respect to L. Then for $1 \leq m \leq n$ the product $\delta_m = \alpha_1 \ldots \alpha_m$ is a gcd of the m-th order minors of the matrix whose columns are the coordinate vectors of the $x_i$ with respect to the basis $(u_j)$.

By Th. 1 it is clear that $M \subset \alpha_1 L$; hence the coordinates of any element of M are multiples of $a_1$. On the other hand, there exists an element $x$ of M for which $a_1$ is a content in L. Expressing $x$ as a linear combination of the $x_i$, it follows that $a_1$ is an element of the ideal generated by the coordinates of the $x_i$. As these are all multiples of $a_1$, it follows that $a_1$ is indeed their gcd, and our assertion is proved in the case $m = 1$.

For general $m$, consider the m-th exterior power $\bigwedge^m M$ of M (III, p. 507). In the notation of Th. 1 there exists a basis $(a_i)$ for M where $a_i = \alpha_i e_i$ ($1 \leq i \leq n$); thus $\bigwedge^m M$ has a basis consisting of the elements $a_{i_1} \wedge \ldots \wedge a_{i_m}$, as $(i_1, \ldots, i_m)$ runs through the set of strictly increasing sequences of $m$ elements of $(1, n)$. Now the elements $e_{i_1} \wedge \ldots \wedge e_{i_m}$ belong to a basis $B_m$ of $\bigwedge^m L$. Hence the canonical map from $\bigwedge^m M$ into $\bigwedge^m L$ is an isomorphism from $\bigwedge^m M$ onto the submodule of $\bigwedge^m L$ having as a basis the elements $(\alpha_{i_1} \ldots \alpha_{i_m})\, e_{i_1} \wedge \ldots \wedge e_{i_m}$, and we identify this submodule with $\bigwedge^m M$. Since $\alpha_j$ is a multiple of $\alpha_k$ for $j \geq k$, the elements $\alpha_{i_1} \ldots \alpha_{i_m}$ are all multiples of $\delta_m = \alpha_1 \ldots \alpha_m$, and one of them is equal to $\delta_m$; thus $\delta_m$ is a gcd of the coordinates with respect to $B_m$ of the elements of a system of generators of $\bigwedge^m M$. The first part of the argument shows that then $\delta_m$ is a gcd of the set of coordinates of any system of generators of $\bigwedge^m M$, with respect to any basis of $\bigwedge^m L$. Taking the basis for $\bigwedge^m L$ induced from the basis $(u_j)$ of $L$, and the system of generators for $\bigwedge^m M$ consisting of exterior products of the $(x_i)$, the expression for the coordinates of these products in terms of determinants (III, p. 528, Prop. 9) gives the stated result.