Exercise 9, § 4

Let A be a principal ideal domain and K its field of fractions; let E be a vector space over K, let M be a finitely generated A-module of rank $n$ contained in E, let N be a finitely generated A-module of rank p contained in KM, and let P be a finitely generated $A$-module of rank q contained in KN.

a) Let $A\alpha_i$ ($1 \leq i \leq p$) be the invariant factors of N with respect to M, arranged in decreasing order (Ex. 3). Show that, for $1 \leq k \leq p$, the element $\alpha_k$ is a gcd of those elements $A \in K$ such that the rank of the quotient module $(N + \lambda(KN \cap M))/N$ is $\leq p - k$ (cf. Ex. 8, d)).

b) Let $A\beta_j$ ($1 \leq j \leq q$) be the invariant factors of P with respect to M and Ay, ($1 \leq j \leq q$) the invariant factors of P with respect to N, arranged in decreasing order. Show that $\alpha_1 \gamma_j$ divides $\beta_j$ for $1 \leq j \leq q$ (cf. Ex. 8).

c) Show that $\gamma_1 \alpha_j$ divides $\beta_j$ for $1 \leq j \leq q$. (First consider the case $n = p = q$, and apply Ex. 8. In general, show that we may always assume $p = n$, and consider a submodule Q of M such that $P + \rho Q$ has rank $p$ for all $\rho \in A$; then choose the ideal $Ap$ sufficiently small, and apply Prop. 4 of VII, p. 20.)

d) For any submodule H of N of rank $k \leq p$, let $(\mu_{H})$ be the fractional ideal consisting of $\mu \in K$ such that $\mu(M \cap KH) \subset H$. Deduce from c) that for each $1 \leq k \leq p$ the element $a_{k}$ is a gcd of the $\mu_{H}$ as H runs through all submodules of N of rank k.