Exercise 14, § 4

Let $A$ be a Prüferian domain ($§ 2$, Exercise 12) and $M, N$ two finitely generated lattices in a vector space $V$ of finite rank $n$ over the field of fractions of $A$. For all $c \neq 0$ in $A$ such that $cN \subset M$, consider the determinantal ideals $\delta_k(M/cN)$ (Exercise 10) and write:

(a) Show that $\delta_k(N, M)$ does not depend on the choice of $c$ such that $cN \subset M$; these are called the determinantal ideals of $N$ with respect to $M$. (b) $\delta_k(N, M) \subset \delta_{k+1}(N, M)$ and $\delta_n(N, M) = A$. Show that, if we write $e_k(N, M) = \delta_{n-k}(N, M)(\delta_{n-k+1}(N, M))^{-1}$, the integral ideals $e_k(N, M)$ are such that $e_k(N, M) \supset e_{k+1}(N, M)$ for $1 \leq k \leq n$; the finitely generated ideals $e_k(N, M)$ are called the invariant factors of $N$ with respect to $M$ (consider the $A_m$-modules $M_m$ and $N_m$ for every maximal ideal $m$ of $A$). (c) If $M$ is a lattice of $V$ but $N$ is a lattice in a subspace $W$ of $V$ of rank $q < n$, the invariant factors of $N$ with respect to $M \cap W$ are called the invariant factors of $N$ with respect to $M$. Show how Exercises 8 to 10 and 14 to 16 of Algebra, Chapter VII, $§ 4$ may be extended.