Exercise 11, § 4

Let $A$ be a Krull domain 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 $d_k(M/cN)$ (Exercise 10) and write:

(a) Show that $d_k(N, M)$ does not depend on the choice of $c$ such that $cN \subset M$; $d_k(N, M)$ is called the determinantal divisor of index $k$ of $N$ with respect to $M$. (b) $d_k(N, M) \geq d_{k+1}(N, M)$ and $d_n(N, M) = 0$. Show that, if we write then for $1 \leq k \leq n$; the divisors $e_k(N, M)$ are called the invariant factors of $N$ with respect to $M$ (reduce it to the case where $A$ is a principal ideal domain). (c) Show that $d_0(N, M) = \chi(M, N)$ (no. 5 ). (d) If $M$ is a lattice in $V$ 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 (assuming if need be in certain cases that $A$ is an integrally closed Noetherian domain).