Exercise 10, § 4

Let $A$ be a commutative ring. For every submodule $R$ of a finitely generated free $A$-module $L = A^n$, let $c_1(R)$ denote the ideal generated by the $(x, x^*)$, where $x$ runs through $R$ and $x^*$ runs through the dual $L^*$; write $c_k(R) = c_1(\mathrm{Im}(\bigwedge^k R))$, $\mathrm{Im}(\bigwedge^k R)$ being the canonical image of the $k$-th exterior power of $R$ in $\bigwedge^k L$. If $R_1, R_2 \subset R$, are two submodules of $L$, then $c_k(R_1) \subset c_k(R_2)$ for all $k$.

(a) Let $M$ be a finitely generated $A$-module; $M$ is isomorphic to a quotient module $L/R$, where $L = A^n$ for a suitable $n$. Show that the sequence of ideals $(a_k)_{k \geq 0}$ such that $a_k = c_{n-k}(R)$ for $k \leq n$ and $a_k = A$ for $k > n$ is independent of the expression of $M$ in the form $L/R$. (Consider first the case where $L/R$ and $L/R'$ are isomorphic; then note that $A^n/R$ is isomorphic to $A^{n+h}/(R \times A^h)$ for all $h > 0$.) We write $d_k(M) = a_k$ for all $k \geq 0$ and say that these are the determinantal ideals associated with $M$. Then $d_k(M) \subset d_{k+1}(M)$ for $k \geq 0$.

(b) If $A$ is a principal ideal domain and $M = L/R$ where $L$ is free and finitely generated, show that the ideals $c_{k+1}(R)(c_k(R))^{-1}$ are the invariant factors of $R$ in $L$.

(c) Let $r = \gamma(M)$ be the least of the cardinals of the systems of generators of $M$ and let $r_0$ be the least integer $h$ such that $d_h(M) = A$. Show that $r_0 \leq r$ and give an example where $r_0 < r$ (take $A$ to be a Dedekind domain).

(d) If $a$ is the annihilator of $M$, show that $d_0(M) \subset a$ and $a^{r-k} \subset d_k(M)$ for $k \leq r = \gamma(M)$.

(e) Suppose that $M$ is a direct sum of submodules $M_i$ ($1 \leq i \leq h$). Show that

where the sum is over the finite sequences $(k_i)_{1 \leq i \leq h}$ such that $\sum k_i = k$.

(f) If $N$ is a finitely generated submodule of $M$, then $d_k(M/N)^{i=1} \subset d_k(M)$ for all $k \geq 0$. Show that:

(g) Let $K$ be a field, $A$ the polynomial ring $K[X, Y, Z]$, $m$ the maximal ideal $AX + AY + AZ$ and $q$ the ideal generated by $X, Y^2, YZ$ and $Z^2$. Consider the $A$-module $M = A/q$ and its submodule $N = m/q$. Show that $d_0(M) = q$, $d_1(M) = A$, $d_0(N) = m^2$ and $d_1(N) = m$.