Exercise 22, § 4

Let A be a local ring, m its maximal ideal and E a finitely generated A-module.

(a) If $r = \gamma(E)$ is the least of the cardinals of a system of generators of E, r is also the least integer h such that the determinantal ideal $d_h(E)$ is equal to A and also the least of the integers k such that $\bigwedge^{k+1} E = 0$ (note that r is the rank of E/mE over A/m).

(b) If we write $e(E) = d_{r-1}(E)$, show that $\bigwedge^r E$ is isomorphic to $A/e(E)$. For an ideal a of A to contain e(E), it is necessary and sufficient that E/aE be a free (A/a)-module (reduce it to the case where $a = 0$).

(c) If e(E) is a principal ideal $A\alpha$, show that E contains a direct factor isomorphic to $A/A\alpha$ (write E as a quotient of $A^r$ by a submodule R and note that there is in R an element of the form $\alpha y$, where y is an element of a basis of $A^r$).

(d) Let $d'_h(E)$ be the annihilator of $\bigwedge^{h+1} E$ for $0 \leq h \leq r-1$. Show that the following conditions are equivalent: (α) The ideals $d_h(E)$ ($0 \leq h \leq r-1$) are principal. (β) The ideals $d'_h(E)$ ($0 \leq h \leq r-1$) are principal. (γ) E is the direct sum of r modules isomorphic to $A/A\lambda_n$, where $\lambda_{n+1}$ divides $\lambda_n$ for $0 \leq h \leq r-1$. Moreover, if these conditions hold, then $d'_h(E) = A\lambda_h$ for $0 \leq h \leq r-1$ and $d_h(E) = \lambda_n \lambda_{n+1} \ldots \lambda_{r-1} A$ for $0 \leq h \leq r-1$. (Proceed by induction on r, using (c).)

(e) Suppose further that A is an integral domain. Show that, if p is a prime ideal of A such that $E/pE$ is a free (A/p)-module and E, a free A,-module, then E is a free A-module (using (b), show that $\gamma(E_p) = \gamma(E)$ and that $e(E_p) = e(E)$).