Exercise 4, § 6

Let $(F_\alpha, f_{\beta\alpha})$ be a direct system of A-modules. (a) For every A-module E, define a canonical $\mathbf{Z}$-module homomorphism

(b) Show that if E is finitely generated, $\varepsilon$ is injective. Give an example where $\varepsilon$ is not injective (cf. Exercise 3, taking $E = \lim \rightarrow F_\alpha$).

(c) Show that if E is finitely generated and the $f_{\beta \alpha}$ are injective, $\varepsilon$ is surjective. Give an example where $E = \lim_{\longrightarrow} F_\alpha$ is free, the $f_{\beta \alpha}$ are injective and $\varepsilon$ is not surjective.

(d) Show that if E is projective and finitely generated, $\varepsilon$ is bijective.

(e) Let A be an integral domain and $a$ an ideal of A which is not finitely generated. If $(a_\alpha)$ is the family of finitely generated ideals contained in $a$, show that $E = A/a$ is canonically isomorphic to the direct limit of the $F_\alpha = A/a_\alpha$, but that the homomorphism $\varepsilon$ is not surjective.