Exercise 3, § 3

Let $A$ be a local ring (not necessarily commutative, cf. Exercise 1) and $m$ its Jacobson radical.

(a) Let $L$ be a free left $A$-module and $x \neq 0$ an element of $L$. Let $(e_i)_{i \in I}$ be a basis of $L$ for which the number of components $\neq 0$ of $x$ is the least possible; if $x = \sum_{i \in I} \xi_i e_i$, let $J$ be the set of $i \in I$ such that $\xi_i \neq 0$. Show that none of the $\xi_i$ ($i \in J$) belongs to the right ideal of $A$ generated by the others.

(b) With the hypotheses and notation of (a), let P, Q be two complementary submodules of L such that $x \in P$; for all $t \in I$, let $e_t = y_t + z_t$, where $y_t \in P$, $z_t \in Q$. Show that the family of $y_t$ for $t \in J$ and $e_t$ for $t \notin J$ is a basis of L. (Prove, using (a), that, if $z_k = \sum_{t \in I} \zeta_{kt} e_t$, the components $\zeta_{kt}$ such that $k \in J$ and $t \in J$ necessarily belong to m and use Corollary 2 to Proposition 4.) Deduce that there exists a free submodule of P which contains x and is a direct factor of P.

(c) Show that every projective A-module P is free (using Kaplansky's Theorem (Algebra, Chapter 11, § 2, Exercise 4), reduce it to the case where P is generated by a countable family of elements and apply (b) to this case).

(d) Give an example of a local ring A and a flat A-module M which is not free and satisfies $mM = M$ (take $A = \mathbf{Z}_{(p)}$). (In Chapter III, § 3, we shall give examples of faithfully flat A-modules which are not free over a Noetherian local ring A.)

(e) Let A be a local ring and M a finitely generated flat A-module. Show that M is a free A-module (use Exercise 23 (e) of Chapter I, § 2).