Exercise 14, § 1

a) Let $0 \to R \to L \to E \to 0$ be an exact sequence of A-modules, where L is a free A-module; let $(e_a)$ be a basis of L. Now prove that the following conditions are equivalent: α) E is flat. β) For every $x \in R$, if $a_x$ is the right ideal generated by the components of x on the basis $(e_a)$, one has $x \in a_x R$. γ) For every $x \in R$, there exists a homomorphism $u_x : L \to R$ such that $u_x(x) = x$. δ) For every finite sequence $(x_i)_{1 \leq i \leq n}$ of elements of R, there exists a homomorphism $u : L \to R$ such that $u(x_i) = x_i$ for $1 \leq i \leq n$. (Use th. 1, p. 14.) b) Let r be the radical of A, and let $0 \to R \to L \to E \to 0$ be an exact sequence of A-modules, such that L is free. Suppose that E is flat and that R is contained in rL. Now prove that one has $R = 0$ (with the notations of a), observe that $a_x$ is an ideal of finite type and that one has $a_x = a_x r$. c) Let E be a flat A-module of finite type; suppose that there exists a two-sided ideal b of A contained in the radical of A, such that $E/bE$ is a $(A/b)$-module free. Now prove that E is then a free A-module (observe that there exists a free A-module of finite type L such that $L/bL$ is isomorphic to $E/bE$ and apply b)). In particular, every flat module of finite type over a local ring is free.