Exercise 4, § 4

Let a be a two-sided ideal of A. Let M be an A-module, such that Tor^A_1(A/a, M) = 0 and such that the (A/a)-module M/aM is projective. (a) Suppose either that M is of finite presentation and a is contained in the radical of A, or that a is nilpotent. Now prove that M is projective (use VIII, § 8, No. 3, cor. 3). (b) Suppose that M is a finitely generated module, that one has $\bigcap_n a^n = 0$ and that the (A/a)-module M/aM is free. Now prove that the A-module M is free. (c) Give an example of a local ring A, of maximal ideal m, and of a non-flat finitely generated A-module M such that Tor^A_1(A/m, M) = 0.