Exercise 7, § 14

Let A be a commutative ring, M and N A-modules, and $u: M\rightarrow N$ an A-linear mapping. For every maximal ideal $\mathfrak{m}$ of A, denote by $u(\mathfrak{m})$ the homomorphism $M/\mathfrak{m}M\rightarrow N/\mathfrak{m}N$ induced by $u$.

a) Suppose that N is finitely generated and that $u(\mathfrak{m})$ is surjective for every maximal ideal $\mathfrak{m}$ of A. Prove that $u$ is surjective (apply Lemma 1 of VIII, p. 81 to the cokernel of $u$).

b) Suppose that M and N are finitely generated, that N is projective, and that $u(\mathfrak{m})$ is bijective for every maximal ideal $\mathfrak{m}$ of A. Prove that $u$ is bijective (use the same method).

c) Suppose that M and N are projective and finitely generated and that $u(\mathfrak{m})$ is injective for every maximal ideal $\mathfrak{m}$ of A. Prove that $u$ induces an isomorphism from M to a direct factor submodule of N (apply a) to the homomorphism $^tu)$.

d) Let B be an A-algebra; suppose that the A-module B is faithful, projective, and finitely generated. Prove that A is a direct factor of the A-module B. Prove that if the B-module homomorphism $u_{(B)}: M_{(B)}\rightarrow N_{(B)}$ is bijective, then so is $u$. Prove that if the B-module $M_{(B)}$ is projective, then so is the A-module M.