Exercise 4, § 2

a) Let E, E’ be two A-modules, N a submodule of E, N’ a submodule of E’, such that E is projective, and that E/N and E’/N’ are isomorphic. Now prove that there exists an exact sequence

(Observe that there exists a homomorphism $f : E \to E'$ such that $f(N) \subset N'$, which gives by passing to the quotients an isomorphism $E/N \to E'/N'$; define u and v in a manner analogous to that used for the exact sequence (29) of II, p. 17.) b) Deduce from a) that if E’ is projective, $E \oplus N'$ and $E' \oplus N$ are isomorphic. c) Let $0 \to E_m \to E_{m-1} \to \cdots \to E_1 \to E_0 \to 0$ be an exact sequence of A-modules, such that $E_0, E_1, \ldots, E_{m-2}$ are projective. Now prove that the A-modules $\bigoplus_{h \geq 0} E_{m-2h}$ and $\bigoplus_{h \geq 0} E_{m-2h+1}$ are isomorphic (with the convention that $E_i = \{0\}$ for $i < 0$).