Exercise 4, § 2

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

(Observe that there exists a homomorphism $f : E \to E'$ such that $f(N) \subset N'$, which gives when passing to the quotients an isomorphism $E/N \to E'/N'$; define u and v in an analogous way to that used for the exact sequence (29) of § 1, no. 7.)

(b) Deduce from (a) that if E' is projective, $E \oplus N'$ and $E' \oplus N$ are isomorphic.

(c) Let $0 \to E_m \to E_{m-1} \to \ldots \to E_1 \to E_0 \to 0$ be an exact sequence of A-modules such that $E_0, E_1, \ldots, E_{m-2}$ are projective. Show 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 $i < 0$).