Exercise 5, § 3

a) Let $0 \to K \to P \to M \to 0,\ 0 \to K' \to P' \to M \to 0$ be two exact sequences of $A$-modules, with $P$ and $P'$ projective. Now prove that there exists a commutative diagram:

where the horizontal lines are exact, and where $u$ and $v$ are isomorphisms (cf. 11, p. 183, exercise 4).

b) Let $n$ be an integer $\geqslant 0$, $(P, p)$ and $(P', p')$ be two left resolutions of the $A$-module $M$, such that $P_j = P'_j = 0$ for $j > n$ and that the $A$-modules $P_i$ and $P'_i$ are projective for $0 \leqslant i \leqslant n - 1$. Now prove that the $A$-modules $\bigoplus_{i \geqslant 0} (P_{2i} \oplus P'_{2i+1})$ and $\bigoplus_{j \geqslant 0} (P'_{2j} \oplus P_{2j+1})$ are isomorphic.

c) Under the hypotheses of b), now prove that there exist two projective complexes $R, R'$, of null homology, such that $R_i = R'_i = 0$ for $i \notin [0, n]$, and an isomorphism of resolutions $u : P \oplus R \to P' \oplus R'$.