Exercise 4, § 2

a) Let C be a bounded on the right A-complex, such that C and H(C) are projective. Now prove that C is split. b) On the ring B = A(\varepsilon) (X, p. 27), consider the complex C defined by $C_n = B$ for all $n$, and $d(b) = \varepsilon b$ for $b \in B$. The complex C is free with zero homology, but is not split.