Exercise 6, § 2

We retain the notations of Exercise 5; suppose in addition that A is commutative, that the complexes C and C’/B(C’) are flat and that there exists an ideal m of A satisfying $\delta(C) \subset mC$ and $\bigcap_n (B(C) + m^n C) = B(C)$ (* this last condition is always satisfied if A is local noetherian with maximal ideal m and if $C_n$ is an A-module of finite type for every n, cf. AC, III, § 3, No. 3, prop. 6). \* Now prove that $s \theta^{-1}$ is a splitting of C’. (The flatness hypotheses imply $\mathrm{Im}\,(d + \delta) \cap m^n C = (d + \delta)(m^n C)$ for every $n$; if $x \in Z(C)$, construct by induction elements $y_n \in C, z_n \in Z(C) \cap m^n C$ such that $\delta(x) = \theta d(y_n) + \delta(z_n)$. Conclude using Exercise 5, a).)