Exercise 1, § 2

One assumes the ring A principal; let C be a complex of free A-modules. (a) Now prove that C is direct sum of a family of complexes $(^{p}E)_p \in \mathbf{Z}$ satisfying: $^pE_n = 0$ for $n \neq p$, $p - 1 ; Z_p(^pE) = 0$. (b) One assumes in addition that $C_n$ is of finite type for every $n \in \mathbf{Z}$. Now prove that C is direct sum of complexes having only, either one non-zero component, or two consecutive non-zero components, these components being isomorphic to A.