Assume that a complex (C, d) of R-modules has a contracting homotopy in which the maps sn: Cn →C

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First, we need to show that the complex $(u005Cmathbf{C}, d)$ is exact, which means that for all $n$

Assume that a complex $(\mathrm{C}, d)$ of $R$-modules has a contracting homotopy in which the maps $s_{n}: C_{n} \rightarrow C_{n+1}$ satisfying $$ 1_{C_{v}}=d_{n+1} s_{n}+s_{n-1} d_{n} $$ are only Z-maps. Prowe that $(\mathbf{C}, d)$ is an exact sequence.