Let $R$ be a ring, let $(C, d_C)$ and $(D, d_D)$ be chain complexes of left $R$-modules, and let $f: C \to D$ be a chain map. Define the mapping cone $\operatorname{Cone}(f)$ by
Let $C[-1]$ denote the shifted chain complex with $C[-1]_n = C_{n-1}$ and differential $d_{C[-1],n} = -d_{C,n-1}$. Then there is a short exact sequence of chain complexes