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A short exact sequence of complexes is a chain of complex homomorphisms

$$0\xrightarrow{} A_\ast\xrightarrow{f}B_\ast\xrightarrow{g}C_\ast\xrightarrow{} 0$$

such that $0\xrightarrow{} A_k\xrightarrow{f_k}B_k\xrightarrow{g_k}C_k\xrightarrow{} 0$ is a short exact sequence for each $k$


[Cohomological version]

A short exact sequence of complexes is a chain of complex homomorphisms

$$0\xrightarrow{} A^\ast\xrightarrow{f}B^\ast\xrightarrow{g}C^\ast\xrightarrow{} 0$$

such that $0\xrightarrow{} A^k\xrightarrow{f^k}B^k\xrightarrow{g^k}C^k\xrightarrow{} 0$ is a short exact sequence for each $k$