Given a short exact sequence of complexes, there exists a connecting homomorphism $\partial_\ast:H_k(C_\ast)\longrightarrow H_{k-1}(A_*)$, defined by

such that the complex

is an exact sequence

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[Cohomological version]

Given a short exact sequence of complexes, there exists a connecting homomorphism $\mathrm{d}^\ast:H^k(C^\ast)\longrightarrow H^{k+1}(A^*)$, defined by

such that the complex

is an exact sequence

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