The Stacks project

Lemma 12.13.6. Let $\mathcal{A}$ be an abelian category. Suppose that

\[ 0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 \]

is a short exact sequence of chain complexes of $\mathcal{A}$. Then there is a canonical long exact homology sequence

\[ \xymatrix{ \ldots & \ldots & \ldots \ar[lld] \\ H_ i(A_\bullet ) \ar[r] & H_ i(B_\bullet ) \ar[r] & H_ i(C_\bullet ) \ar[lld] \\ H_{i - 1}(A_\bullet ) \ar[r] & H_{i - 1}(B_\bullet ) \ar[r] & H_{i - 1}(C_\bullet ) \ar[lld] \\ \ldots & \ldots & \ldots \\ } \]

Proof. Omitted. The maps come from the Snake Lemma 12.5.17 applied to the diagrams

\[ \xymatrix{ & A_ i/\mathop{\mathrm{Im}}(d_{A, i + 1}) \ar[r] \ar[d]^{d_{A, i}} & B_ i/\mathop{\mathrm{Im}}(d_{B, i + 1}) \ar[r] \ar[d]^{d_{B, i}} & C_ i/\mathop{\mathrm{Im}}(d_{C, i + 1}) \ar[r] \ar[d]^{d_{C, i}} & 0 \\ 0 \ar[r] & \mathop{\mathrm{Ker}}(d_{A, i - 1}) \ar[r] & \mathop{\mathrm{Ker}}(d_{B, i - 1}) \ar[r] & \mathop{\mathrm{Ker}}(d_{C, i - 1}) & } \]
$\square$

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