Short exact sequences of complexes give rise to long exact sequences of (co)homology.
Lemma 12.13.12. 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 cochain complexes of $\mathcal{A}$. Then there is a long exact cohomology sequence
\[ \xymatrix{ \ldots & \ldots & \ldots \ar[lld] \\ H^ i(A^\bullet ) \ar[r] & H^ i(B^\bullet ) \ar[r] & H^ i(C^\bullet ) \ar[lld] \\ H^{i + 1}(A^\bullet ) \ar[r] & H^{i + 1}(B^\bullet ) \ar[r] & H^{i + 1}(C^\bullet ) \ar[lld] \\ \ldots & \ldots & \ldots \\ } \]
The construction produces long exact cohomology sequences which are functorial in the short exact sequence and compatible with shifts as in Definition 12.14.8.
Proof.
For the horizontal maps $H^ i(A^\bullet ) \to H^ i(B^\bullet )$ and $H^ i(B^\bullet ) \to H^ i(C^\bullet )$ we use the fact that $H^ i$ is a functor, see above. For the “boundary map” $H^ i(C^\bullet ) \to H^{i + 1}(A^\bullet )$ we use the map $\delta $ of the Snake Lemma 12.5.17 applied to the diagram
\[ \xymatrix{ & A^ i/\mathop{\mathrm{Im}}(d_ A^{i - 1}) \ar[r] \ar[d]^{d_ A^ i} & B^ i/\mathop{\mathrm{Im}}(d_ B^{i - 1}) \ar[r] \ar[d]^{d_ B^ i} & C^ i/\mathop{\mathrm{Im}}(d_ C^{i - 1}) \ar[r] \ar[d]^{d_ C^ i} & 0 \\ 0 \ar[r] & \mathop{\mathrm{Ker}}(d_ A^{i + 1}) \ar[r] & \mathop{\mathrm{Ker}}(d_ B^{i + 1}) \ar[r] & \mathop{\mathrm{Ker}}(d_ C^{i + 1}) & } \]
This works as the kernel of the right vertical map is equal to $H^ i(C^\bullet )$ and the cokernel of the left vertical map is $H^{i + 1}(A^\bullet )$. The exactness of the long sequence is the exactnesss in part (2) of Lemma 12.5.17. The functoriality is Lemma 12.5.18. Compatibility with shifts is immediate from the definitions.
$\square$
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