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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