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.

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 $H^{i + 1}(A^\bullet )$. We omit the verification that we obtain a long exact sequence and we omit the verification of the properties mentioned at the end of the statement of the lemma. $\square$

Comment #1223 by David Corwin on

Suggested slogan: Short exact sequences of complexes give rise to long exact sequences of homology

Comment #8353 by on

Suggestion: substituting "we omit the verification that we obtain a long exact sequence" by "the exactness of the long sequence is the exactnesss in part (2) of 12.5.17", and substituting "we omit the verification of the properties mentioned at the end of the statement of the lemma" by "the functoriality is Lemma 12.5.18, whereas compatibility with shifts comes from Definition 12.14.8".

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