The Stacks project

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. $\square$


Comments (3)

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".

There are also:

  • 5 comment(s) on Section 12.13: Complexes

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0117. Beware of the difference between the letter 'O' and the digit '0'.