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

\[ \xymatrix{ \ldots & \ldots & \ldots \ar[lld] \\ H^ i(A^\bullet ) \ar[r] & H^ i(B^\bullet ) \ar[r] & H^ i(C^\bullet ) \ar[lld] \\ H^{i + 1}(A^\bullet ) \ar[r] & H^{i + 1}(B^\bullet ) \ar[r] & H^{i + 1}(C^\bullet ) \ar[lld] \\ \ldots & \ldots & \ldots \\ } \]

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

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

Comments (1)

Comment #1223 by David Corwin on

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

There are also:

  • 1 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Your email address will not be published.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'.