The Stacks project

Lemma 13.27.7. Let $\mathcal{A}$ be an abelian category. Let $0 \to A \to Z \to B \to 0$ and $0 \to B \to Z' \to C \to 0$ be short exact sequences in $\mathcal{A}$. Denote $[Z] \in \mathop{\mathrm{Ext}}\nolimits ^1(B, A)$ and $[Z'] \in \mathop{\mathrm{Ext}}\nolimits ^1(C, B)$ their classes. Then $[Z] \circ [Z'] \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {A}(C, A)$ is $0$ if and only if there exists a commutative diagram

\[ \xymatrix{ & & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & A \ar[r] \ar[d]^1 & Z \ar[r] \ar[d] & B \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A \ar[r] & W \ar[r] \ar[d] & Z' \ar[r] \ar[d] & 0 \\ & & C \ar[r]^1 \ar[d]& C \ar[d]\\ & & 0 & 0 } \]

with exact rows and columns in $\mathcal{A}$.

Proof. Omitted. Hints: You can argue this using the result of Lemma 13.27.5 and working out what it means for a $2$-extension class to be zero. Or you can use that if $[Z] \circ [Z'] \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {A}(C, A)$ is zero, then by the long exact cohomology sequence of $\mathop{\mathrm{Ext}}\nolimits $ the element $[Z] \in \mathop{\mathrm{Ext}}\nolimits ^1(B, A)$ is the image of some element in $\mathop{\mathrm{Ext}}\nolimits ^1(W', A)$. $\square$

Comments (2)

Comment #7367 by Shizhang on

Maybe the vertical arrows should all go downward?

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Your email address will not be published.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 0GSM. Beware of the difference between the letter 'O' and the digit '0'.