The Stacks project

Lemma 76.12.5. Let $B \to S$ as in Section 76.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then the category of quasi-coherent modules on $(U, R, s, t, c)$ is abelian.

Proof. Let $\varphi : (\mathcal{F}, \alpha ) \to (\mathcal{G}, \beta )$ be a homomorphism of quasi-coherent modules on $(U, R, s, t, c)$. Since $s$ is flat we see that

\[ 0 \to s^*\mathop{\mathrm{Ker}}(\varphi ) \to s^*\mathcal{F} \to s^*\mathcal{G} \to s^*\mathop{\mathrm{Coker}}(\varphi ) \to 0 \]

is exact and similarly for pullback by $t$. Hence $\alpha $ and $\beta $ induce isomorphisms $\kappa : t^*\mathop{\mathrm{Ker}}(\varphi ) \to s^*\mathop{\mathrm{Ker}}(\varphi )$ and $\lambda : t^*\mathop{\mathrm{Coker}}(\varphi ) \to s^*\mathop{\mathrm{Coker}}(\varphi )$ which satisfy the cocycle condition. Then it is straightforward to verify that $(\mathop{\mathrm{Ker}}(\varphi ), \kappa )$ and $(\mathop{\mathrm{Coker}}(\varphi ), \lambda )$ are a kernel and cokernel in the category of quasi-coherent modules on $(U, R, s, t, c)$. Moreover, the condition $\mathop{\mathrm{Coim}}(\varphi ) = \mathop{\mathrm{Im}}(\varphi )$ follows because it holds over $U$. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 76.12: Quasi-coherent sheaves on groupoids

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Your email address will not be published.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 06VZ. Beware of the difference between the letter 'O' and the digit '0'.