The Stacks project

Lemma 39.14.6. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. 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 $U$. $\square$

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