Lemma 64.18.3. The category of $\mathbf{Z}_\ell$-sheaves on $X$ is abelian.

Proof. Let $\Phi = \left\{ \varphi _ n\right\} _{n\geq 1} : \left\{ \mathcal{F}_ n\right\} \to \left\{ \mathcal{G}_ n\right\}$ be a morphism of $\mathbf{Z}_\ell$-sheaves. Set

$\mathop{\mathrm{Coker}}(\Phi ) = \left\{ \mathop{\mathrm{Coker}}\left(\mathcal{F}_ n \xrightarrow {\varphi _ n} \mathcal{G}_ n\right) \right\} _{n\geq 1}$

and $\mathop{\mathrm{Ker}}(\Phi )$ is the result of Lemma 64.18.2 applied to the inverse system

$\left\{ \bigcap _{m\geq n} \mathop{\mathrm{Im}}\left(\mathop{\mathrm{Ker}}(\varphi _ m) \to \mathop{\mathrm{Ker}}(\varphi _ n)\right) \right\} _{n \geq 1}.$

That this defines an abelian category is left to the reader. $\square$

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