Definition 63.18.6. A $\mathbf{Z}_\ell$-sheaf $\mathcal{F}$ is torsion if $\ell ^ n : \mathcal{F} \to \mathcal{F}$ is the zero map for some $n$. The abelian category of $\mathbf{Q}_\ell$-sheaves on $X$ is the quotient of the abelian category of $\mathbf{Z}_\ell$-sheaves by the Serre subcategory of torsion sheaves. In other words, its objects are $\mathbf{Z}_\ell$-sheaves on $X$, and if $\mathcal{F}, \mathcal{G}$ are two such, then

$\mathop{\mathrm{Hom}}\nolimits _{\mathbf{Q}_\ell } \left(\mathcal{F}, \mathcal{G} \right) = \mathop{\mathrm{Hom}}\nolimits _{\mathbf{Z}_\ell } \left(\mathcal{F}, \mathcal{G}\right) \otimes _{\mathbf{Z}_\ell } \mathbf{Q}_\ell .$

We denote by $\mathcal{F} \mapsto \mathcal{F} \otimes \mathbf{Q}_\ell$ the quotient functor (right adjoint to the inclusion). If $\mathcal{F} = \mathcal{F}' \otimes \mathbf{Q}_\ell$ where $\mathcal{F}'$ is a $\mathbf{Z}_\ell$-sheaf and $\bar x$ is a geometric point, then the stalk of $\mathcal{F}$ at $\bar x$ is $\mathcal{F}_{\bar x} = \mathcal{F}'_{\bar x} \otimes \mathbf{Q}_\ell$.

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