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Definition 64.18.6. 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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