64.18 On l-adic sheaves
Definition 64.18.1. Let X be a Noetherian scheme. A \mathbf{Z}_\ell -sheaf on X, or simply an \ell -adic sheaf \mathcal{F} is an inverse system \left\{ \mathcal{F}_ n\right\} _{n\geq 1} where
\mathcal{F}_ n is a constructible \mathbf{Z}/\ell ^ n\mathbf{Z}-module on X_{\acute{e}tale}, and
the transition maps \mathcal{F}_{n+1}\to \mathcal{F}_ n induce isomorphisms \mathcal{F}_{n+1} \otimes _{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}} \mathbf{Z}/\ell ^ n\mathbf{Z} \cong \mathcal{F}_ n.
We say that \mathcal{F} is lisse if each \mathcal{F}_ n is locally constant. A morphism of such is merely a morphism of inverse systems.
Lemma 64.18.2. Let \{ \mathcal{G}_ n\} _{n\geq 1} be an inverse system of constructible \mathbf{Z}/\ell ^ n\mathbf{Z}-modules. Suppose that for all k\geq 1, the maps
\mathcal{G}_{n+1}/\ell ^ k \mathcal{G}_{n+1}\to \mathcal{G}_ n /\ell ^ k \mathcal{G}_ n
are isomorphisms for all n\gg 0 (where the bound possibly depends on k). In other words, assume that the system \{ \mathcal{G}_ n/\ell ^ k\mathcal{G}_ n\} _{n\geq 1} is eventually constant, and call \mathcal{F}_ k the corresponding sheaf. Then the system \left\{ \mathcal{F}_ k\right\} _{k\geq 1} forms a \mathbf{Z}_\ell -sheaf on X.
Proof.
The proof is obvious.
\square
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
Example 64.18.4. Let X=\mathop{\mathrm{Spec}}(\mathbf{C}) and \Phi : \mathbf{Z}_\ell \to \mathbf{Z}_\ell be multiplication by \ell . More precisely,
\Phi = \left\{ \mathbf{Z}/\ell ^ n\mathbf{Z} \xrightarrow {\ell } \mathbf{Z}/\ell ^ n\mathbf{Z}\right\} _{n \geq 1}.
To compute the kernel, we consider the inverse system
\ldots \to \mathbf{Z}/\ell \mathbf{Z}\xrightarrow {0} \mathbf{Z}/\ell \mathbf{Z}\xrightarrow {0}\mathbf{Z}/\ell \mathbf{Z}.
Since the images are always zero, \mathop{\mathrm{Ker}}(\Phi ) is zero as a system.
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 .
Definition 64.18.8. If X is a separated scheme of finite type over an algebraically closed field k and \mathcal{F} = \left\{ \mathcal{F}_ n\right\} _{n\geq 1} is a \mathbf{Z}_\ell -sheaf on X, then we define
H^ i(X, \mathcal{F}) := \mathop{\mathrm{lim}}\nolimits _ n H^ i(X, \mathcal{F}_ n) \quad \text{and}\quad H_ c^ i(X, \mathcal{F}) := \mathop{\mathrm{lim}}\nolimits _ n H_ c^ i(X, \mathcal{F}_ n).
If \mathcal{F} = \mathcal{F}'\otimes \mathbf{Q}_\ell for a \mathbf{Z}_\ell -sheaf \mathcal{F}' then we set
H_ c^ i(X , \mathcal{F}) := H_ c^ i(X, \mathcal{F}')\otimes _{\mathbf{Z}_\ell }\mathbf{Q}_\ell .
We call these the \ell -adic cohomology of X with coefficients \mathcal{F}.
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