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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