The Stacks project

63.18 On l-adic sheaves

Definition 63.18.1. 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

  1. $\mathcal{F}_ n$ is a constructible $\mathbf{Z}/\ell ^ n\mathbf{Z}$-module on $X_{\acute{e}tale}$, and

  2. 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 63.18.2. 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 63.18.3. 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 63.18.2 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 63.18.4. 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.

Remark 63.18.5. If $\mathcal{F} = \left\{ \mathcal{F}_ n\right\} _{n\geq 1}$ is a $\mathbf{Z}_\ell $-sheaf on $X$ and $\bar x$ is a geometric point then $M_ n = \left\{ \mathcal{F}_{n, \bar x}\right\} $ is an inverse system of finite $\mathbf{Z}/\ell ^ n\mathbf{Z}$-modules such that $M_{n+1}\to M_ n$ is surjective and $M_ n = M_{n+1}/\ell ^ n M_{n+1}$. It follows that

\[ M = \mathop{\mathrm{lim}}\nolimits _ n M_ n = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, \bar x} \]

is a finite $\mathbf{Z}_\ell $-module. Indeed, $M/\ell M= M_1$ is finite over $\mathbf{F}_\ell $, so by Nakayama $M$ is finite over $\mathbf{Z}_\ell $. Therefore, $M\cong \mathbf{Z}_\ell ^{\oplus r} \oplus \oplus _{i = 1}^ t \mathbf{Z}_\ell /\ell ^{e_ i}\mathbf{Z}_\ell $ for some $r, t\geq 0$, $e_ i\geq 1$. The module $M = \mathcal{F}_{\bar x}$ is called the stalk of $\mathcal{F}$ at $\bar x$.

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 $.

Remark 63.18.7. Since a $\mathbf{Z}_\ell $-sheaf is only defined on a Noetherian scheme, it is torsion if and only if its stalks are torsion.

Definition 63.18.8. 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 $\mathcal{F}$.


Comments (2)

Comment #2440 by sdf on

I guess the brace { in definition 50.95.1 should be {\it. Also you refer to when you define lisse in definition 50.95.1, but you haven't said what is, I guess you meant to write instead of just in the first sentence.

Comment #2483 by on

OK, I sort of fixed this, but the whole section needs a rewrite. See fix here. Thanks!


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