# The Stacks Project

## Tag 03UL

Definition 53.102.1. 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 53.102.2. 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 53.102.3. 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 $$\mathop{\rm Coker}(\Phi) = \left\{ \mathop{\rm Coker}\left(\mathcal{F}_n \xrightarrow{\varphi_n} \mathcal{G}_n\right) \right\}_{n\geq 1}$$ and $\mathop{\rm Ker}(\Phi)$ is the result of Lemma 53.102.2 applied to the inverse system $$\left\{ \bigcap_{m\geq n} \mathop{\rm Im}\left(\mathop{\rm Ker}(\varphi_m) \to \mathop{\rm Ker}(\varphi_n)\right) \right\}_{n \geq 1}.$$ That this defines an abelian category is left to the reader. $\square$

Example 53.102.4. Let $X=\mathop{\rm Spec}(\mathbf{C})$ 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, $\mathop{\rm Ker}(\Phi)$ is zero as a system.

Remark 53.102.5. 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 $$M = \mathop{\rm lim}\nolimits_n M_n = \mathop{\rm lim}\nolimits \mathcal{F}_{n, \bar x}$$ 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 53.102.6. 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 $$\mathop{\rm Hom}\nolimits_{\mathbf{Q}_\ell} \left(\mathcal{F}, \mathcal{G} \right) = \mathop{\rm Hom}\nolimits_{\mathbf{Z}_\ell} \left(\mathcal{F}, \mathcal{G}\right) \otimes_{\mathbf{Z}_\ell} \mathbf{Q}_\ell.$$ 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 53.102.7. 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 53.102.8. 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 $$H^i(X, \mathcal{F}) := \mathop{\rm lim}\nolimits_n H^i(X, \mathcal{F}_n) \quad\text{and}\quad H_c^i(X, \mathcal{F}) := \mathop{\rm lim}\nolimits_n H_c^i(X, \mathcal{F}_n).$$ 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 $\mathcal{F}$.

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 17914–18071 (see updates for more information).

\section{On l-adic sheaves}

\begin{definition}
Let $X$ be a noetherian scheme. A {\it $\mathbf{Z}_\ell$-sheaf} on $X$, or
simply an {\it $\ell$-adic sheaf} $\mathcal{F}$ is an
inverse system $\left\{\mathcal{F}_n\right\}_{n\geq 1}$ where
\begin{enumerate}
\item
$\mathcal{F}_n$ is a constructible $\mathbf{Z}/\ell^n\mathbf{Z}$-module on
$X_\etale$, and
\item
the transition maps $\mathcal{F}_{n+1}\to \mathcal{F}_n$ induce isomorphisms
$\mathcal{F}_{n+1} \otimes_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} \cong \mathcal{F}_n$.
\end{enumerate}
We say that $\mathcal{F}$ is {\it lisse} if each $\mathcal{F}_n$ is locally
constant. A {\it morphism} of such is merely a morphism of inverse systems.
\end{definition}

\begin{lemma}
\label{lemma-eventually-constant}
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 $X$.
\end{lemma}

\begin{proof}
The proof is obvious.
\end{proof}

\begin{lemma}
The category of $\mathbf{Z}_\ell$-sheaves on $X$ is abelian.
\end{lemma}

\begin{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
$$\Coker(\Phi) = \left\{ \Coker\left(\mathcal{F}_n \xrightarrow{\varphi_n} \mathcal{G}_n\right) \right\}_{n\geq 1}$$
and $\Ker(\Phi)$ is the result of
Lemma \ref{lemma-eventually-constant}
applied to the inverse system
$$\left\{ \bigcap_{m\geq n} \Im\left(\Ker(\varphi_m) \to \Ker(\varphi_n)\right) \right\}_{n \geq 1}.$$
That this defines an abelian category is left to the reader.
\end{proof}

\begin{example}
\label{example-kernel}
Let $X=\Spec(\mathbf{C})$ 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, $\Ker(\Phi)$ is zero as a system.
\end{example}

\begin{remark}
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
$$M = \lim_n M_n = \lim \mathcal{F}_{n, \bar x}$$
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 {\it stalk} of
$\mathcal{F}$ at $\bar x$.
\end{remark}

\begin{definition}
A $\mathbf{Z}_\ell$-sheaf $\mathcal{F}$ is {\it 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
$$\Hom_{\mathbf{Q}_\ell} \left(\mathcal{F}, \mathcal{G} \right) = \Hom_{\mathbf{Z}_\ell} \left(\mathcal{F}, \mathcal{G}\right) \otimes_{\mathbf{Z}_\ell} \mathbf{Q}_\ell.$$
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
{\it stalk} of $\mathcal{F}$ at $\bar x$ is $\mathcal{F}_{\bar x} = \mathcal{F}'_{\bar x} \otimes \mathbf{Q}_\ell$.
\end{definition}

\begin{remark}
\label{remark-torsion-stalks}
Since a $\mathbf{Z}_\ell$-sheaf is only defined on a noetherian scheme, it is
torsion if and only if its stalks are torsion.
\end{remark}

\begin{definition}
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
$$H^i(X, \mathcal{F}) := \lim_n H^i(X, \mathcal{F}_n) \quad\text{and}\quad H_c^i(X, \mathcal{F}) := \lim_n H_c^i(X, \mathcal{F}_n).$$
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 {\it $\ell$-adic cohomology} of $X$ with coefficients
$\mathcal{F}$.
\end{definition}

Comment #2440 by sdf on February 28, 2017 a 9:41 pm UTC

I guess the brace { in definition 50.95.1 should be {\it. Also you refer to $\mathcal{F}$ when you define lisse in definition 50.95.1, but you haven't said what $\mathcal{F}$ is, I guess you meant to write $\mathcal{F}=\{\mathcal{F}_n\}_{n\ge 1}$ instead of just $\{\mathcal{F}_n\}_{n\ge 1}$ in the first sentence.

Comment #2483 by Johan (site) on April 13, 2017 a 10:34 pm UTC

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

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