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Tag 03UL

53.102. On l-adic sheaves

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 17911–18068 (see updates for more information).

    \section{On l-adic sheaves}
    \label{section-l-adic-sheaves}
    
    \begin{definition}
    \label{definition-l-adic-sheaf}
    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}
    \label{lemma-l-adic-abelian}
    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}
    \label{remark-stalk-l-adic-sheaf}
    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}
    \label{definition-torsion-l-adic-sheaf}
    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}
    \label{definition-cohomology-l-adic}
    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}

    Comments (2)

    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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