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Tag 03SH

53.65. Méthode de la trace

A reference for this section is [SGA4, Exposé IX, \S 5]. The material here will be used in the proof of Lemmas 53.77.8 below.

Let $f : Y \to X$ be an étale morphism of schemes. There is a sequence $$ f_!, f^{-1}, f_* $$ of adjoint functors between $\textit{Ab}(X_{\acute{e}tale})$ and $\textit{Ab}(Y_{\acute{e}tale})$. The adjunction map $\text{id} \to f_* f^{-1}$ is called restriction. The adjunction map $f_! f^{-1} \to \text{id}$ is often called the trace map. If $f$ is finite, then $f_* = f_!$ and we can view this as a map $f_*f^{-1} \to \text{id}$.

Definition 53.65.1. Let $f : Y \to X$ be a finite étale morphism of schemes. The map $f_* f^{-1} \to \text{id}$ described above is called the trace.

Let $f : Y \to X$ be a finite étale morphism. The trace map is characterized by the following two properties:

  1. it commutes with étale localization and
  2. if $Y = \coprod_{i = 1}^d X$ then the trace map is the sum map $f_*f^{-1} \mathcal{F} = \mathcal{F}^{\oplus d} \to \mathcal{F}$.

It follows that if $f$ has constant degree $d$, then the composition $$ \mathcal{F} \xrightarrow{res} f_* f^{-1} \mathcal{F} \xrightarrow{trace} \mathcal{F} $$ is multiplication by $d$. The ''méthode de la trace'' is the following observation: if $\mathcal{F}$ is an abelian sheaf on $X_{\acute{e}tale}$ such that multiplication by $d$ on $\mathcal{F}$ is an isomorphism, then the map $$ H^n_{\acute{e}tale}(X, \mathcal{F}) \longrightarrow H^n_{\acute{e}tale}(Y, f^{-1}\mathcal{F}) $$ is injective. Namely, we have $$ H^n_{\acute{e}tale}(Y, f^{-1}\mathcal{F}) = H^n_{\acute{e}tale}(X, f_*f^{-1}\mathcal{F}) $$ by the vanishing of the higher direct images (Proposition 53.54.2) and the Leray spectral sequence (Proposition 53.53.2). Thus we can consider the maps $$ H^n_{\acute{e}tale}(X, \mathcal{F}) \to H^n_{\acute{e}tale}(Y, f^{-1}\mathcal{F})= H^n_{\acute{e}tale}(X, f_*f^{-1}\mathcal{F}) \xrightarrow{trace} H^n_{\acute{e}tale}(X, \mathcal{F}) $$ and the composition is an isomorphism (under our assumption on $\mathcal{F}$ and $f$). In particular, if $H_{\acute{e}tale}^q(Y, f^{-1}\mathcal{F}) = 0$ then $H_{\acute{e}tale}^q(X, \mathcal{F}) = 0$ as well. Indeed, multiplication by $d$ induces an isomorphism on $H_{\acute{e}tale}^q(X, \mathcal{F})$ which factors through $H_{\acute{e}tale}^q(Y, f^{-1}\mathcal{F})= 0$.

This is often combined with the following.

Lemma 53.65.2. Let $S$ be a connected scheme. Let $\ell$ be a prime number. Let $\mathcal{F}$ a finite type, locally constant sheaf of $\mathbf{F}_\ell$-vector spaces on $S_{\acute{e}tale}$. Then there exists a finite étale morphism $f : T \to S$ of degree prime to $\ell$ such that $f^{-1}\mathcal{F}$ has a finite filtration whose successive quotients are $\underline{\mathbf{Z}/\ell\mathbf{Z}}_T$.

Proof. Choose a geometric point $\overline{s}$ of $S$. Via the equivalence of Lemma 53.64.1 the sheaf $\mathcal{F}$ corresponds to a finite dimensional $\mathbf{F}_\ell$-vector space $V$ with a continuous $\pi_1(S, \overline{s})$-action. Let $G \subset \text{Aut}(V)$ be the image of the homomorphism $\rho : \pi_1(S, \overline{s}) \to \text{Aut}(V)$ giving the action. Observe that $G$ is finite. The surjective continuous homomorphism $\overline{\rho} : \pi_1(S, \overline{s}) \to G$ corresponds to a Galois object $Y \to S$ of $\textit{FÉt}_S$ with automorphism group $G = \text{Aut}(Y/S)$, see Fundamental Groups, Remark 52.6.3. Let $H \subset G$ be an $\ell$-Sylow subgroup. We claim that $T = Y/H \to S$ works. Namely, let $\overline{t} \in T$ be a geometric point over $\overline{s}$. The image of $\pi_1(T, \overline{t}) \to \pi_1(S, \overline{s})$ is $(\overline{\rho})^{-1}(H)$ as follows from the functorial nature of fundamental groups. Hence the action of $\pi_1(T, \overline{t})$ on $V$ corresponding to $f^{-1}\mathcal{F}$ is through the map $\pi_1(T, \overline{t}) \to H$, see Remark 53.64.2. As $H$ is a finite $\ell$-group, the irreducible constituents of the representation $\rho|_{\pi_1(T, \overline{t})}$ are each trivial of rank $1$ (this is a simple lemma on representation theory of finite groups; insert future reference here). Via the equivalence of Lemma 53.64.1 this means $f^{-1}\mathcal{F}$ is a successive extension of constant sheaves with value $\underline{\mathbf{Z}/\ell\mathbf{Z}}_T$. Moreover the degree of $T = Y/H \to S$ is prime to $\ell$ as it is equal to the index of $H$ in $G$. $\square$

    The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 9428–9552 (see updates for more information).

    \section{M\'ethode de la trace}
    \label{section-trace-method}
    
    \noindent
    A reference for this section is \cite[Expos\'e IX, \S 5]{SGA4}.
    The material here will be used in the proof of
    Lemmas \ref{lemma-vanishing-easier} below.
    
    \medskip\noindent
    Let $f : Y \to X$ be an \'etale morphism of schemes. There
    is a sequence
    $$
    f_!, f^{-1}, f_*
    $$
    of adjoint functors between
    $\textit{Ab}(X_\etale)$ and $\textit{Ab}(Y_\etale)$. The
    adjunction map $\text{id} \to f_* f^{-1}$ is called {\it restriction}.
    The adjunction map $f_! f^{-1} \to \text{id}$ is often
    called the {\it trace map}. If $f$ is finite, then $f_* = f_!$ and
    we can view this as a map $f_*f^{-1} \to \text{id}$.
    
    \begin{definition}
    \label{definition-trace-map}
    Let $f : Y \to X$ be a finite \'etale morphism of schemes.
    The map $f_* f^{-1} \to \text{id}$ described above is called the {\it trace}.
    \end{definition}
    
    \noindent
    Let $f : Y \to X$ be a finite \'etale morphism. The trace map is
    characterized by the following two properties:
    \begin{enumerate}
    \item it commutes with \'etale localization and
    \item if $Y = \coprod_{i = 1}^d X$ then the trace map is
    the sum map $f_*f^{-1} \mathcal{F} = \mathcal{F}^{\oplus d} \to \mathcal{F}$.
    \end{enumerate}
    It follows that if $f$ has constant degree $d$, then the composition
    $$
    \mathcal{F} \xrightarrow{res}
    f_* f^{-1} \mathcal{F} \xrightarrow{trace}
    \mathcal{F}
    $$
    is multiplication by $d$. The ``m\'ethode de la trace''
    is the following observation: if $\mathcal{F}$
    is an abelian sheaf on $X_\etale$ such that multiplication by $d$
    on $\mathcal{F}$ is an isomorphism, then the map
    $$
    H^n_\etale(X, \mathcal{F}) \longrightarrow H^n_\etale(Y, f^{-1}\mathcal{F})
    $$
    is injective. Namely, we have
    $$
    H^n_\etale(Y, f^{-1}\mathcal{F}) = H^n_\etale(X, f_*f^{-1}\mathcal{F})
    $$
    by the vanishing of the higher direct images
    (Proposition \ref{proposition-finite-higher-direct-image-zero})
    and the Leray spectral sequence
    (Proposition \ref{proposition-leray}).
    Thus we can consider the maps
    $$
    H^n_\etale(X, \mathcal{F}) \to
    H^n_\etale(Y, f^{-1}\mathcal{F})= H^n_\etale(X, f_*f^{-1}\mathcal{F})
    \xrightarrow{trace}
    H^n_\etale(X, \mathcal{F})
    $$
    and the composition is an isomorphism (under our assumption on $\mathcal{F}$
    and $f$). In particular, if
    $H_\etale^q(Y, f^{-1}\mathcal{F}) = 0$ then
    $H_\etale^q(X, \mathcal{F}) = 0$ as well.
    Indeed, multiplication by $d$ induces an
    isomorphism on $H_\etale^q(X, \mathcal{F})$ which factors through
    $H_\etale^q(Y, f^{-1}\mathcal{F})= 0$.
    
    \medskip\noindent
    This is often combined with the following.
    
    \begin{lemma}
    \label{lemma-pullback-filtered}
    Let $S$ be a connected scheme. Let $\ell$ be a prime number. Let
    $\mathcal{F}$ a finite type, locally constant sheaf of
    $\mathbf{F}_\ell$-vector spaces on $S_\etale$.
    Then there exists a finite \'etale morphism
    $f : T \to S$ of degree prime to $\ell$ such that $f^{-1}\mathcal{F}$
    has a finite filtration whose successive quotients are
    $\underline{\mathbf{Z}/\ell\mathbf{Z}}_T$.
    \end{lemma}
    
    \begin{proof}
    Choose a geometric point $\overline{s}$ of $S$.
    Via the equivalence of Lemma \ref{lemma-locally-constant-on-connected}
    the sheaf $\mathcal{F}$ corresponds to a finite dimensional
    $\mathbf{F}_\ell$-vector space $V$ with a continuous
    $\pi_1(S, \overline{s})$-action.
    Let $G \subset \text{Aut}(V)$ be the image of the homomorphism
    $\rho : \pi_1(S, \overline{s}) \to \text{Aut}(V)$ giving the action.
    Observe that $G$ is finite.
    The surjective continuous homomorphism
    $\overline{\rho} : \pi_1(S, \overline{s}) \to G$
    corresponds to a Galois object $Y \to S$ of
    $\textit{F\'Et}_S$ with automorphism group $G = \text{Aut}(Y/S)$, see
    Fundamental Groups, Remark \ref{pione-remark-finite-etale-under-galois}.
    Let $H \subset G$ be an $\ell$-Sylow subgroup.
    We claim that $T = Y/H \to S$ works. Namely, let $\overline{t} \in T$
    be a geometric point over $\overline{s}$. The image of
    $\pi_1(T, \overline{t}) \to \pi_1(S, \overline{s})$
    is $(\overline{\rho})^{-1}(H)$ as follows from the functorial
    nature of fundamental groups. Hence the action of $\pi_1(T, \overline{t})$
    on $V$ corresponding to $f^{-1}\mathcal{F}$ is through
    the map $\pi_1(T, \overline{t}) \to H$, see
    Remark \ref{remark-functorial-locally-constant-on-connected}. As
    $H$ is a finite $\ell$-group, the irreducible constituents of the
    representation $\rho|_{\pi_1(T, \overline{t})}$
    are each trivial of rank $1$ (this is a simple lemma on
    representation theory of finite groups; insert future reference here).
    Via the equivalence of
    Lemma \ref{lemma-locally-constant-on-connected}
    this means $f^{-1}\mathcal{F}$ is a successive extension of
    constant sheaves with value $\underline{\mathbf{Z}/\ell\mathbf{Z}}_T$.
    Moreover the degree of $T = Y/H \to S$ is prime to $\ell$
    as it is equal to the index of $H$ in $G$.
    \end{proof}

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