## 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 thetrace 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:

- it commutes with étale localization and
- 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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