\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

54.65 Méthode de la trace

A reference for this section is [Exposé IX, §5, SGA4]. The material here will be used in the proof of Lemma 54.77.8 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 54.65.1. 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 54.54.2) and the Leray spectral sequence (Proposition 54.53.2). 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 54.65.2. 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 54.64.1 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 53.6.3. 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 54.64.2. 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 54.64.1 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 $G$. $\square$

Comments (1)

Comment #3396 by Dongryul Kim on

At the end of the second paragraph, it is stated that if is finite etale. Can we refer to Lemma 03S7 (54.69.5) here? I had a hard time finding out that this is proved later.

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