Definition 59.66.1. Let $f : Y \to X$ be a finite étale morphism of schemes. The map $f_* f^{-1} \to \text{id}$ described above and explicitly below is called the trace.
59.66 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 59.83.9 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 functor $f_!$ is discussed in Section 59.70. 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 étale, then $f_* = f_!$ (Lemma 59.70.7) and we can view this as a map $f_*f^{-1} \to \text{id}$.
Let $f : Y \to X$ be a finite étale morphism of schemes. The trace map is characterized by the following two properties:
it commutes with étale localization on $X$ and
if $Y = \coprod _{i = 1}^ d X$ then the trace map is the sum map $f_*f^{-1} \mathcal{F} = \mathcal{F}^{\oplus d} \to \mathcal{F}$.
By Étale Morphisms, Lemma 41.18.3 every finite étale morphism $f : Y \to X$ is étale locally on $X$ of the form given in (2) for some integer $d \geq 0$. Hence we can define the trace map using the characterization given; in particular we do not need to know about the existence of $f_!$ and the agreement of $f_!$ with $f_*$ in order to construct the trace map. This description shows 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 59.55.2) and the Leray spectral sequence (Proposition 59.54.2). 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 59.66.2. Let $S$ be a connected scheme. Let $\ell $ be a prime number. Let $\mathcal{F}$ be 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 59.65.1 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, Section 58.7. 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 59.65.3. 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 59.65.1 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 $G$. $\square$
Lemma 59.66.3. Let $\Lambda $ be a Noetherian ring. Let $\ell $ be a prime number and $n \geq 1$. Let $H$ be a finite $\ell $-group. Let $M$ be a finite $\Lambda [H]$-module annihilated by $\ell ^ n$. Then there is a finite filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_ t = M$ by $\Lambda [H]$-submodules such that $H$ acts trivially on $M_{i + 1}/M_ i$ for all $i = 0, \ldots , t - 1$.
Proof. Omitted. Hint: Show that the augmentation ideal $\mathfrak m$ of the noncommutative ring $\mathbf{Z}/\ell ^ n\mathbf{Z}[H]$ is nilpotent. $\square$
Lemma 59.66.4. Let $S$ be an irreducible, geometrically unibranch scheme. Let $\ell $ be a prime number and $n \geq 1$. Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a finite type, locally constant sheaf of $\Lambda $-modules on $S_{\acute{e}tale}$ which is annihilated by $\ell ^ n$. 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 of the form $\underline{M}_ T$ for some finite $\Lambda $-modules $M$.
Proof. Choose a geometric point $\overline{s}$ of $S$. Via the equivalence of Lemma 59.65.2 the sheaf $\mathcal{F}$ corresponds to a finite $\Lambda $-module $M$ with a continuous $\pi _1(S, \overline{s})$-action. Let $G \subset \text{Aut}(V)$ be the image of the homomorphism $\rho : \pi _1(S, \overline{s}) \to \text{Aut}(M)$ giving the action. Observe that $G$ is finite as $M$ is a finite $\Lambda $-module (see proof of Lemma 59.65.2). 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, Section 58.7. 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 $M$ corresponding to $f^{-1}\mathcal{F}$ is through the map $\pi _1(T, \overline{t}) \to H$, see Remark 59.65.3. Let $0 = M_0 \subset M_1 \subset \ldots \subset M_ t = M$ be as in Lemma 59.66.3. This induces a filtration $0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ t = f^{-1}\mathcal{F}$ such that the successive quotients are constant with value $M_{i + 1}/M_ i$. Finally, the degree of $T = Y/H \to S$ is prime to $\ell $ as it is equal to the index of $H$ in $G$. $\square$
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