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Give a reference to lower shriek + extra

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aisejohan committed Aug 3, 2018
1 parent 5bf6c60 commit b144eb840b997f3fb4dab09b3548ce30bfc6f91e
Showing with 17 additions and 7 deletions.
  1. +17 −7 etale-cohomology.tex
@@ -9450,27 +9450,37 @@ \section{M\'ethode de la trace}
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}.
$\textit{Ab}(X_\etale)$ and $\textit{Ab}(Y_\etale)$.
The functor $f_!$ is discussed in Section \ref{section-extension-by-zero}.
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
called the {\it trace map}. If $f$ is finite \'etale, then $f_* = f_!$
(Lemma \ref{lemma-shriek-equals-star-finite-etale}) 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}.
The map $f_* f^{-1} \to \text{id}$ described above and below
is called the {\it trace}.
\end{definition}

\noindent
Let $f : Y \to X$ be a finite \'etale morphism. The trace map is
Let $f : Y \to X$ be a finite \'etale morphism of schemes. The trace map is
characterized by the following two properties:
\begin{enumerate}
\item it commutes with \'etale localization and
\item it commutes with \'etale localization on $X$ 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
By \'Etale Morphisms, Lemma \ref{etale-lemma-finite-etale-etale-local}
every finite \'etale morphism $f : Y \to X$ is \'etale 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}

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