# stacks/stacks-project

Give a reference to lower shriek + extra

Thanks to Dongryul Kim
https://stacks.math.columbia.edu/tag/03SH#comment-3396
 @@ -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}