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\section*{sheaf with transfer}

[[!redirects Nisnevich sheaves with transfer]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{motivic_cohomology}{}\paragraph*{{Motivic cohomology}}\label{motivic_cohomology}

[[!include motivic cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given some [[category]] ([[site]]) $S$ of test spaces, suppose one fixes some category $Corr_p(S)$ of [[correspondences]] in $S$ equipped with certain cohomological data on their correspondence space. Then a \emph{sheaf with transfer} on $S$ is a contravariant functor on $Corr_p(S)$ such that the restriction along the canonical embedding $S \to Corr_p(S)$ makes the resulting [[presheaf]] a [[sheaf]].

Traditionally this is considered for $S$ the [[Nisnevich site]] and $Corr_p(S)$ constructed from correspondences equipped with algebraic cycles as discussed at \emph{[[pure motive]]}, (e.g. \hyperlink{Voevodsky00}{Voevodsky 00, 2.1 and def. 3.1.1}).

The idea is that, looking at it the other way around, the extension of a sheaf to a sheaf with transfer defines a kind of [[Umkehr map]]/[[fiber integration]] by which the sheaf is not only pulled back along maps, but also pushed forward, hence ``transferred'' (this concept of course makes sense rather generally in [[cohomology]], see e.g. \hyperlink{Piacenza84}{Piacenza 84, 1.1}).

The [[derived categories]] of those [[abelian sheaves]] with transfers for the [[Nisnevich site]] with are [[A1-homotopy theory|A1-homotopy invariant]] provides a model for [[motive|motives]] known as \emph{\href{motive#VoevodskyMotives}{Voevodsky motives}} or similar (\hyperlink{Voevodsky00}{Voevodsky 00, p. 20}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

On this page we construct the category of Nisnevich sheaves with transfer over a base scheme $S$ which is assumed to be [[regular scheme|regular]] and [[noetherian scheme|noetherian]], following \hyperlink{Deglise}{D\'e{}glise}. Nisnevich sheaves with transfer play an important role in the theory of [[mixed motives]].

\begin{enumerate}%
\item Let $S$ be a base [[scheme]] which is [[regular scheme|regular]] and [[noetherian scheme|noetherian]]. Let $Sm_S$ denote the category of [[schemes]] that are [[smooth]] and of [[finite type]] over $S$. For $X,Y \in \Sm_S$, one defines the group of \textbf{finite $S$-[[correspondences]]} $C_S(X, Y)$ as the [[free abelian group]] generated by closed integral subschemes $Z \subset X \times_S Y$ such that the projection $Z \to X$ is [[finite]] and [[equidimensional morphism|equidimensional]]. This defines an [[additive category]] $Sm_S^{cor}$ whose objects are the same as $Sm_S$ and whose morphisms are finite $S$-correspondences.


\item Note that there is a canonical functor $\gamma_S : \Sm_S \to \Sm_S^cor$ which acts on a [[morphism]] by taking the finite correspondence induced by its [[graph]].


\item Recall that the [[fibred product]] over $S$ defines a [[symmetric monoidal structure]] on $\Sm_S$. One can check that this also induces a [[symmetric monoidal structure]] on $\Sm_S^{cor}$. The functor $\gamma_S$ respects these structures.


\item Let $P_S$ (resp. $P_S^{tr}$) denote the category of [[abelian presheaves]] on $\Sm_S$ (resp. on $\Sm_S^{cor}$). Objects of $P_S^{tr}$ are called \textbf{presheaves with transfer} on $S$. We will write $L_S[X]$ for the [[presheaf]] with transfers [[representable|represented]] by an object $X \in \Sm_S$.


\item Let $N_S$ denote the category of [[sheaves]] on $\Sm_S$ with respect to the [[Nisnevich topology]]. Let $N_S^{tr} \subset P_S^{tr}$ be the [[full subcategory]] spanned by [[presheaves]] $F : \Sm_S^{cor} \to Ab$ whose [[composition]] with $\gamma_S : \Sm_S \to \Sm_S^{cor}$ is a [[sheaf]] on $Sm_S$ (with respect to the [[Nisnevich topology]]). Objects of $N_S^{tr}$ are called \textbf{Nisnevich sheaves with transfer}.


\item \textbf{Proposition.} The category $N_S^{tr}$ is [[cocomplete]] and [[Grothendieck abelian category|Grothendieck abelian]] with an [[essentially small]] set of [[generators]] given by the objects $L_S[X]$ for $X \in \Sm_S$.


\item This allows one to define a [[symmetric monoidal structure]] on $N_S^{tr}$, which is in fact the unique one for which the functor $\Sm_S \to \N_S^{tr}$ is [[symmetric monoidal]] and $- \otimes_S^{tr} -$ preserves [[colimits]]. Given $F$ and $G$ in $N_S^{tr}$, we can write

\begin{displaymath}
F = \colim_{X \to F} L_S[X] ; \qquad G = \colim_{Y \to G} L_S[Y]
\end{displaymath}
where the [[colimits]] are taken over the family of morphisms $L_S[X] \to F$ (resp. $L_S[Y] \to G$. Then we define

\begin{displaymath}
F \otimes_S^{tr} G = \colim_{X \to F ; Y \to G} L_S[X \times_S Y].
\end{displaymath}

\item This [[symmetric monoidal structure]] is further [[closed monoidal structure|closed]]: one defines

\begin{displaymath}
\underline{\Hom}_{N_S^{tr}} (F,G) (X) = \Hom_{N_S^{tr}} (F \otimes_S^{tr} L_S[X], G)
\end{displaymath}
for all $X \in \Sm_S$.



\end{enumerate}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Becker-Gottlieb transfer]], [[transfer context]]


\item [[Mackey functor]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Frédéric Déglise]], \emph{Finite correspondences and transfers over a regular base}, \href{http://www.math.uiuc.edu/K-theory/0765/regular_base.pdf}{pdf}.


\item [[Vladimir Voevodsky]], section 3.1 of \emph{Triangulated categories of motives over a field}, K-Theory, 74 (2000) (\href{http://math.illinois.edu/K-theory/0368/s5.pdf}{pdf}, \href{http://www.math.uiuc.edu/K-theory/0074/}{web})


\item [[Marc Levine]], section 3.2 of \emph{Six lectures on motives}, ICTP lectures 2006 (\href{http://users.ictp.it/~pub_off/lectures/lns023/Levine/Levine.pdf}{pdf})



\end{itemize}
Relation to [[Mackey functors]]:

\begin{itemize}%
\item [[Bruno Kahn]], Takao Yamazaki, \emph{Voevodsky's motives and Weil reciprocity}, Duke Mathematical Journal 162, 14 (2013) 2751-2796 (\href{http://arxiv.org/abs/1108.2764}{arXiv:1108.2764})

\end{itemize}
In the more general context of [[abelian sheaf cohomology]] a kind of ``transfer'' is discussed in

\begin{itemize}%
\item [[Robert Piacenza]], \emph{Transfer in generalized sheaf cohomology}, Proceedings of the AMS, Volume 90, Number 4 (1984)

\end{itemize}
[[!redirects sheaves with transfer]] [[!redirects sheaf with transfer]]

[[!redirects sheaves with transfers]] [[!redirects sheaf with transfers]]

[[!redirects presheaves with transfer]] [[!redirects presheaf with transfer]]

[[!redirects presheaves with transfers]] [[!redirects presheaf with transfers]]

[[!redirects Nisnevich sheaf with transfer]]



\end{document}