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\begin{document}

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\section*{Fourier-Mukai transform}

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

\hypertarget{higher_linear_algebra}{}\paragraph*{{Higher linear algebra}}\label{higher_linear_algebra}

[[!include homotopy - contents]]

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

[[!include motivic cohomology - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{enhancements}{Enhancements}\dotfill \pageref*{enhancements} \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}

The \emph{Fourier-Mukai transform} is a [[categorification|categorified]] [[integral transform]] roughly similar to the standard [[Fourier transform]].

Generally, for $X,Y$ two suitably well-behaved [[schemes]] (e.g. affine, smooth, complex) and with $D(X)$, $D(Y)$ their [[derived categories]] of [[quasicoherent sheaves]], then a \emph{Fourier-Mukai transform} with [[integral kernel]] $E \in D(X\times Y)$ is a [[functor]] (of [[triangulated categories]]/[[stable (infinity,1)-categories]])

\begin{displaymath}
\Phi \colon D(X)\longrightarrow D(Y)
\end{displaymath}
which is given as the composite of the ([[derived functor|derived]]) operations of

\begin{enumerate}%
\item pull ([[inverse image]]) along the [[projection]] $p_X\colon X\times Y \to X$


\item [[tensor product]] with $E$;


\item push ([[direct image]]) along the other projection $p_Y \colon X\times Y \to Y$



\end{enumerate}
i.e.

\begin{displaymath}
\Phi(A) \coloneqq (p_Y)_\ast (E\otimes p_X^\ast A)
\end{displaymath}
(where here we implicitly understand all operations as [[derived functors]]). (e.g. \hyperlink{Huybrechts08}{Huybrechts 08, page 4})

Hence this is a pull-tensor-push [[integral transform]] through the product [[correspondence]]

\begin{displaymath}
\itexarray{
    && X \times Y
    \\
    & \swarrow && \searrow
    \\
    X && && Y
  }
\end{displaymath}
with twist $E$ on the correspondence space.

Such concept of [[integral transform]] is rather general and may be considered also in [[derived algebraic geometry]] (e.g. \hyperlink{BenZviNadlerPreygel13}{BenZvi-Nadler-Preygel 13}) and lots of other contexts.

As discussed at \emph{[[integral transforms on sheaves]]} this kind of [[integral transform]] is a [[categorification]] of an integral transform/[[matrix multiplication]] of functions induced by an [[integral kernel]], the role of which here is played by $E\in D(X \times Y)$.

Indeed, the central kind of result of the theory (theorem \ref{OrlovTheorem}) says that every suitable linear functor $D(X)\to D(Y)$ arises as a Fourier-Mukai transform for some $E$, a statement which is the [[categorification]] of the standard fact from [[linear algebra]] that every [[linear function]] between finite dimensional [[vector spaces]] is represented by a [[matrix]].

The original Fourier-Mukai transform proper is the special case of the above where $X$ is an [[abelian variety]], $Y = A^\vee$ its [[dual abelian variety]] and $E$ is the corresponding [[Poincaré line bundle]].

If $X$ is a [[moduli space of bundles|moduli space of line bundles]] over a suitable [[algebraic curve]], then a slight variant of the Fourier-Mukai transform is the [[geometric Langlands correspondence]] in the abelian case (\hyperlink{Frenkel05}{Frenkel 05, section 4.4, 4.5}).

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

Let $X$ and $Y$ be [[schemes]] over a [[field]] $K$. Let $E \in D(QCoh(O_{X \times Y}))$ be an object in the [[triangulated categories of sheaves|derived category of quasi-coherent sheaves]] over their [[product]]. (This is a [[correspondence]] between $X$ and $Y$ equipped with a [[chain complex]] $E$ of [[quasi-coherent sheaves]]).

The functor $\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y))$ defined by

\begin{displaymath}
F \mapsto \mathbf{R}q_*(\mathbf{L}p^*(F) \otimes^{\mathbf{L}} E),
\end{displaymath}
where $p$ and $q$ are the [[projections]] from $X \times Y$ onto $X$ and $Y$, respectively, is called the \textbf{Fourier-Mukai transform of $E$}, or the \textbf{Fourier-Mukai functor induced by $E$}.

When $F : D(QCoh(O_X)) \to D(QCoh(O_Y))$ is isomorphic to $\Phi(E)$ for some $E \in D(QCoh(O_{X \times Y}))$, one also says that $F$ is \textbf{represented by $E$} or simply that $F$ is \textbf{of Fourier-Mukai type}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The key fact is as follows

\begin{theorem}
\label{OrlovTheorem}\hypertarget{OrlovTheorem}{}
Let $X$ and $Y$ be smooth projective [[varieties]] over a [[field]] $K$. Let $F : D(X) \to D(Y)$ be a [[triangulated functor|triangulated]] [[fully faithful functor]]. Then $F$ is represented by some object $E \in D(X \times Y)$ which is unique up to isomorphism.

\end{theorem}
See \hyperlink{OrlovSurvey}{Orlov 2003, 3.2.1} for a proof.

\begin{remark}
\label{}\hypertarget{}{}
Though theorem \ref{OrlovTheorem} is stated there for $F$ admitting a [[right adjoint]], it follows from \hyperlink{BondalBergh2002}{Bondal-van den Bergh 2002} that every [[triangulated functor|triangulated]] [[fully faithful functor]] admits a [[right adjoint]] automatically (see e.g. \hyperlink{Huybrechts08}{Huybrechts 08, p. 6}).

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
It was believed that theorem \ref{OrlovTheorem} should be true for \emph{all} [[triangulated functors]] (e.g. \hyperlink{Huybrechts08}{Huybrechts 08, p. 5}). However according to \hyperlink{RVdB2015}{(RVdB 2015)} this is not true.

\end{remark}
\hypertarget{enhancements}{}\subsection*{{Enhancements}}\label{enhancements}

On the level of the [[DG enhancements]], it is true for all smooth proper $K$-[[schemes]] that, in the [[homotopy category]] of [[DG categories]], \emph{every} functor corresponds bijectively to an isomorphism class of objects on $D(X \times Y)$. See \hyperlink{Toen2006}{(Toen 2006)}.

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

\begin{itemize}%
\item [[triangulated categories of sheaves]]


\item [[Grothendieck duality]]



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

\begin{itemize}%
\item [[Shigeru Mukai]], \emph{Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves}. Nagoya Mathematical Journal 81: 153--175. (1981)


\item [[Alexei Bondal]], [[Michel van den Bergh]]. \emph{Generators and representability of functors in commutative and noncommutative geometry}, 2002, \href{http://arxiv.org/abs/math/0204218}{arXiv}


\item [[Dmitri Orlov]], \emph{Derived categories of coherent sheaves and equivalences between them}, Russian Math. Surveys, 58 (2003), 3, 89-172, \href{http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf}{translation}.


\item Lutz Hille, Michel van den Bergh, \emph{Fourier-Mukai transforms} (\href{http://arxiv.org/abs/math/0402043}{arXiv:0402043})


\item [[Daniel Huybrechts]], \emph{Fourier-Mukai transforms}, 2008 (\href{http://www.math.uni-bonn.de/people/huybrech/Garda2.pdf}{pdf})


\item Alice Rizzardo, [[Michel Van den Bergh]], \emph{An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves} (\href{http://arxiv.org/abs/1410.4039}{arXiv:1410.4039})


\item [[Pieter Belmans]], section 2.2 of \emph{Grothendieck duality: lecture 3}, 2014 ([[BelmansDuality.pdf:file]])



\end{itemize}
Banerjee and Hudson have defined Fourier-Mukai functors analogously on [[algebraic cobordism]].

\begin{itemize}%
\item Anandam Banerjee, Thomas Hudson, \emph{Fourier-Mukai transformation on algebraic cobordism}, \href{https://sites.google.com/site/anandamb/research/fourier-mukai.pdf?attredirects=0}{pdf}.

\end{itemize}
Discussion of [[internal homs]] of [[dg-categories]] in terms of refined Fourier-Mukai transforms is in

\begin{itemize}%
\item [[Bertrand Toën]], \emph{The homotopy theory of dg-categories and derived Morita theory}, Invent. Math. 167 (2007), 615--667


\item Alberto Canonaco, Paolo Stellari, \emph{Internal Homs via extensions of dg functors} (\href{http://arxiv.org/abs/1312.5619}{arXiv:1312.5619})



\end{itemize}
Discussion in the context of [[geometric Langlands duality]] is in

\begin{itemize}%
\item [[Edward Frenkel]], \emph{Lectures on the Langlands Program and Conformal Field Theory} (\href{http://arxiv.org/abs/hep-th/0512172}{arXiv:hep-th/0512172})

\end{itemize}
For a discussion of Fourier-Mukai transforms in the setting of $(\infty,1)$-enhancements, see

\begin{itemize}%
\item [[David Ben-Zvi]], [[David Nadler]], Anatoly Preygel. \emph{Integral transforms for coherent sheaves}, \href{http://arxiv.org/abs/1312.7164}{arXiv:1312.7164}

\end{itemize}
[[!redirects Fourier-Mukai transforms]] [[!redirects Fourier-Mukai functor]] [[!redirects Fourier-Mukai functors]] [[!redirects Fourier-Mukai transformation]]



\end{document}