cohomology

# Contents

## Idea

The Fourier-Mukai transform is a categorified integral transform analogous to the standard Fourier transform.

## 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 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

(1)$F \mapsto \mathbf{R}q_*(\mathbf{L}p^*(F) \otimes^{\mathbf{L}} E),$

where $p$ and $q$ are the projections from $X \times Y$ onto $X$ and $Y$, respectively, is called the Fourier-Mukai transform of $E$, or the 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 represented by $E$ or simply that $F$ is of Fourier-Mukai type.

## Properties

The key fact is as follows

###### Theorem (Orlov)

Let $X$ and $Y$ be smooth projective varieties over a field $K$. Let $F : D(X) \to D(Y)$ be a triangulated fully faithful functor. Then $F$ is represented by some object $E \in D(X \times Y)$ which is unique up to isomorphism.

See Orlov 2003, 3.2.1 for a proof. Though the theorem is stated there for $F$ admitting a right adjoint, it follows from Bondal-van den Bergh 2002 that every triangulated fully faithful functor admits a right adjoint automatically.

It is generally believed that this theorem should be true for all triangulated functors?.

## 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, every functor corresponds bijectively to an isomorphism class of objects on $D(X \times Y)$. See (Toen 2006).

## References

• Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys, 58 (2003), 3, 89-172, translation.

Banerjee and Hudson have defined Fourier-Mukai functors analogously on algebraic cobordism.

• Anandam Banerjee, Thomas Hudson, Fourier-Mukai transformation on algebraic cobordism, pdf.

Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in

• Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667

• Alberto Canonaco, Paolo Stellari, Internal Homs via extensions of dg functors (arXiv:1312.5619)

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

• David Ben-Zvi, David Nadler, Anatoly Preygel. Integral transforms for coherent sheaves. arXiv

Revised on February 25, 2014 00:12:47 by Anandam Banerjee? (143.248.25.234)