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?.

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.
• Lutz Hille, Michel van den Bergh, Fourier-Mukai transforms (arXiv)

• Bertrand Toen, The homotopy theory of dg-categories and derived Morita theory, 2006, arXiv.

• Alexei Bondal, Michel van den Bergh?. Generators and representability of functors in commutative and noncommutative geometry, 2002, arXiv

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

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

Revised on November 4, 2013 01:16:02 by Adeel Khan (132.252.63.205)