nLab
Fourier-Mukai transform

Context

Motivic cohomology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

Definition

Let XX and YY be schemes over a field KK. Let ED(QCoh(O X×Y))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 XX and YY equipped with a chain complex EE of quasi-coherent sheaves).

The functor Φ(E):D(QCoh(O X))D(QCoh(O Y))\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y)) defined by

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

where pp and qq are the projections from X×YX \times Y onto XX and YY, respectively, is called the Fourier-Mukai transform of EE, or the Fourier-Mukai functor induced by EE.

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

Properties

The key fact is as follows

Theorem (Orlov)

Let XX and YY be smooth projective varieties over a field KK. Let F:D(X)D(Y)F : D(X) \to D(Y) be a triangulated fully faithful functor. Then FF is represented by some object ED(X×Y)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 FF 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 KK-schemes that, in the homotopy category of DG categories, every functor corresponds bijectively to an isomorphism class of objects on D(X×Y)D(X \times Y). See (Toen 2006).

See also

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 (,1)(\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)