## Equivalences of triangulated categories and Fourier-Mukai transforms (1999)

Venue: | Bull. London Math. Soc |

Citations: | 81 - 6 self |

### BibTeX

@ARTICLE{Bridgeland99equivalencesof,

author = {Tom Bridgeland},

title = {Equivalences of triangulated categories and Fourier-Mukai transforms},

journal = {Bull. London Math. Soc},

year = {1999}

}

### OpenURL

### Abstract

Abstract. We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to Fourier-Mukai transforms are discussed. In particular we obtain a large number of such transforms for K3 surfaces. 1.

### Citations

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Citation Context ...show that F is fully faithful. 3. Equivalences of triangulated categories Here we give a condition for a fully faithful exact functor between triangulated categories to be an equivalence. We refer to =-=[9]-=-, VIII.2 for the notion of biproducts in an additive category. Definition 3.1. A triangulated category A will be called indecomposable if whenever A1 and A2 are full subcategories of A satisfying (a) ... |

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Citation Context ...y)) has a left-inverse. For any point z iz ֒→ Y , there are isomorphisms of vector spaces Lpi ∗ z (GFOy) ∼ = Hom p D(Y ) (GFOy, Oz) ∼ = Hom p D(X) (FOy, FOz) coming from the adjunctions i ∗ z ⊣ iz,∗ (=-=[7]-=-, Corollary 5.11), and G ⊣ F. Thus, by [3], Proposition 1.5, GFOy is a sheaf supported at the point y. Furthermore, there is a unique morphism GFOy → Oy. If K is the kernel of this morphism, one has a... |

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Citation Context ...). Following Mukai, we call F an integral functor. Here we derive various general properties of such functors. Most of these appeared in some form in the original papers of Mukai on Abelian varieties =-=[10]-=-,[11]. Given a scheme S, one can define a relative version of F over S. This is the functor FS : D(S × Y ) −→ D(S × X), given by the formula FS(−) = RπS×Y,∗(PS L ⊗ π ∗ S×X(−)), where S ×X πS×X ←− S ×X... |

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Citation Context ...is surjective, so that Q is the structure sheaf of a closed subscheme of S × Y . Let P be the (constant) Hilbert polynomial of the sheaf Qs on Y . By the general existence theorem for Hilbert schemes =-=[6]-=-, there is a scheme Hilb P (Y ) representing the functor which assigns to a scheme S the set of S-flat quotients Q of OS×Y with Hilbert polynomial P. Let E be the universal quotient on Hilb P (Y ) × Y... |

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Citation Context ...unctor F is fully faithful if, and only if, P is strongly simple over Y . It is an equivalence of categories precisely when one also has Py = Py ⊗ ωX for all y ∈ Y . The first statement is well-known =-=[3]-=-,[8], but the second part is new. In this paper we shall prove Theorem 1.1, along with some more general results concerning exact functors between triangulated categories. As an example of the use of ... |

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Fourier functor and its application to the moduli of bundles on an abelian variety
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Citation Context ...ections maps. Functors of this type which are equivalences of categories are called Fourier-Mukai transforms, and have proved to be powerful tools for studying moduli spaces of vector bundles [4],[5],=-=[11]-=-. A vector bundle P on X × Y is called strongly simple over Y if for each point y ∈ Y , the bundle Py on X is simple, and if for any two distinct points y1, y2 of Y , and any integer i, one has Ext i ... |

43 | Fourier-Mukai transforms for elliptic surfaces - Bridgeland - 1998 |

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On the moduli space of bundles on K3 surfaces
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Citation Context ...able vector bundles on X. Then Y is also a K3 surface, and if P is a universal bundle on X ×Y , the functor F is an equivalence of categories. Proof. The fact that Y is a K3 surface is Theorem 1.4 of =-=[12]-=-. Since ωX is trivial, it is enough to check that P is strongly simple over Y . This follows from [12], Proposition 3.12, because any stable sheaf which moves in a 2-dimensional moduli is semi-rigid. ... |

18 | Generalized Fourier-Mukai transforms
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Citation Context ...or F is fully faithful if, and only if, P is strongly simple over Y . It is an equivalence of categories precisely when one also has Py = Py ⊗ ωX for all y ∈ Y . The first statement is well-known [3],=-=[8]-=-, but the second part is new. In this paper we shall prove Theorem 1.1, along with some more general results concerning exact functors between triangulated categories. As an example of the use of Ther... |

9 | Hilbert schemes of points on some K3 surfaces and Gieseker stable bundles
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Citation Context ...projections maps. Functors of this type which are equivalences of categories are called Fourier-Mukai transforms, and have proved to be powerful tools for studying moduli spaces of vector bundles [4],=-=[5]-=-,[11]. A vector bundle P on X × Y is called strongly simple over Y if for each point y ∈ Y , the bundle Py on X is simple, and if for any two distinct points y1, y2 of Y , and any integer i, one has E... |

6 |
Equivalences of derived categories and K3 surfaces, preprint alggeom/9606006
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Citation Context ...on of the paper. 2. Fully faithful functors In this section we give a general criterion for an exact functor between triangulated categories to be fully faithful. Its proof is very similar to that of =-=[14]-=-, Lemma 2.15. Definition 2.1. Let A be a triangulated category. A subclass Ω of the objects of A will be called a spanning class for A, if for any object a of A Hom i A(ω, a) = 0 ∀ω ∈ Ω ∀i ∈ Z =⇒ a ∼ ... |

1 | Duality of polarized K3 surfaces, to appear - Mukai - 1996 |