Equivalences of Triangulated Categories and Fourier-Mukai Transforms

Abstract

where XKL π X X¬YMN π Y Y are the projection 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 0 on X¬Y is called strongly simple over Y if, for each point y `Y, the bundle 0 y on X is simple, and if, for any two distinct points y , y # of Y, and any integer i, one has Exti X (0 y ,0 y # ) 0. One might think of the family 20 y :y `Y as an ‘orthonormal’ set of bundles on X. The following basic result allows one to construct many examples of Fourier– Mukai transforms.