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Fourier-Mukai Transforms in Algebraic Geometry$
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D. Huybrechts

Print publication date: 2006

Print ISBN-13: 9780199296866

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199296866.001.0001

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Fourier–Mukai Transforms

Fourier–Mukai Transforms

Chapter:
(p.113) 5 Fourier–Mukai Transforms
Source:
Fourier-Mukai Transforms in Algebraic Geometry
Author(s):

D. Huybrechts

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199296866.003.0005

This chapter introduces the central notion of a Fourier-Mukai transform between derived categories. It is the derived version of the notion of a correspondence, which has been studied for all kinds of cohomology theories for many decades. In fact, Orlov's celebrated result, which is stated but not proved, says that any equivalence between derived categories of smooth projective varieties is of Fourier-Mukai type. Fourier-Mukai functors behave well in many respects: they are exact, admit left and right adjoints, can be composed, etc. The cohomological Fourier-Mukai transform behaves with respect to grading, Hodge structure, and Mukai pairing.

Keywords:   Orlov's existence, Gabriel's result, cohomological Fourier-Mukai, Mukai pairing

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