Perfect correspondences and Chow motives

A. Kh. Yusufzai

Published 2013-10-01Version 1

It is an open conjecture of Orlov that the bounded derived category of coherent sheaves of a smooth projective variety determines its Chow motive with rational coefficients. In this master's thesis we introduce a category of \emph{perfect correspondences}, whose objects are smooth projective varieties and morphisms $X \to Y$ are perfect complexes on $X \times Y$. We show that isomorphism in this category is the same as equivalence of derived categories, and use this to show that the derived category determines the noncommutative Chow motive (in the sense of Tabuada) and, up to Tate twists, the commutative Chow motive with rational coefficients. In particular, all additive invariants like K-theory and Hochschild or cyclic homology depend only on the derived category.