On derived categories and noncommutative motives of varieties

2014/01/28 by Yusufzai, A. Khan
#14A22 #14C15 #14F05 #18D20 #18E30 #19E08 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1401.7222

Abstract

In this short note we show how results of Orlov and Toën imply that any equivalence between the derived categories of coherent sheaves on two varieties lifts to an equivalence at the level of dg-categories. This establishes the link between the noncommutative geometry practised by the school of Bondal-Orlov, and the variant developed by Kontsevich and Tabuada. As an application we recover Orlov's result that the derived category determines the Chow motive with rational coefficients up to Tate twists.

Related

• Perfect correspondences and Chow motives
• Recent developments on noncommutative motives
• Derived categories of coherent sheaves and motives of K3 surfaces
• Relations between the Chow motive and the noncommutative motive of a smooth projective variety
• Motives of Noncommutative Tori