Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives
Author(s)Trigo Neri Tabuada, Goncalo Jorge
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I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev–Panin's computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneous varieties. As a second application, we show that the noncommutative motive of a twisted projective homogeneous variety is trivial if and only if the Brauer classes of the associated central simple algebras are trivial. Along the way we construct a fully-faithful ⊗-functor from Merkurjev–Panin's motivic category to Kontsevich's category of noncommutative Chow motives, which is of independent interest.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of Algebra
Tabuada, Gonçalo. “Additive Invariants of Toric and Twisted Projective Homogeneous Varieties via Noncommutative Motives.” Journal of Algebra 417 (November 2014): 15–38.