2013/10/15 by Gonçalo Tabuada, Tabuada, Goncalo · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1310.4063
I. Panin proved in the nineties that the algebraic K-theory of twisted\nprojective homogeneous varieties can be expressed in terms of central simple\nalgebras. Later, Merkurjev and Panin described the algebraic K-theory of toric\nvarieties as a direct summand of the algebraic K-theory of separable algebras.\nIn this article, making use of the recent theory of noncommutative motives, we\nextend Panin and Merkurjev-Panin computations from algebraic K-theory to every\nadditive invariant. As a first application, we fully compute the cyclic\nhomology (and all its variants) of twisted projective homogeneous varieties. As\na second application, we show that the noncommutative motive of a twisted\nprojective homogeneous variety is trivial if and only if the Brauer classes of\nthe associated central simple algebras are trivial. Along the way we construct\na fully-faithful tensor functor from Merkurjev-Panin's motivic category to\nKontsevich's category of noncommutative Chow motives, which is of independent\ninterest.\n