Goncalo Tabuada: Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives.


Submission: 2013, Oct 15

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 the recent theory of noncommutative motives, we extend Panin and Merkurjev-Panin 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 tensor functor from Merkurjev-Panin's motivic category to Kontsevich's category of noncommutative Chow motives, which is of independent interest.

2010 Mathematics Subject Classification: 11E81, 14A22, 14L17, 14M25, 18F25, 19D55

Keywords and Phrases: Homogeneous varieties, toric varieties, twisted forms, torsors, noncommutative motives, algebraic K-theory, cyclic homology, noncommutative algebraic geometry.

Full text: dvi.gz 50 k, dvi 192 k, ps.gz 1257 k, pdf.gz 237 k, pdf 303 k.

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