Noncommutative motives of separable algebras
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1411.7970.pdf
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Author(s)
Van den Bergh, Michel
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
September 2016
Journal
Advances in Mathematics
Publisher
Elsevier BV
Citation
Tabuada, Gonçalo, and Michel Van den Bergh. “Noncommutative Motives of Separable Algebras.” Advances in Mathematics 303 (November 2016): 1122–1161 © 2016 Elsevier
Version
Original manuscript
Abstract
In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory of commutative separable algebras CSep(k). Making use of these models, we then explain how the category Sep(k) can be described as a "fibered Z-order" over CSep(k). This viewpoint leads to several computations and structural properties of the category Sep(k). For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (= CSA) and sequences of elements in the Brauer group of k. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic K-theory, cyclic homology, and topological Horhschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras. Keywords: Noncommutative motives; Separable algebra; Brauer group; Twisted flag variety; Hecke algebra; Convolution; Cyclic sieving phenomenon;dg Azumaya algebra
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/J.AIM.2016.08.031
