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Title: | Noncommutative motives of separable algebras | Authors: | Tabuada, Goncalo VAN DEN BERGH, Michel |
Issue Date: | 2016 | Publisher: | ACADEMIC PRESS INC ELSEVIER SCIENCE | Source: | ADVANCES IN MATHEMATICS, 303, p. 1122-1161 | 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. | Notes: | [Tabuada, Goncalo] MIT, Dept Math, Cambridge, MA 02139 USA. [Tabuada, Goncalo] Univ Nova Lisboa, FCT, Dept Matemat, Lisbon, Portugal. [Tabuada, Goncalo] Univ Nova Lisboa, FCT, CMA, Lisbon, Portugal. [Van den Bergh, Michel] Univ Hasselt, Dept WNI, B-3590 Diepenbeek, Belgium. [Van den Bergh, Michel] FWO Flanders, Res, Brussels, Belgium. | Keywords: | Noncommutative motives; Separable algebra; Brauer group; Twisted flag variety; Hecke algebra; Convolution;noncommutative motives; separable algebra; Brauer group; twisted flag variety; Hecke algebra; convolution; cyclic sieving phenomenon; dg Azumaya algebra | Document URI: | http://hdl.handle.net/1942/22743 | ISSN: | 0001-8708 | e-ISSN: | 1090-2082 | DOI: | 10.1016/j.aim.2016.08.031 | ISI #: | 000386192700027 | Rights: | (C) 2016 Elsevier Inc. All rights reserved. | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2017 |
Appears in Collections: | Research publications |
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tabuada 1.pdf Restricted Access | Published version | 712.28 kB | Adobe PDF | View/Open Request a copy |
Noncummutative.pdf | Peer-reviewed author version | 538.49 kB | Adobe PDF | View/Open |
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