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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/22743

Title: Noncommutative motives of separable algebras
Authors: Tabuada, Goncalo
Van den Bergh, Michel
Issue Date: 2016
Citation: 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.
URI: http://hdl.handle.net/1942/22743
DOI: 10.1016/j.aim.2016.08.031
ISI #: 000386192700027
ISSN: 0001-8708
Category: A1
Type: Journal Contribution
Validation: ecoom, 2017
Appears in Collections: Research publications

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