Vol. 3, No. 1, 2018

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Equivariant noncommutative motives

Gonçalo Tabuada

Vol. 3 (2018), No. 1, 125–156
Abstract

Given a finite group G, we develop a theory of G-equivariant noncommutative motives. This theory provides a well-adapted framework for the study of G-schemes, Picard groups of schemes, G-algebras, 2-cocycles, G-equivariant algebraic K-theory, etc. Among other results, we relate our theory with its commutative counterpart as well as with Panin’s theory. As a first application, we extend Panin’s computations, concerning twisted projective homogeneous varieties, to a large class of invariants. As a second application, we prove that whenever the category of perfect complexes of a G-scheme X admits a full exceptional collection of G-invariant (G-equivariant) objects, the G-equivariant Chow motive of X is of Lefschetz type. Finally, we construct a G-equivariant motivic measure with values in the Grothendieck ring of G-equivariant noncommutative Chow motives.

Keywords
$\mathrm G$-scheme, $2$-cocycle, semidirect product algebra, twisted group algebra, equivariant algebraic $K\mkern-2mu$-theory, twisted projective homogeneous scheme, full exceptional collection, equivariant motivic measure, noncommutative algebraic geometry
Mathematical Subject Classification 2010
Primary: 14A22, 14L30, 16S35, 19L47, 55N32
Milestones
Received: 23 August 2016
Revised: 5 April 2017
Accepted: 19 April 2017
Published: 7 September 2017
Authors
Gonçalo Tabuada
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Departamento de Matemática
Faculdade de Ciência e Tecnologia
Universidade Nova de Lisboa
Portugal
Centro de Matemática e Aplicações
Faculdade de Ciência e Tecnologia
Universidade Nova de Lisboa
Portugal