Weil transfer of noncommutative motives

TL;DR

This paper introduces a noncommutative analogue of the Weil restriction functor for motives, extending classical concepts to dg algebras and NC motives, and applies it to compute motives of schemes and relate to Brauer groups.

Contribution

It develops the noncommutative Weil restriction functor for NC motives, extending classical functors and comparing them, with applications to motives of schemes and central simple algebras.

Findings

Computed NC Chow motives for Weil restrictions of schemes with full exceptional collections

Extended Karpenko's functor to NC numerical motives and compared with classical

Explicitly described the NC Weil restriction for central simple algebras using field extension degree

Abstract

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integral coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with its NC analogue. As an application, we compute the (NC) Chow motive of the Weil restriction of every smooth projective scheme whose category of perfect complexes admits a full exceptional collection. Finally, in the case of central simple algebras, we describe explicitly the NC analogue of the Weil restriction functor using solely the degree of the field extension. This leads to a "categorification" of the classical corestriction homomorphism between Brauer groups.