Equivariant noncommutative motives

TL;DR

This paper develops a theory of G-equivariant noncommutative motives, connecting it with classical and motivic theories, and applies it to compute invariants and analyze G-schemes with exceptional collections.

Contribution

It introduces a new framework for G-equivariant noncommutative motives, relating it to existing theories and extending computations for twisted projective varieties.

Findings

Extended Panin's computations to new invariants

Proved G-equivariant Chow motive is of Lefschetz type under certain conditions

Constructed a G-equivariant motivic measure

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, equivariant algebraic K-theory, orbifold cohomology theory, etc. Among other results, we relate our theory with its commutative counterpart as well as with Panin's motivic 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 (different from 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 group of G-equivariant noncommutative Chow motives.