Equivariant noncommutative motives

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 ($\ne G$-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.