Given a finite group
, we
develop a theory of
-equivariant
noncommutative motives. This theory provides a well-adapted framework for the study of
-schemes, Picard groups
of schemes,
-algebras,
-cocycles,
-equivariant
algebraic
-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
-scheme
admits a full exceptional
collection of
-invariant
(-equivariant) objects,
the
-equivariant
Chow motive of
is of Lefschetz type. Finally, we construct a
-equivariant
motivic measure with values in the Grothendieck ring of
-equivariant
noncommutative Chow motives.
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