This paper sets out basic properties of motivic twisted
–theory
with respect to degree three motivic cohomology classes of weight one. Motivic twisted
–theory is
defined in terms of such motivic cohomology classes by taking pullbacks along the universal
principal
–bundle
for the classifying space of the multiplicative group scheme
.
We show a Künneth isomorphism for homological motivic twisted
–groups computing the latter
as a tensor product of
–groups
over the
–theory
of
. The proof
employs an Adams Hopf algebroid and a trigraded Tor-spectral sequence for motivic twisted
–theory. By adapting
the notion of an
–ring
spectrum to the motivic homotopy theoretic setting, we construct
spectral sequences relating motivic (co)homology groups to twisted
–groups.
It generalizes various spectral sequences computing the algebraic
–groups of
schemes over fields. Moreover, we construct a Chern character between motivic twisted
–theory
and twisted periodized rational motivic cohomology, and show that it is a rational
isomorphism. The paper includes a discussion of some open problems.