TRANSCHROMATIC EXTENSIONS IN MOTIVIC BORDISM

We show a number of Toda brackets in the homotopy of the mo-tivic bordism spectrum MGL and of the Real bordism spectrum MU_R.These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height n, the bracket itself will be of chromatic height pn ` 1q. Using these, we deduce a family of exotic multiplications in the π_p˚,˚qMGL-module structure of the motivic Morava K-theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the π_‹MU_R-module structure on the Real Morava K-theories.