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Abstract

In classical homotopy theory, the Periodicity theorem by Devinatz-Hopkins-Smith gives a complete answer as to what types of periodicities can occur in stable homotopy theory. In motivic homotopy theory, there are additional "exotic" periodicities, but a precise classification is unknown. This work is a first step towards such a classification in the stable motivic homotopy category over the complex numbers.
Our approach is to reduce the understanding of periodicity in the motivic homotopy category to corresponding questions in a category of derived comodules over the BP Hopf algebroid, through a theorem of Gheorghe, Wang and Xu. This algebraic question in turn can be inductively reduced to simpler Hopf algebroids.
In addition to the classical family of periodicities and one previously known family of exotic motivic periodicities due to Gheorghe, we are able to show the existence of an infinite number of analogous such families. Corresponding periodic patterns in motivic homotopy groups are discussed. Interesting byproducts of the approach include analogous results in the setting of derived comodules over the BP Hopf algebroid, as well as an explicit periodicity statement about the Adams-Novikov E2 page akin to classical Adams periodicity.