Motivic Landweber Exactness

TL;DR

This paper extends Landweber's exact functor theorem to the motivic setting, establishing a framework for motivic homology theories, their representability, and related operations, including a motivic Chern character.

Contribution

It proves a motivic version of Landweber's theorem, showing that certain homology theories are representable by cellular spectra and deriving formulas for motivic operations.

Findings

Motivic Landweber spectra are representable by Tate-like spectra.

Formulas for motivic operations including homotopy algebraic K-theory.

Construction of a motivic Chern character map.

Abstract

We prove a motivic version of Landweber's exact functor theorem from topology. The main result is that the assignment given by a Landweber-type formula using the MGL-homology of a motivic spectrum defines a homology theory on the stable motivic homotopy category and is representable by a Tate-like (or cellular) spectrum. Using the universal coefficient spectral sequence of Dugger-Isaksen we deduce formulas for operations of motivic Landweber spectra of a certain type including homotopy algebraic K-theory. Finally we construct a Chern character as a map between motivic spectra.