Preorientations of the derived motivic multiplicative group

Abstract

We establish several new model structures and Quillen adjunctions both in the classical and in the motivic case for algebras over operads and for modules over strictly commutative ring spectra. As an application, we provide a proof in the language of model categories and symmetric spectra of Lurie’s Theorem that topological complex $K$–theory represents orientations of the derived multiplicative group. Then we generalize this result to the motivic situation.