On the motivic spectra representing algebraic cobordism and algebraic $K$-theory

Abstract

We show that the motivic spectrum representing algebraic K -theory is a localization of the suspension spectrum of \mathbb{P}^\infty , and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of BGL . In particular, working over \mathbb{C} and passing to spaces of \mathbb{C} -valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we show that algebraic K -theory and periodic algebraic cobordism are E_\infty motivic spectra.