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.