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

### David Gepner

### Victor Snaith

## Abstract

We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $P_{∞}$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $C$ and passing to spaces of $C$-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 $E_{∞}$ motivic spectra.

## Cite this article

David Gepner, Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic $K$-theory. Doc. Math. 14 (2009), pp. 359–396

DOI 10.4171/DM/276