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

TL;DR

This paper demonstrates that motivic spectra for algebraic K-theory and periodic algebraic cobordism are localizations of suspension spectra of classifying spaces, providing new proofs and applications in motivic homotopy theory.

Contribution

It establishes that these motivic spectra are localizations of suspension spectra, offering new proofs of classical theorems and showing they are $E_$-motivic spectra.

Findings

Motivic spectrum for algebraic K-theory is a localization of suspension spectrum of ^

Motivic spectrum for periodic algebraic cobordism is a localization of suspension spectrum of BGL

Provides new proofs of classical theorems and shows spectra are E_ motivic

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.