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

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

2007-12-17

arxiv.org/abs/0712.2817v3

Doc. Math. 14 (2009), 359-396

Mathematics

Algebraic Geometry

28 pages; minor revisions and added applications

Scientific paper

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.

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