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Motivic strict ring models for $$K$$-theory. (English) Zbl 1209.14018
Let $$\mathcal S$$ be a noetherian base scheme of finite Krull dimension with multiplicative group scheme $${\mathbf{ {G}}}_{\mathfrak m}$$ and let $$KGL$$ be the motivic spectrum representing $$K$$-theory V. Voevodsky, [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. I, 579–604 (1998; Zbl 0907.19002)]. It is known [cf. M. Spitzweck, P. A. Østvaer, Bull. Lond. Math. Soc. 41, No. 2, 281–292 (2009; Zbl 1213.55006)] that after inverting a homotopy class of the Bott map $$\beta \in {\pi}_{2,1}{\Sigma}^{\infty}B{\mathbf{ {G}}}_{{\mathfrak m} +}$$ one obtains a natural isomorphism in the motivic stable category: $\begin{tikzcd}\Sigma^\infty B\mathbf G_{\mathfrak m +}[\beta^{-1}] \rar["\cong"] & KGL\end{tikzcd}$ Using this the authors construct the spectrum $$KGL^{\beta}$$ which is a commutative monoid in the category of motivic symmetric spectra and has the homotopy type of $$K$$-theory. They also show that the multiplicative structures on $$KGL^{\beta}$$ and $$KGL$$ coincide in the motivic stable homotopy category.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 55P43 Spectra with additional structure ($$E_\infty$$, $$A_\infty$$, ring spectra, etc.) 19E08 $$K$$-theory of schemes
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##### References:
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