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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.

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