×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] D.-C. Cisinski, F. Déglise. Triangulated categories of motives.
[2] Bjørn Ian Dundas, Oliver Röndigs, and Paul Arne Østvær, Motivic functors, Doc. Math. 8 (2003), 489 – 525. · Zbl 1042.55006
[3] David Gepner and Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic \?-theory, Doc. Math. 14 (2009), 359 – 396. · Zbl 1232.55010
[4] J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445 – 553. · Zbl 0969.19004
[5] Fabien Morel and Vladimir Voevodsky, \?\textonesuperior -homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45 – 143 (2001). · Zbl 0983.14007
[6] N. Naumann, M. Spitzweck, P. A. Østvær. Motivic Landweber exactness. Doc. Math. 14:551-593 (electronic), 2009. · Zbl 1230.55005
[7] N. Naumann, M. Spitzweck, P. A. Østvær. Chern classes, \( {K}\)-theory and Landweber exactness over nonregular base schemes, in Motives and Algebraic Cycles: A Celebration in Honour of Spencer J. Bloch, Fields Institute Communications, Vol. 56, 307-317, AMS, Providence, RI, 2009. · Zbl 1190.14022
[8] I. Panin, K. Pimenov, O. Röndigs. On Voevodsky’s algebraic \( K\)-theory spectrum, in Algebraic Topology, Abel Symposium 2007, 279-330, Springer-Verlag, Berlin, 2009. · Zbl 1179.14022
[9] Oliver Röndigs and Paul Arne Østvær, Motives and modules over motivic cohomology, C. R. Math. Acad. Sci. Paris 342 (2006), no. 10, 751 – 754 (English, with English and French summaries). · Zbl 1097.14016 · doi:10.1016/j.crma.2006.03.013 · doi.org
[10] Oliver Röndigs and Paul Arne Østvær, Modules over motivic cohomology, Adv. Math. 219 (2008), no. 2, 689 – 727. · Zbl 1180.14015 · doi:10.1016/j.aim.2008.05.013 · doi.org
[11] S. Schwede. An untitled book project about symmetric spectra. Available on the author’s homepage, http://www.math.uni-bonn.de/Tschwede.
[12] Markus Spitzweck and Paul Arne Østvær, The Bott inverted infinite projective space is homotopy algebraic \?-theory, Bull. Lond. Math. Soc. 41 (2009), no. 2, 281 – 292. · Zbl 1213.55006 · doi:10.1112/blms/bdn124 · doi.org
[13] M. Spitzweck, P. A. Østvær. A Bott inverted model for equivariant unitary topological \( {K}\)-theory. To appear in Math. Scand. · Zbl 1213.19004
[14] Vladimir Voevodsky, \?\textonesuperior -homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579 – 604. · Zbl 0907.19002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.