Motivic strict ring models for $K$-theory

Authors:
Oliver Röndigs, Markus Spitzweck and Paul Arne Østvær

Journal:
Proc. Amer. Math. Soc. **138** (2010), 3509-3520

MSC (2010):
Primary 14F42, 55P43; Secondary 19E08

DOI:
https://doi.org/10.1090/S0002-9939-10-10394-3

Published electronically:
May 10, 2010

MathSciNet review:
2661551

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the $K$-theory of every noetherian base scheme of finite Krull dimension is represented by a commutative strict ring object in the setting of motivic stable homotopy theory. The adjective ‘strict’ is used here in order to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. An analogous topological result follows by running the same type of arguments as in the motivic setting.

- D.-C. Cisinski, F. Déglise. Triangulated categories of motives.
*arXiv:0912.2110.* - Bjørn Ian Dundas, Oliver Röndigs, and Paul Arne Østvær,
*Motivic functors*, Doc. Math.**8**(2003), 489–525. MR**2029171** - David Gepner and Victor Snaith,
*On the motivic spectra representing algebraic cobordism and algebraic $K$-theory*, Doc. Math.**14**(2009), 359–396. MR**2540697** - J. F. Jardine,
*Motivic symmetric spectra*, Doc. Math.**5**(2000), 445–552. MR**1787949** - Fabien Morel and Vladimir Voevodsky,
*${\bf A}^1$-homotopy theory of schemes*, Inst. Hautes Études Sci. Publ. Math.**90**(1999), 45–143 (2001). MR**1813224** - N. Naumann, M. Spitzweck, P. A. Østvær. Motivic Landweber exactness.
*Doc. Math.*14:551–593 (electronic), 2009. - 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. - 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. - 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). MR**2227753**, DOI https://doi.org/10.1016/j.crma.2006.03.013 - Oliver Röndigs and Paul Arne Østvær,
*Modules over motivic cohomology*, Adv. Math.**219**(2008), no. 2, 689–727. MR**2435654**, DOI https://doi.org/10.1016/j.aim.2008.05.013 - S. Schwede. An untitled book project about symmetric spectra. Available on the author’s homepage, http://www.math.uni-bonn.de/<monospace></monospace>̃schwede.
- Markus Spitzweck and Paul Arne Østvær,
*The Bott inverted infinite projective space is homotopy algebraic $K$-theory*, Bull. Lond. Math. Soc.**41**(2009), no. 2, 281–292. MR**2496504**, DOI https://doi.org/10.1112/blms/bdn124 - M. Spitzweck, P. A. Østvær. A Bott inverted model for equivariant unitary topological ${K}$-theory. To appear in
*Math. Scand.* - Vladimir Voevodsky,
*$\mathbf A^1$-homotopy theory*, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604. MR**1648048**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
14F42,
55P43,
19E08

Retrieve articles in all journals with MSC (2010): 14F42, 55P43, 19E08

Additional Information

**Oliver Röndigs**

Affiliation:
Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany

Email:
oroendig@math.uni-osnabrueck.de

**Markus Spitzweck**

Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

Email:
markussp@math.uio.no

**Paul Arne Østvær**

Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

Email:
paularne@math.uio.no

Received by editor(s):
October 13, 2009

Received by editor(s) in revised form:
January 19, 2010

Published electronically:
May 10, 2010

Communicated by:
Brooke Shipley

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.