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Brown representability in 1-homotopy theory

Published online by Cambridge University Press:  05 May 2011

Niko Naumann  and
Markus Spitzweck
Niko Naumann
Affiliation:
NWF I- Mathematik, Universität Regensburg, 93040 Regensburg, Germany, [email protected]
Markus Spitzweck
Affiliation:
NWF I- Mathematik, Universität Regensburg, 93040 Regensburg, Germany, [email protected]
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Abstract

We prove the following result announced by V. Voevodsky. If S is a finite dimensional noetherian scheme such that S = ∪αSpec(Rα) for countable rings Rα, then the stable motivic homotopy category over S satisfies Brown representability.

Keywords

Motivic homotopy theorytriangulated categoriesBrown representability
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 3 , June 2011 , pp. 527 - 539
Copyright
Copyright © ISOPP 2011

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References

B.Blander, B., Local projective model structures on simplicial presheaves, K-Theory 24 (2001), no. 3, 283–301.CrossRefGoogle Scholar
DRø.Dundas, B. I., Röndigs, O., Østvær, P. A., Motivic Functors, Documenta Math 8, 2003, 527–546.Google Scholar
Hi.Hirschhorn, P.S., Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI, 2003, xvi+457 pp.Google Scholar
Ho1.Hovey, M., Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001), no. 1, 63–127.CrossRefGoogle Scholar
Ho2.Hovey, M., Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society, Providence, RI, 1999Google Scholar
HPS.Hovey, M., Palmieri, J., Strickland, N., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114 pp.Google Scholar
I.Isaksen, D., Flasque model structures for simplicial presheaves, K-Theory 36 (2005), no. 3-4, 371–395 (2006).CrossRefGoogle Scholar
J.Jardine, J.F., Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553.CrossRefGoogle Scholar
Ma.May, J.P., Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London 1967.Google Scholar
Mo.Morel, F., An introduction to 1-homotopy theory, Contemporary developments in algebraic K-theory, 357–441, ICTP Lect. Notes XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.Google Scholar
MV.Morel, F., Voevodsky, V., 1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001).CrossRefGoogle Scholar
NSø.Naumann, N., Spitzweck, M., Østvær, P. A., Motivic Landweber Exactness, Doc. Math. 14 (2009), 551–593.CrossRefGoogle Scholar
N1.Neeman, A., On a theorem of Brown and Adams, Topology 36 (1997), no. 3, 619–645.CrossRefGoogle Scholar
N2.Neeman, A., The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236.CrossRefGoogle Scholar
N3.Neeman, A., Triangulated Categories, Annals of Mathematics Studies 148, Princeton University Press, 2001.Google Scholar
PPR.Panin, I., Pimenov, K., Röndigs, O., On Voevodsky's algebraic K-theory spectrum, Algebraic topology, 279–330, Abel Symp. 4, Springer, Berlin, 2009.CrossRefGoogle Scholar
Ra.Morera, O. Raventós, Adams Representability in Triangulated Categories, PhD Thesis.Google Scholar
V.Voevodsky, V., A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Doc. Math. 1998, Extra Vol. I, 579–604.Google Scholar