Abstract
By a theorem of Mandell, May, Schwede and Shipley [21] the stable homotopy theory of classical S1-spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic spectra are introduced and studied. It is shown that stable homotopy theory of motivic spectra is recovered from each of these types of spectra. An application is given for the localization functor \({C *}{\cal F}r:{\rm{S}}{{\rm{H}}_{{\rm{nis}}}}(k) \to {\rm{S}}{{\rm{H}}_{{\rm{nis}}}}(k)\) in the sense of [15] that converts Morel–Voevodsky stable motivic homotopy theory SH(k) into the equivalent local theory of framed bispectra [15].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Alonso Tarrío, A. Jeremías López and M. J. Souto Salorio, Localization in categories of complexes and unbounded resolutions, Canadian Journal of Mathematics 52 (2000), 225–247.
A. Asok, B. Doran and J. Fasel, Smooth models of motivic spheres and the clutching construction, International Mathematics Research Notices 2017 (2017), 1890–1925.
A. Asok and J. Fasel, An explicit KO-degree map and applications, Journal of Topology 10 (2017), 268–300.
A. Asok, J. Fasel and B. Williams, Motivic spheres and the image of Suslin’s Hurewicz map, Inventiones Mathematicae 219 (2020), 39–73.
A. Asok, M. Hoyois and M. Wendt, Affine representability results in \({\mathbb{A}^1}\)-homotopy theory II: Principal bundles and homogeneous spaces, Geometry & Topology 22 (2018), 1181–1225.
A. Asok, M. Hoyois and M. Wendt, Affine representability results in \({\mathbb{A}^1}\)-homotopy theory III: finite fields and complements, Algebraic Geometry 7 (2020), 634–644.
F. Borceux, Handbook of Categorical Algebra. 2, Encyclopedia of Mathematics and its Applications, Vol. 51, Cambridge University Press, Cambridge, 1994.
B. Conrad, Reductive group schemes, in Autour des schémas en groupes. Vol. I, Panoramas et Synthèses, Vol. 42/43, Société mathématique de France, Paris, 2014, pp. 93–444.
B. Day, On closed categories of functors, in Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 1–38.
B. I. Dundas, O. Röndigs and P. A. Østvær, Enriched functors and stable homotopy theory, Documenta Mathematica 8 (2003), 409–488.
G. Garkusha and A. Neshitov, Fibrant resolutions for motivic Thom spectra, https://arxiv.org/abs/1804.07621.
G. Garkusha, A. Neshitov and I. Panin, Framed motives of relative motivic spheres, Transactions of the American Mathematical Society 374 (2021), 5131–5161.
G. Garkusha and I. Panin, On the motivic spectral sequence, Journal of the Institute of Mathematics of Jussieu 17 (2018), 137–170.
G. Garkusha and I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), Journal of the American Mathematical Society 34 (2021), 261–313.
G. Garkusha and I. Panin, Triangulated categories offramed bispectra and framed motives, Algebra i Analiz 34 (2022), 135–169.
J. P. C. Greenlees and J. P. May, Equivariant stable homotopy theory, in Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995, pp. 277–323.
M. Hovey, Model Categories, Mathematical Surveys and Monographs, Vol. 63, American mathematical Society, Providence, RI, 1999.
M. Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra 165 (2001), 63–127.
D. Isaksen, Flasque model structures for simplicial presheaves, K-Theory 36 (2005), 371–395.
J. F. Jardine, Motivic symmetric spectra, Documenata Mathematica 5 (2000), 445–552.
M. A. Mandell, J. P. May, S. Schwede and B. Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society 82 (2001), 441–512.
F. Morel, The stable \({\mathbb{A}^1}\)-connectivity theorems, K-theory 35 (2006), 1–68.
F. Morel and V. Voevodsky, \({\mathbb{A}^1}\)-homotopytheory of schemes, Institut des Hautes Etudes Scientifiques. Publications mathématiques 90 (1999), 45–143.
P. A. Østvær, Lectures on homotopy theory of schemes, given at the Summer school on Motives and Milnor Conjecture, Paris, 2011.
I. Panin and C. Walter, On the algebraic cobordism spectra MSL and MSp, Algebra i Analiz 34 (2022), 144–187.
O. Röndigs, M. Spitzweck and P. A. Østvær, Motivic strict ring models for K-theory, Proceedings of the American Mathematical Society 138 (2010), 3509–3520.
S. Schwede, On the homotopy groups of symmetric spectra, Geometry & Topology 12 (2008), 1313–1344.
S. Schwede, An untitled book project about symmetric spectra (version April 2012).
S. Schwede and B. Shipley, Algebras and modules in monoidal model categories, Proceedings of the London Mathematical Society 80 (2000), 491–511.
V. Voevodsky, Notes on framed correspondences, https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/framed.pdf.
Acknowledgements
The author is very grateful to Alexey Ananyevskiy, Semen Podkorytov and Matthias Wendt for numerous helpful discussions. He also thanks Aravind Asok, Andrei Druzhinin and Sergey Gorchinsky for various comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by EPSRC grant EP/W012030/1.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made.
About this article
Cite this article
Garkusha, G. Group schemes and motivic spectra. Isr. J. Math. 259, 727–758 (2024). https://doi.org/10.1007/s11856-023-2492-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-023-2492-x