Group schemes and motivic spectra. (English) Zbl 07834191
Summary: By a theorem of Mandell, May, Schwede and Shipley [21] the stable homotopy theory of classical \(S^1\)-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 \(C_* \mathcal{F}r : \mathrm{SH}_{\mathrm{nis}} (k) \to \mathrm{SH}_{\mathrm{nis}} (k)\) in the sense of [15] that converts Morel-Voevodsky stable motivic homotopy theory \(\mathrm{SH}(k)\) into the equivalent local theory of framed bispectra [15].
MSC:
| 14Fxx | (Co)homology theory in algebraic geometry |
| 55Pxx | Homotopy theory |
| 55Uxx | Applied homological algebra and category theory in algebraic topology |
References:
| [1] | Alonso Tarrío, L.; Jeremías López, A.; Souto Salorio, M. J., Localization in categories of complexes and unbounded resolutions, Canadian Journal of Mathematics, 52, 225-247, 2000 · Zbl 0948.18008 · doi:10.4153/CJM-2000-010-4 |
| [2] | Asok, A.; Doran, B.; Fasel, J., Smooth models of motivic spheres and the clutching construction, International Mathematics Research Notices, 2017, 1890-1925, 2017 · Zbl 1405.14050 |
| [3] | Asok, A.; Fasel, J., An explicit KO-degree map and applications, Journal of Topology, 10, 268-300, 2017 · Zbl 1400.14060 · doi:10.1112/topo.12007 |
| [4] | Asok, A.; Fasel, J.; Williams, B., Motivic spheres and the image of Suslin’s Hurewicz map, Inventiones Mathematicae, 219, 39-73, 2020 · Zbl 1444.19004 · doi:10.1007/s00222-019-00907-z |
| [5] | Asok, A.; Hoyois, M.; Wendt, M., Affine representability results in \({\mathbb{A}^1} \)-homotopy theory II: Principal bundles and homogeneous spaces, Geometry & Topology, 22, 1181-1225, 2018 · Zbl 1400.14061 · doi:10.2140/gt.2018.22.1181 |
| [6] | Asok, A.; Hoyois, M.; Wendt, M., Affine representability results in \({\mathbb{A}^1} \)-homotopy theory III: finite fields and complements, Algebraic Geometry, 7, 634-644, 2020 · Zbl 1505.14050 · doi:10.14231/AG-2020-023 |
| [7] | Borceux, F., Handbook of Categorical Algebra. 2, 1994, Cambridge: Cambridge University Press, Cambridge · Zbl 0911.18001 · doi:10.1017/CBO9780511525872 |
| [8] | Conrad, B., Reductive group schemes, Autour des schémas en groupes. Vol. I, 93-444, 2014, Paris: Société mathématique de France, Paris · Zbl 1349.14151 |
| [9] | Day, B., On closed categories of functors, Reports of the Midwest Category Seminar, IV, 1-38, 1970, Berlin: Springer, Berlin · Zbl 0203.31402 |
| [10] | Dundas, B. I.; Röndigs, O.; A. Østvær, P., Enriched functors and stable homotopy theory, Documenta Mathematica, 8, 409-488, 2003 · Zbl 1040.55002 · doi:10.4171/dm/147 |
| [11] | G. Garkusha and A. Neshitov, Fibrant resolutions for motivic Thom spectra, https://arxiv.org/abs/1804.07621. · Zbl 1527.14048 |
| [12] | Garkusha, G.; Neshitov, A.; Panin, I., Framed motives of relative motivic spheres, Transactions of the American Mathematical Society, 374, 5131-5161, 2021 · Zbl 1484.14048 · doi:10.1090/tran/8386 |
| [13] | Garkusha, G.; Panin, I., On the motivic spectral sequence, Journal of the Institute of Mathematics of Jussieu, 17, 137-170, 2018 · Zbl 1390.19006 · doi:10.1017/S1474748015000419 |
| [14] | Garkusha, G.; Panin, I., Framed motives of algebraic varieties (after V. Voevodsky), Journal of the American Mathematical Society, 34, 261-313, 2021 · Zbl 1491.14034 · doi:10.1090/jams/958 |
| [15] | Garkusha, G.; Panin, I., Triangulated categories offramed bispectra and framed motives, Algebra i Analiz, 34, 135-169, 2022 · Zbl 1535.14058 |
| [16] | Greenlees, J. P C.; May, J. P., Equivariant stable homotopy theory, Handbook of Algebraic Topology, 277-323, 1995, Amsterdam: North-Holland, Amsterdam · Zbl 0866.55013 · doi:10.1016/B978-044481779-2/50009-2 |
| [17] | Hovey, M., Model Categories, 1999, Providence, RI: American mathematical Society, Providence, RI · Zbl 0909.55001 |
| [18] | Hovey, M., Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra, 165, 63-127, 2001 · Zbl 1008.55006 · doi:10.1016/S0022-4049(00)00172-9 |
| [19] | Isaksen, D., Flasque model structures for simplicial presheaves, K-Theory, 36, 371-395, 2005 · Zbl 1116.18008 · doi:10.1007/s10977-006-7113-z |
| [20] | Jardine, J. F., Motivic symmetric spectra, Documenata Mathematica, 5, 445-552, 2000 · Zbl 0969.19004 · doi:10.4171/dm/86 |
| [21] | Mandell, M. A.; May, J. P.; Schwede, S.; Shipley, B., Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82, 441-512, 2001 · Zbl 1017.55004 · doi:10.1112/S0024611501012692 |
| [22] | Morel, F., The stable \({\mathbb{A}^1} \)-connectivity theorems, K-theory, 35, 1-68, 2006 · Zbl 1117.14023 · doi:10.1007/s10977-005-1562-7 |
| [23] | Morel, F.; Voevodsky, V., \({\mathbb{A}^1} \)-homotopytheory of schemes, Institut des Hautes Etudes Scientifiques. Publications mathématiques, 90, 45-143, 1999 · Zbl 0983.14007 · doi:10.1007/BF02698831 |
| [24] | Østvær, P. A., Lectures on homotopy theory of schemes, 2011, Paris: Summer school on Motives and Milnor Conjecture, Paris |
| [25] | Panin, I.; Walter, C., On the algebraic cobordism spectra MSL and MSp, Algebra i Analiz, 34, 144-187, 2022 · Zbl 1511.14037 |
| [26] | Röndigs, O.; Spitzweck, M.; A. Østvær, P., Motivic strict ring models for K-theory, Proceedings of the American Mathematical Society, 138, 3509-3520, 2010 · Zbl 1209.14018 · doi:10.1090/S0002-9939-10-10394-3 |
| [27] | Schwede, S., On the homotopy groups of symmetric spectra, Geometry & Topology, 12, 1313-1344, 2008 · Zbl 1146.55005 · doi:10.2140/gt.2008.12.1313 |
| [28] | S. Schwede, An untitled book project about symmetric spectra (version April 2012). |
| [29] | Schwede, S.; Shipley, B., Algebras and modules in monoidal model categories, Proceedings of the London Mathematical Society, 80, 491-511, 2000 · Zbl 1026.18004 · doi:10.1112/S002461150001220X |
| [30] | V. Voevodsky, Notes on framed correspondences, https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/framed.pdf. |
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