Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T15:16:01.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Reconstructing rational stable motivic homotopy theory

Part of: (Co)homology theory

Published online by Cambridge University Press:  25 June 2019

Grigory Garkusha
Grigory Garkusha*
Affiliation:
Department of Mathematics, Swansea University, Fabian Way, Swansea SA1 8EN, UK email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor–Witt correspondences in the sense of Calmès and Fasel.

Keywords

motivic homotopy theorygeneralized correspondencestriangulated categories of motives

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 7 , July 2019 , pp. 1424 - 1443
Copyright
© The Author 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Al Hwaeer, H. and Garkusha, G., Grothendieck categories of enriched functors , J. Algebra 450 (2016), 204–241.10.1016/j.jalgebra.2015.09.052Google Scholar
Ananyevskiy, A., Garkusha, G. and Panin, I., Cancellation theorem for framed motives of algebraic varieties, Preprint (2016), arXiv:1601.06642.Google Scholar
Ananyevskiy, A., Levine, M. and Panin, I., Witt sheaves and the 𝜂-inverted sphere spectrum , J. Topology 10 (2017), 370–385.10.1112/topo.12015Google Scholar
Bachmann, T., Motivic and real étale stable homotopy theory , Compos. Math. 154 (2018), 883–917.10.1112/S0010437X17007710Google Scholar
Calmès, B. and Fasel, J., The category of finite $MW$ -correspondences, Preprint (2014),arXiv:1412.2989v2.Google Scholar
Cisinski, D. C. and Déglise, F., Local and stable homological algebra in Grothendieck abelian categories , Homology Homotopy Appl. 11 (2009), 219–260.10.4310/HHA.2009.v11.n1.a11Google Scholar
Cisinski, D. C. and Déglise, F., Triangulated categories of mixed motives, Preprint (2009), arXiv:0912.2110v3.Google Scholar
Day, B., On closed categories of functors , in Reports of the midwest category seminar, IV, Lecture Notes in Mathematics, vol. 137 (Springer, Berlin, 1970), 1–38.Google Scholar
Dundas, B., Röndigs, O. and Østvær, P. A., Motivic functors , Doc. Math. 8 (2003), 489–525.Google Scholar
Déglise, F. and Fasel, J., $MW$ -motivic complexes, Preprint (2017), arXiv:1708.06095.Google Scholar
Elmanto, E. and Kolderup, H., Modules over Milnor–Witt Motivic cohomology, Preprint (2017), arXiv:1708.05651.Google Scholar
Fasel, J., Groupes de Chow–Witt , Mém. Soc. Math. Fr. (N.S.) 113 (2008).Google Scholar
Fasel, J. and Østvær, P. A., A cancellation theorem for Milnor–Witt correspondences, Preprint (2017), arXiv:1708.06098.Google Scholar
Garkusha, G., Comparing motives of smooth algebraic varieties , C. R. Math. Acad. Sci. Paris 356 (2018), 1100–1105.10.1016/j.crma.2018.11.006Google Scholar
Garkusha, G., Neshitov, A. and Panin, I., Framed motives of relative motivic spheres, Preprint (2016), arXiv:1604.02732.Google Scholar
Garkusha, G. and Panin, I., K-motives of algebraic varieties , Homology Homotopy Appl. 14 (2012), 211–264.10.4310/HHA.2012.v14.n2.a13Google Scholar
Garkusha, G. and Panin, I., The triangulated category of K-motives DK - eff(k) , J. K-theory 14 (2014), 103–137.10.1017/is014004013jkt263Google Scholar
Garkusha, G. and Panin, I., Framed motives of algebraic varieties (after V. Voevodsky), Preprint (2014), arXiv:1409.4372v2.Google Scholar
Garkusha, G. and Panin, I., On the motivic spectral sequence , J. Inst. Math. Jussieu 17 (2018), 137–170.10.1017/S1474748015000419Google Scholar
Hovey, M., Spectra and symmetric spectra in general model categories , J. Pure Appl. Algebra 165 (2001), 63–127.10.1016/S0022-4049(00)00172-9Google Scholar
Hoyois, M., Kelly, S. and Østvær, P. A., The motivic Steenrod algebra in positive characteristic , J. Eur. Math. Soc. (JEMS) 19 (2017), 3813–3849.10.4171/JEMS/754Google Scholar
Jardine, J. F., Motivic symmetric spectra , Doc. Math. 5 (2000), 445–552.Google Scholar
Kolderup, H., Homotopy invariance of Nisnevich sheaves with Milnor–Witt transfers, Preprint (2017), arXiv:1708.04229.Google Scholar
Levine, M., Yang, Y. and Zhao, G., Algebraic elliptic cohomology theory and flops, I, Preprint (2013), arXiv:1311.2159.Google Scholar
Morel, F., An introduction to A1 -homotopy theory, ICTP Lecture Notes Series, vol. 15 (Abdus Salam International Centre for Theoretical Physics, Trieste, 2003), 357–442.Google Scholar
Morel, F., On the motivic 𝜋0 of the sphere spectrum , in Enriched and motivic homotopy theory, NATO Science Series, vol. 131 (Kluwer, Dordrecht, 2004), 219–260.10.1007/978-94-007-0948-5_7Google Scholar
Morel, F., The stable A1 -connectivity theorems , K-theory 35 (2006), 1–68.10.1007/s10977-005-1562-7Google Scholar
Morel, F. and Voevodsky, V., A1 -homotopy theory of schemes , Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143.10.1007/BF02698831Google Scholar
Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , J. Amer. Math. Soc. 9 (1996), 205–236.10.1090/S0894-0347-96-00174-9Google Scholar
Robinson, A., The extraordinary derived category , Math. Z. 196 (1987), 231–238.10.1007/BF01163657Google Scholar
Röndigs, O. and Østvær, P. A., Modules over motivic cohomology , Adv. Math. 219 (2008), 689–727.10.1016/j.aim.2008.05.013Google Scholar
Röndigs, O. and Østvær, P. A., Rigidity in motivic homotopy theory , Math. Ann. 341 (2008), 651–675.10.1007/s00208-008-0208-5Google Scholar
Schwede, S. and Shipley, B., Algebras and modules in monoidal model categories , Proc. Lond. Math. Soc. (3) 80 (2000), 491–511.10.1112/S002461150001220XGoogle Scholar
Serre, J.-P., Groupes d’homotopie et classes de groupes abéliens , Ann. of Math. (2) 58 (1953), 258–294.10.2307/1969789Google Scholar
Suslin, A., On the Grayson spectral sequence , Tr. Mat. Inst. Steklova 241 (2003), 218–253; (Russian). English transl. in Proc. Steklov Inst. Math. 241 (2003), 202–237.Google Scholar
Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients , in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Science Series C: Mathematics, Physics and Science, vol. 548 (Kluwer, Dordrecht, 2000), 117–189.10.1007/978-94-011-4098-0_5Google Scholar
Voevodsky, V., Cohomological theory of presheaves with transfers , in Cycles, transfers and motivic homology theories, Ann. Math. Studies, eds Voevodsky, V., Suslin, A. and Friedlander, E. (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Voevodsky, V., Triangulated category of motives over a field , in Cycles, transfers and motivic homology theories, Ann. Math. Studies, eds Voevodsky, V., Suslin, A. and Friedlander, E. (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Voevodsky, V., Notes on framed correspondences, unpublished, 2001. Also available atmath.ias.edu/vladimir/files/framed.pdf.Google Scholar
Walker, M. E., Motivic cohomology and the K-theory of automorphisms, PhD Thesis, University of Illinois at Urbana-Champaign, 1996.Google Scholar