Duke Mathematical Journal

On the conservativity of the functor assigning to a motivic spectrum its motive

Tom Bachmann

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Abstract

Given a 0-connective motivic spectrum ESH(k) over a perfect field k, we determine h̲0 of the associated motive MEDM(k) in terms of π̲0(E). Using this, we show that if k has finite 2-étale cohomological dimension, then the functor M:SH(k)DM(k) is conservative when restricted to the subcategory of compact spectra and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual 2-étale cohomological dimension by considering what we call real motives.

Article information

Source
Duke Math. J., Volume 167, Number 8 (2018), 1525-1571.

Dates
Received: 11 March 2016
Revised: 21 December 2017
First available in Project Euclid: 28 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1522224100

Digital Object Identifier
doi:10.1215/00127094-2018-0002

Mathematical Reviews number (MathSciNet)
MR3807316

Zentralblatt MATH identifier
06896952

Subjects
Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]

Keywords
motivic homotopy theory conservativity slice filtration motives

Citation

Bachmann, Tom. On the conservativity of the functor assigning to a motivic spectrum its motive. Duke Math. J. 167 (2018), no. 8, 1525--1571. doi:10.1215/00127094-2018-0002. https://projecteuclid.org/euclid.dmj/1522224100


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References

  • [1] A. Ananyevskiy, M. Levine, and I. Panin, Witt sheaves and the $\eta$-inverted sphere spectrum, J. Topol. 10 (2017), 370–385.
  • [2] T. Bachmann, On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363–395.
  • [3] T. Bachmann, Motivic and real étale stable homotopy theory, preprint, arXiv:1608.08855v3 [math.KT].
  • [4] A. A. Beilinson, J. Bernstein, and P. Deligne, “Faisceaux pervers” in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982, 5–171.
  • [5] M. V. Bondarko, On infinite effectivity of motivic spectra and the vanishing of their motives, preprint, arXiv:1602.04477v3 [math.AG].
  • [6] D.-C. Cisinski and F. Déglise, Local and stable homological algebra in Grothendieck abelian categories, Homology Homotopy Appl. 11 (2009), 219–260.
  • [7] D.-C. Cisinski and F. Déglise, Étale motives, Compos. Math. 152 (2016), 556–666.
  • [8] F. Déglise, Orientable homotopy modules, Amer. J. Math. 135 (2013), 519–560.
  • [9] H. Delfs, Homology of Locally Semialgebraic Spaces, Lecture Notes in Math. 1484, Springer, Berlin, 1991.
  • [10] R. Elman and C. Lum, On the cohomological $2$-dimension of fields, Comm. Algebra 27 (1999), 615–620.
  • [11] S. I. Gelfand and Y. I. Manin, Methods of Homological Algebra, 2nd ed., Springer Monogr. Math., Springer, Berlin, 2003.
  • [12] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967).
  • [13] M. Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001), 63–127.
  • [14] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610.
  • [15] M. Hoyois, From algebraic cobordism to motivic cohomology, J. Reine Angew. Math. 702 (2015), 173–226.
  • [16] P. Hu, On the Picard group of the stable $\mathbb{A}^{1}$-homotopy category, Topology 44 (2005), 609–640.
  • [17] M. Karoubi, M. Schlichting, and C. Weibel, The Witt group of real algebraic varieties, J. Topol. 9 (2016), 1257–1302.
  • [18] M. Knebusch, “Symmetric bilinear forms over algebraic varieties” in Conference on Quadratic Forms—1967 (Kingston, Ont., 1976), Queen’s Papers Pure Appl. Math. 46, Queen’s Univ., Kingston, Ont., 1977, 103–283.
  • [19] M. Knebusch and M. Kolster, Wittrings, Aspects Math. 2, Vieweg, Braunschweig, 1982.
  • [20] M. Levine, The slice filtration and Grothendieck-Witt groups, Pure Appl. Math. Q. 7 (2011), 1543–1584.
  • [21] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxf. Grad. Texts Math. 6, Oxford Univ. Press, Oxford, 2002.
  • [22] J. P. May, Picard groups, Grothendieck rings, and Burnside rings of categories, Adv. Math. 163 (2001), 1–16.
  • [23] C. Mazza, V. Voevodsky, and C. Weibel, Lecture Notes on Motivic Cohomology, Clay Math. Monogr. 2, Amer. Math. Soc., Providence, 2006.
  • [24] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Ergeb. Math. Grenzgeb (3) 73, Springer, New York, 1973.
  • [25] F. Morel, Voevodsky’s proof of Milnor’s conjecture, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 123–143.
  • [26] F. Morel, “An introduction to $\mathbb{A}^{1}$-homotopy theory” in Contemporary Developments in Algebraic $K$-theory, ICTP Lect. Notes 15, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, 357–441.
  • [27] F. Morel, “On the motivic $\pi_{0}$ of the sphere spectrum” in Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer, Dordrecht, 2004, 219–260.
  • [28] F. Morel, The stable $\mathbb{A}^{1}$-connectivity theorems, $K$-Theory 35 (2005), 1–68.
  • [29] F. Morel, $\mathbb{A}^{1}$-Algebraic Topology over a Field, Lecture Notes in Math. 2052, Springer, Heidelberg, 2012.
  • [30] F. Morel and V. Voevodsky, $\mathbb{A}^{1}$-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143.
  • [31] A. Neeman, The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 547–566.
  • [32] M. Ojanguren and I. Panin, A purity theorem for the Witt group, Ann. Sci. Éc. Norm. Supér. (4) 32 (1999), 71–86.
  • [33] O. Röndigs and P. A. Østvær, Modules over motivic cohomology, Adv. Math. 219 (2008), 689–727.
  • [34] O. Röndigs and P. A. Østvær, Rigidity in motivic homotopy theory, Math. Ann. 341 (2008), 651–675.
  • [35] W. Scharlau, Quadratic reciprocity laws, J. Number Theory 4 (1972), 78–97.
  • [36] C. Scheiderer, Real and Étale Cohomology, Lecture Notes in Math. 1588, Springer, Berlin, 1994.
  • [37] S. S. Shatz, Profinite Groups, Arithmetic, and Geometry, Ann. of Math. Stud. 67, Princeton Univ. Press, Princeton, 1972.
  • [38] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu (accessed 3 February 2018).
  • [39] V. Voevodsky, “Open problems in the motivic stable homotopy theory, I” in Motives, Polylogarithms and Hodge Theory, Part I (Irvine, Calif., 1998), Int. Press Lect. Sr. 3, Int. Press, Somerville, Mass., 2002, 3–34.