Annals of K-Theory

Stable $\mathbb{A}^1$-connectivity over Dedekind schemes

Johannes Schmidt and Florian Strunk

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Abstract

We show that A 1 -localization decreases the stable connectivity by at most one over a Dedekind scheme with infinite residue fields. For the proof, we establish a version of Gabber’s geometric presentation lemma over a henselian discrete valuation ring with infinite residue field.

Article information

Source
Ann. K-Theory, Volume 3, Number 2 (2018), 331-367.

Dates
Received: 19 December 2016
Revised: 4 April 2017
Accepted: 19 April 2017
First available in Project Euclid: 4 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.akt/1522807265

Digital Object Identifier
doi:10.2140/akt.2018.3.331

Mathematical Reviews number (MathSciNet)
MR3781430

Zentralblatt MATH identifier
06861676

Subjects
Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]
Secondary: 55P42: Stable homotopy theory, spectra

Keywords
$\mathbb{A}^1$-homotopy theory motivic homotopy theory

Citation

Schmidt, Johannes; Strunk, Florian. Stable $\mathbb{A}^1$-connectivity over Dedekind schemes. Ann. K-Theory 3 (2018), no. 2, 331--367. doi:10.2140/akt.2018.3.331. https://projecteuclid.org/euclid.akt/1522807265


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References

  • J. Ayoub, “Un contre-exemple à la conjecture de $\mathbb A^1$-connexité de F. Morel”, C. R. Math. Acad. Sci. Paris 342:12 (2006), 943–948.
  • J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I, Astérisque 314, 2007.
  • J.-L. Colliot-Thélène, R. T. Hoobler, and B. Kahn, “The Bloch–Ogus–Gabber theorem”, pp. 31–94 in Algebraic $K$-theory (Toronto, 1996), edited by V. P. Snaith, Fields Inst. Communications 16, American Mathematical Society, Providence, RI, 1997.
  • D. Dugger, “Replacing model categories with simplicial ones”, Trans. Amer. Math. Soc. 353:12 (2001), 5003–5027.
  • B. I. Dundas, O. Röndigs, and P. A. Østvær, “Motivic functors”, Doc. Math. 8 (2003), 489–525.
  • S. P. Dutta, “On Chow groups and intersection multiplicity of modules, II”, J. Algebra 171:2 (1995), 370–382.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III”, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255.
  • H. Fausk, P. Hu, and J. P. May, “Isomorphisms between left and right adjoints”, Theory Appl. Categ. 11 (2003), No. 4, 107–131.
  • S. I. Gelfand and Y. I. Manin, Methods of homological algebra, 2nd ed., Springer, 2003.
  • H. Gillet and M. Levine, “The relative form of Gersten's conjecture over a discrete valuation ring: the smooth case”, J. Pure Appl. Algebra 46:1 (1987), 59–71.
  • D. R. Grayson, “Projections, cycles, and algebraic $K$-theory”, Math. Ann. 234:1 (1978), 69–72.
  • P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI, 2003.
  • M. Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society, Providence, RI, 1999.
  • M. Hovey, “Spectra and symmetric spectra in general model categories”, J. Pure Appl. Algebra 165:1 (2001), 63–127.
  • M. Hoyois, “The six operations in equivariant motivic homotopy theory”, Adv. Math. 305 (2017), 197–279.
  • P. Hu, “Base change functors in the $\mathbb A^1$-stable homotopy category”, Homology Homotopy Appl. 3:2 (2001), 417–451.
  • J. F. Jardine, “Motivic symmetric spectra”, Doc. Math. 5 (2000), 445–552.
  • W. Kai, “A moving lemma for algebraic cycles with modulus and contravariance”, preprint, 2015.
  • J. Lurie, “Higher algebra”, preprint, 2017, http://www.math.harvard.edu/~lurie/papers/HA.pdf.
  • H. R. Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library 29, North-Holland Publishing Co., Amsterdam, 1983.
  • J. P. May, “The additivity of traces in triangulated categories”, Adv. Math. 163:1 (2001), 34–73.
  • F. Morel, “An introduction to $\mathbb A^1$-homotopy theory”, pp. 357–441 in Contemporary developments in algebraic $K$-theory (Trieste, 2002), edited by M. Karoubi et al., ICTP Lecture Notes 15, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.
  • F. Morel, “The stable ${\mathbb A}^1$-connectivity theorems”, $K$-Theory 35:1-2 (2005), 1–68.
  • F. Morel, $\mathbb A^1$-algebraic topology over a field, Lecture Notes in Math. 2052, Springer, 2012.
  • F. Morel and V. Voevodsky, “${\mathbb A}^1$-homotopy theory of schemes”, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143.
  • D. Quillen, “Higher algebraic $K$-theory, I”, pp. 85–147 in Algebraic $K$-theory, I: Higher $K$-theories (Seattle, 1972), edited by H. Bass, Lecture Notes in Math. 341, Springer, 1973.
  • I. R. Shafarevich, Basic algebraic geometry 1: Varieties in projective space, 2nd ed., Springer, 1994.
  • M. Spitzweck, “Algebraic cobordism in mixed characteristic”, preprint, 2014.
  • R. W. Thomason and T. Trobaugh, “Higher algebraic $K$-theory of schemes and of derived categories”, pp. 247–435 in The Grothendieck Festschrift, vol. III, edited by P. Cartier et al., Progress in Math. 88, Birkhäuser, Boston, 1990.