Duke Mathematical Journal

Descent Properties of Homotopy K-Theory

Christian Haesemeyer

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Abstract

In this paper, we show that the widely held expectation that Weibel's homotopy K-theory satisfies cdh-descent is indeed fulfilled for schemes over a field of characteristic zero. The main ingredient in the proof is a certain factorization of the resolution of hypersurface singularities. Some consequences are derived. Finally, some evidence for a conjecture of Weibel concerning negative K-theory is given.

Article information

Source
Duke Math. J., Volume 125, Number 3 (2004), 589-619.

Dates
First available in Project Euclid: 18 November 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1100793680

Digital Object Identifier
doi:10.1215/S0012-7094-04-12534-5

Mathematical Reviews number (MathSciNet)
MR2166754

Zentralblatt MATH identifier
1079.19001

Subjects
Primary: 19D35: Negative $K$-theory, NK and Nil 19E08: $K$-theory of schemes [See also 14C35]
Secondary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]

Citation

Haesemeyer, Christian. Descent Properties of Homotopy K -Theory. Duke Math. J. 125 (2004), no. 3, 589--619. doi:10.1215/S0012-7094-04-12534-5. https://projecteuclid.org/euclid.dmj/1100793680


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References

  • \lccH. Bass, Algebraic $K$-Theory, Benjamin, New York, 1968.
    Mathematical Reviews (MathSciNet): MR0249491
  • \lccP. Berthelot, A. Grothendieck, and L. Illusie, Théorie des intersections et théorème de Riemann-Roch, Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Math. 225, Springer, Berlin, 1971.
    Mathematical Reviews (MathSciNet): MR0354655
  • \lccB. A. Blander, Local projective model structures on simplicial presheaves, $K$-Theory 24 (2001), 283–301.
    Mathematical Reviews (MathSciNet): MR1876801
    Digital Object Identifier: doi:10.1023/A:1013302313123
  • \lccB. H. Dayton and C. A. Weibel, $K$-theory of hyperplanes, Trans. Amer. Math. Soc. 257 (1980), 119–141.
    Mathematical Reviews (MathSciNet): MR0549158
  • \lccD. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
    Mathematical Reviews (MathSciNet): MR1322960
    Zentralblatt MATH: 0819.13001
  • \lccE. M. Friedlander and A. Suslin, The spectral sequence relating algebraic $K$-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 773–875.
    Mathematical Reviews (MathSciNet): MR1949356
    Digital Object Identifier: doi:10.1016/S0012-9593(02)01109-6
  • \lccS. C. Geller and C. A. Weibel, $K\sb{1}(A,\,B,\,I)$, J. Reine Angew. Math. 342 (1983), 12–34.
    Mathematical Reviews (MathSciNet): MR0703484
  • \lccH. Gillet and C. Soulé, Descent, motives and $K$-theory, J. Reine Angew. Math. 478 (1996), 127–176.
    Mathematical Reviews (MathSciNet): MR1409056
  • \lccM. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and Blowing Up: An Algebraic Study, appendix by B. Moonen, Springer, Berlin, 1988.
    Mathematical Reviews (MathSciNet): MR0954831
    Zentralblatt MATH: 0649.13011
  • \lccH. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. of Math. (2) 79 (1964), 109–203; 205–326.
    Mathematical Reviews (MathSciNet): MR0199184
    Zentralblatt MATH: 0122.38603
    Digital Object Identifier: doi:10.2307/1970547
  • \lccJ. F. Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39 (1987), 733–747.
    Mathematical Reviews (MathSciNet): MR0905753
    Digital Object Identifier: doi:10.4153/CJM-1987-035-8
  • ––––, Generalized Étale Cohomology Theories, Progr. Math. 146, Birkhäuser, Basel, 1997.
    Mathematical Reviews (MathSciNet): MR1437604
  • ––––, Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553.
    Mathematical Reviews (MathSciNet): MR1787949
  • \lccF. Morel and V. Voevodsky, ${\bf A}\sp 1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143.
    Mathematical Reviews (MathSciNet): MR1813224
    Digital Object Identifier: doi:10.1007/BF02698831
  • \lccD. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158.
    Mathematical Reviews (MathSciNet): MR0059889
    Zentralblatt MATH: 0057.02601
    Digital Object Identifier: doi:10.1017/S0305004100029194
  • \lccC. Pedrini and C. Weibel, “Divisibility in the Chow group of zero-cycles on a singular surface” in $K$-Theory (Strasbourg, France, 1992), Astérisque 226 (1994), 10–11, 371–409.
    Mathematical Reviews (MathSciNet): MR1317125
  • \lccA. Suslin and V. Voevodsky, “Bloch-Kato conjecture and motivic cohomology with finite coefficients” in The Arithmetic and Geometry of Algebraic Cycles (Banff, Canada, 1998), NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer, Dordrecht, 2000, 117–189.
    Mathematical Reviews (MathSciNet): MR1744945
  • \lccR. W. Thomason, Les $K$-groupes d'un schéma éclaté et une formule d'intersection excédentaire, Invent. Math. 112 (1993), 195–215.
    Mathematical Reviews (MathSciNet): MR1207482
    Digital Object Identifier: doi:10.1007/BF01232430
  • \lccR. W. Thomason and T. Trobaugh, “Higher algebraic $K$-theory of schemes and of derived categories” in The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkhäuser, Boston, 1990, 247–435.
    Mathematical Reviews (MathSciNet): MR1106918
  • \lccV. Voevodsky, “Cohomological theory of presheaves with transfers” in Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud. 143, Princeton Univ. Press, Princeton, 2000, 87–137.
    Mathematical Reviews (MathSciNet): MR1764200
  • ––––, Homotopy theory of simplicial sheaves in completely decomposable topologies, preprint, 2000, http://www.math.uiuc.edu/K-theory/0443/
  • ––––, Unstable motivic homotopy categories in Nisnevich and cdh-topologies, preprint, 2000, http://www.math.uiuc.edu/K-theory/0444/
  • \lccC. A. Weibel, $K$-theory and analytic isomorphisms, Invent. Math. 61 (1980), 177–197.
    Mathematical Reviews (MathSciNet): MR0590161
    Digital Object Identifier: doi:10.1007/BF01390120
  • ––––, “Homotopy algebraic $K$-theory” in Algebraic $K$-Theory and Number Theory, Contemp. Math. 83, Amer. Math. Soc., Providence, 1989, 461–488.
    Mathematical Reviews (MathSciNet): MR0991991
  • ––––, The negative $K$-theory of normal surfaces, Duke Math. J. 108 (2001), 1–35.
    Mathematical Reviews (MathSciNet): MR1831819
    Digital Object Identifier: doi:10.1215/S0012-7094-01-10811-9
    Project Euclid: euclid.dmj/1091737123

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