Descent Properties of Homotopy K-Theory

previous :: next

Descent Properties of Homotopy K-Theory

Christian Haesemeyer

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

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.

Primary Subjects: 19D35, 19E08

Secondary Subjects: 14E15

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1100793680
Digital Object Identifier: doi:10.1215/S0012-7094-04-12534-5
Zentralblatt MATH identifier: 02141303
Mathematical Reviews number (MathSciNet): MR2166754

back to Table of Contents

References

H. Bass, Algebraic $K$-Theory, Benjamin, New York, 1968.

Mathematical Reviews (MathSciNet): MR0249491

P. 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

B. 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

B. H. Dayton and C. A. Weibel, $K$-theory of hyperplanes, Trans. Amer. Math. Soc. 257 (1980), 119--141.

Mathematical Reviews (MathSciNet): MR0549158

Digital Object Identifier: doi:10.2307/1998128

JSTOR: links.jstor.org

D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.

Mathematical Reviews (MathSciNet): MR1322960

E. 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

S. C. Geller and C. A. Weibel, $K\sb1(A,\,B,\,I)$, J. Reine Angew. Math. 342 (1983), 12--34.

Mathematical Reviews (MathSciNet): MR0703484

H. Gillet and C. Soulé, Descent, motives and $K$-theory, J. Reine Angew. Math. 478 (1996), 127--176.

Mathematical Reviews (MathSciNet): MR1409056

M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and Blowing Up: An Algebraic Study, appendix by B. Moonen, Springer, Berlin, 1988.

Mathematical Reviews (MathSciNet): MR0954831

H. 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

Digital Object Identifier: doi:10.2307/1970547

JSTOR: links.jstor.org

J. F. Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39 (1987), 733--747.

Mathematical Reviews (MathSciNet): MR0905753

--------, 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

F. 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

D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145--158.

Mathematical Reviews (MathSciNet): MR0059889

Digital Object Identifier: doi:10.1017/S0305004100029194

C. 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

A. 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

R. 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

R. 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

V. 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/

C. 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

previous :: next