Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Thomason's theorem for varieties over algebraically closed fields

Author(s): Mark E. Walker
Journal: Trans. Amer. Math. Soc. 356 (2004), 2569-2648.
MSC (2000): Primary 19E15, 19E20, 14F20
Posted: October 29, 2003
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We present a novel proof of Thomason's theorem relating Bott inverted algebraic $K$-theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic $K$-homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic $K$-homology and algebraic $K$-theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.

References:

1.
M. Artin, A. Grothendieck, and J. L. Verdier.
SGA4: Théries des Topos et Cohomologie Étale des Schémas, Tome 3, volume 305 of Lecture Notes in Math.
Springer-Verlag, Berlin, Heidelberg, New York, 1973. MR 50:7132
2.
Hyman Bass.
Algebraic $K$-theory.
W. A. Benjamin, New York, Amsterdam, 1968. MR 40:2736

3.
Spencer Bloch and Steven Lichtenbaum.
A spectral sequence for motivic cohomology.
Preprint. Available at http://www.math.uiuc.edu/K-theory/0062/.

4.
A. K. Bousfield.
On the homology spectral sequence of a cosimplicial space.
Amer. J. Math., 109(2):361-394, 1987. MR 88j:55017

5.
A. K. Bousfield and E. M. Friedlander.
Homotopy of $\Gamma$-spaces, spectra and bisimplicial sets.
In M. G. Barret and M. E. Mahowald, editors, Geometric Applications of Homotopy Theory II, volume 658, pages 80-130. Springer-Verlag, 1978. MR 80e:55021

6.
A. K. Bousfield and D. M. Kan.
Homotopy Limits, Completions and Localizations, volume 304 of Lecture Notes in Math.
Springer-Verlag, 1972. MR 51:1825

7.
A. J. de Jong.
Smoothness, semi-stability and alterations.
Inst. Hautes Études Sci. Publ. Math., (83):51-93, 1996. MR 98e:14011

8.
Eric M. Friedlander and Andrei Suslin.
The spectral sequence relating algebraic $K$-theory to motivic cohomology.
Ann. Ec. Norm. Sup., 35:773-875, 2002.

9.
Eric M. Friedlander and Vladimir Voevodsky.
Bivariant cycle cohomology.
In Higher Chow groups and étale cohomology, pages 138-187. Princeton Univ. Press, Princeton, NJ, 2000. MR 2001d:14026

10.
William Fulton and Robert MacPherson.
Categorical framework for the study of singular spaces.
Mem. Amer. Math. Soc., 31(243):vi+165, 1981. MR 83a:55015

11.
Daniel R. Grayson.
Higher algebraic $K$-theory: II [after D. Quillen].
In Algebraic $K$-theory, volume 551 of Lecture Notes in Math. Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 58:28137

12.
Daniel R. Grayson.
Projections, cycles, and algebraic $K$-theory.
Math. Ann., 234:69-72, 1978. MR 58:10891

13.
Daniel R. Grayson.
Weight filtrations in algebraic $K$-theory.
In Motives, volume 55 of Proceedings of Symposia in Pure Mathematics, pages 207-244. American Mathematical Society, 1994. MR 95a:19006

14.
Daniel R. Grayson.
Weight filtrations via commuting automorphisms.
$K$-theory, 9:139-172, 1995. MR 96h:19001

15.
Daniel R. Grayson and Mark E. Walker.
Geometric models for algebraic $K$-theory.
$K$-theory, 20(4):311-330, 2000. MR 2001m:19006

16.
Robin Hartshorne.
Algebraic Geometry.
Springer-Verlag, Berlin, Heibelberg, New York, 1977. MR 57:3116

17.
Marc Levine.
$K$-theory and motivic cohomology of schemes.
Preprint. Available at http: //www.math.uiuc.edu/K-theory/0336/, February 1999.

18.
Hideyuki Matsumura.
Commutative Ring Theory.
Cambridge University Press, Cambridge, 1990. MR 90i:13001

19.
J. S. Milne.
Étale Cohomology.
Princeton University Press, Princeton, New Jersey, 1980. MR 81j:14002

20.
Daniel Quillen.
Higher algebraic $K$-theory: I.
In Algebraic $K$-theory I, volume 341 of Lecture Notes in Math. Springer-Verlag, Berlin, Heidelberg, New York, 1972. MR 49:2895

21.
Michel Raynaud and Laurent Gruson.
Critères de platitude et de projectivité. Techniques de ``platification'' d'un module.
Invent. Math., 13:1-89, 1971. MR 46:7219

22.
J. P. Serre.
Algébra Locale, Multiplicités, volume 11 of Lecture Notes in Math.
Springer-Verlag, Berlin, Heidelberg, New York, 1965. MR 34:1352

23.
A. Suslin.
On the $K$-theory of local fields.
J. Pure. and Appl. Algebra, 34:301-318, 1984. MR 86d:18010

24.
A. Suslin and V. Voevodsky.
Singular homology of abstract algebraic varieties.
Invent. Math., 123:61-94, 1996. MR 97e:14030

25.
Andrei Suslin and Vladimir Voevodsky.
Relative cycles and Chow sheaves.
In Cycles, transfers, and motivic homology theories, pages 10-86. Princeton Univ. Press, Princeton, NJ, 2000. MR 2001d:14026
26.
Andrei A. Suslin.
Higher Chow groups and étale cohomology.
In Cycles, transfers, and motivic homology theories, pages 239-254. Princeton Univ. Press, Princeton, NJ, 2000. MR 2001d:14026

27.
R. Thomason.
Algebraic $K$-theory and étale cohomology.
Ann. Sceint. Éc. Norm. Sup., 18:437-552, 1985. MR 87k:14016

28.
R. W. Thomason and T. Trobaugh.
Higher algebraic $K$-theory of schemes and of derived categories.
In The Grothendieck Festschrift, Volume III, volume 88 of Progress in Math., pages 247-436. Birkhäuser, Boston, Basél, Berlin, 1990. MR 92f:19001

29.
Vladimir Voevodsky.
Cohomological theory of presheaves with transfers.
In Higher Chow groups and étale cohomology, pages 87-137. Princeton Univ. Press, Princeton, NJ, 2000. MR 2001d:14026
30.
Friedhelm Waldhausen.
Algebraic ${K}$-theory of generalized free products. I, II.
Ann. of Math. (2), 108(1):135-204, 1978. MR 58:16845a

31.
Friedhelm Waldhausen.
Algebraic ${K}$-theory of spaces.
In Algebraic and geometric topology (New Brunswick, N.J., 1983), pages 318-419. Springer, Berlin, 1985. MR 86m:18011

32.
Mark E. Walker.
Motivic Complexes and the $K$-theory of Automorphisms.
Ph.D. thesis, University of Illinois, Urbana-Champaign, 1996.
Available at http://www.math.unl.edu/ $\sim$mwalker.

33.
Mark E. Walker.
The primitive topology of a scheme.
J. Algebra, 201(2):656-685, 1998. MR 99b:14019

34.
Mark E. Walker.
Adams operations for bivariant $K$-theory and a filtration using projective lines.
$K$-theory, 21(2):101-140, 2000. MR 2002i:19005

35.
Mark E. Walker.
Weight zero motivic cohomology and the general linear group of a simplicial ring.
J. Pure Appl. Algebra, 147(3):311-319, 2000. MR 2001b:19003

36.
Mark E. Walker.
Semi-topological $K$-homology and Thomason's theorem.
$K$-Theory, 26(3):207-286, 2002. MR 2003k:19004

37.
C. Weibel.
Homotopy algebraic $K$-theory.
In Algebraic $K$-theory and Number Theory, volume 83 of Contemporary Mathematics, pages 461-488. American Mathematical Society, 1989. MR 90d:18006

38.
Charles A. Weibel.
Negative ${K}$-theory of varieties with isolated singularities.
In Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), volume 34, pages 331-342, 1984. MR 86d:14015

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 19E15, 19E20, 14F20

Retrieve articles in all Journals with MSC (2000): 19E15, 19E20, 14F20

Additional Information:

Mark E. Walker
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0323
Email: mwalker@math.unl.edu

DOI: 10.1090/S0002-9947-03-03479-2
PII: S 0002-9947(03)03479-2
Keywords: Algebraic $K$-theory, \'etale cohomology, Thomason's theorem
Received by editor(s): August 24, 2002
Posted: October 29, 2003
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google