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Niveau spectral sequences on singular schemes and failure of generalized Gersten conjecture

Author(s): Paul Balmer
Journal: Proc. Amer. Math. Soc. 137 (2009), 99-106.
MSC (2000): Primary 19E08, 19D35, 18E30
Posted: July 10, 2008
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Abstract: We construct a new local-global spectral sequence for Thomason's non-connective $ K$-theory, generalizing the Quillen spectral sequence to possibly non-regular schemes. Our spectral sequence starts at the $ E_1$-page where it displays Gersten-type complexes. It agrees with Thomason's hypercohomology spectral sequence exactly when these Gersten-type complexes are locally exact, a condition which fails for general singular schemes, as we indicate.

References:

1.
P. Balmer.
Triangular Witt groups. I. The 12-term localization exact sequence.
$ K$-Theory, 19(4):311-363, 2000. MR 1763933 (2002h:19002)

2.
P. Balmer.
Supports and filtrations in algebraic geometry and modular representation theory.
Amer. J. Math., 129(5):1227-1250, 2007. MR 2354319

3.
P. Balmer and M. Schlichting.
Idempotent completion of triangulated categories.
J. Algebra, 236(2):819-834, 2001. MR 1813503 (2002a:18013)

4.
G. Cortiñas, C. Haesemeyer, M. Schlichting, and C. Weibel.
Cyclic homology, cdh-cohomology and negative $ K$-theory, Ann. of Math., to appear.

5.
S. P. Dutta, M. Hochster, and J. E. McLaughlin.
Modules of finite projective dimension with negative intersection multiplicities.
Invent. Math., 79(2):253-291, 1985. MR 778127 (86h:13023)

6.
S. C. Geller.
A note on injectivity of lower $ K$-groups for integral domains.
In Applications of algebraic $ K$-theory to algebraic geometry and number theory, Parts I, II (Boulder, Colo., 1983), volume 55 of Contemp. Math., pages 437-447, Amer. Math. Soc., Providence, RI, 1986.
With an appendix by R. Keith Dennis and Clayton C. Sherman. MR 862647 (87k:18013)

7.
J. Hornbostel and M. Schlichting.
Localization in Hermitian $ K$-theory of rings.
J. London Math. Soc. (2), 70(1):77-124, 2004. MR 2064753 (2005b:19007)

8.
M. Levine.
Localization on singular varieties.
Invent. Math., 91(3):423-464, 1988. MR 928491 (89c:14015a)

9.
S. Mochizuki.
Gersten's conjecture for $ {K}_0$-groups.
www.math.uiuc.edu/K-theory/0842, 2007.

10.
D. Quillen.
Higher algebraic $ K$-theory. I.
Algebraic $ K$-theory, I: Higher $ K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 85-147, Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973. MR 0338129 (49:2895)

11.
M. Schlichting.
Negative $ K$-theory of derived categories.
Math. Z., 253(1):97-134, 2006. MR 2206639 (2006i:19003)

12.
R. W. Thomason and T. Trobaugh.
Higher algebraic $ K$-theory of schemes and of derived categories.
The Grothendieck Festschrift, Vol. III, volume 88 of Progr. Math., pages 247-435. Birkhäuser, Boston, MA, 1990. MR 1106918 (92f:19001)

13.
T. Vorst.
Localization of the $ K$-theory of polynomial extensions.
Math. Ann., 244(1):33-53, 1979.
With an appendix by Wilberd van der Kallen. MR 550060 (80k:18016)

14.
C. A. Weibel.
$ K$-theory and analytic isomorphisms.
Invent. Math., 61(2):177-197, 1980. MR 590161 (83b:13011)

15.
C. 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 772067 (86d:14015)

16.
C. A. Weibel.
A Brown-Gersten spectral sequence for the $ K$-theory of varieties with isolated singularities.
Adv. in Math., 73(2):192-203, 1989. MR 987274 (90j:14012)

17.
C. A. Weibel.
A Quillen-type spectral sequence for the $ K$-theory of varieties with isolated singularities.
$ K$-Theory, 3(3):261-270, 1989. MR 1040402 (91g:14009)

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Additional Information:

Paul Balmer
Affiliation: Department of Mathematics, Box 951555, University of California, Los Angeles, California 90095-1555
Email: balmer@math.ucla.edu

DOI: 10.1090/S0002-9939-08-09496-3
PII: S 0002-9939(08)09496-3
Keywords: Spectral sequence, $K$-theory, singular schemes
Received by editor(s): September 17, 2007,
Received by editor(s) in revised form: January 9, 2008
Posted: July 10, 2008
Additional Notes: The author’s research was supported by NSF grant 0654397.
Communicated by: Paul Goerss
Copyright of article: Copyright 2008, Paul Balmer

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