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

Balmer, Paul

doi:10.1090/S0002-9939-08-09496-3

Niveau spectral sequences on singular schemes and failure of generalized Gersten conjecture
HTML articles powered by AMS MathViewer

by Paul Balmer PDF

Proc. Amer. Math. Soc. 137 (2009), 99-106

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

Paul Balmer, Triangular Witt groups. I. The 12-term localization exact sequence, $K$-Theory 19 (2000), no. 4, 311–363. MR 1763933, DOI 10.1023/A:1007844609552
Paul Balmer, Supports and filtrations in algebraic geometry and modular representation theory, Amer. J. Math. 129 (2007), no. 5, 1227–1250. MR 2354319, DOI 10.1353/ajm.2007.0030
Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819–834. MR 1813503, DOI 10.1006/jabr.2000.8529
G. Cortiñas, C. Haesemeyer, M. Schlichting, and C. Weibel. Cyclic homology, cdh-cohomology and negative $K$-theory, Ann. of Math., to appear.
Sankar P. Dutta, M. Hochster, and J. E. McLaughlin, Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985), no. 2, 253–291. MR 778127, DOI 10.1007/BF01388973
Susan C. Geller, A note on injectivity of lower $K$-groups for integral domains, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 437–447. With an appendix by R. Keith Dennis and Clayton C. Sherman. MR 862647, DOI 10.1090/conm/055.2/1862647
Jens Hornbostel and Marco Schlichting, Localization in Hermitian $K$-theory of rings, J. London Math. Soc. (2) 70 (2004), no. 1, 77–124. MR 2064753, DOI 10.1112/S0024610704005393
Marc Levine, Localization on singular varieties, Invent. Math. 91 (1988), no. 3, 423–464. MR 928491, DOI 10.1007/BF01388780
S. Mochizuki. Gersten’s conjecture for ${K}_0$-groups. www.math.uiuc.edu/K-theory/0842, 2007.
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Marco Schlichting, Negative $K$-theory of derived categories, Math. Z. 253 (2006), no. 1, 97–134. MR 2206639, DOI 10.1007/s00209-005-0889-3
R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
Ton Vorst, Localization of the $K$-theory of polynomial extensions, Math. Ann. 244 (1979), no. 1, 33–53. With an appendix by Wilberd van der Kallen. MR 550060, DOI 10.1007/BF01420335
Charles A. Weibel, $K$-theory and analytic isomorphisms, Invent. Math. 61 (1980), no. 2, 177–197. MR 590161, DOI 10.1007/BF01390120
Charles A. Weibel, Negative $K$-theory of varieties with isolated singularities, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 331–342. MR 772067, DOI 10.1016/0022-4049(84)90045-8
Charles A. Weibel, A Brown-Gersten spectral sequence for the $K$-theory of varieties with isolated singularities, Adv. in Math. 73 (1989), no. 2, 192–203. MR 987274, DOI 10.1016/0001-8708(89)90068-6
Charles A. Weibel, A Quillen-type spectral sequence for the $K$-theory of varieties with isolated singularities, $K$-Theory 3 (1989), no. 3, 261–270. MR 1040402, DOI 10.1007/BF00533372