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Niveau spectral sequences on singular schemes and failure of generalized Gersten conjecture. (English) Zbl 1159.19002

The author constructs a new local-global spectral sequence for Thomason’s non-connective \(K\)-theory, generalizing the Quillen spectral sequences to possibly non-regular schemes. The 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 the author indicates.

MSC:

19E08 \(K\)-theory of schemes
19D35 Negative \(K\)-theory, NK and Nil
18E30 Derived categories, triangulated categories (MSC2010)
18G40 Spectral sequences, hypercohomology
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[1] Paul Balmer, Triangular Witt groups. I. The 12-term localization exact sequence, \?-Theory 19 (2000), no. 4, 311 – 363. · Zbl 0953.18003 · doi:10.1023/A:1007844609552
[2] Paul Balmer, Supports and filtrations in algebraic geometry and modular representation theory, Amer. J. Math. 129 (2007), no. 5, 1227 – 1250. · Zbl 1130.18005 · doi:10.1353/ajm.2007.0030
[3] Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819 – 834. · Zbl 0977.18009 · doi:10.1006/jabr.2000.8529
[4] G. Cortiñas, C. Haesemeyer, M. Schlichting, and C. Weibel. Cyclic homology, cdh-cohomology and negative \( K\)-theory, Ann. of Math., to appear. · Zbl 1191.19003
[5] 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. · Zbl 0588.13020 · doi:10.1007/BF01388973
[6] Susan C. Geller, A note on injectivity of lower \?-groups for integral domains, Applications of algebraic \?-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. · doi:10.1090/conm/055.2/1862647
[7] Jens Hornbostel and Marco Schlichting, Localization in Hermitian \?-theory of rings, J. London Math. Soc. (2) 70 (2004), no. 1, 77 – 124. · Zbl 1061.19003 · doi:10.1112/S0024610704005393
[8] Marc Levine, Localization on singular varieties, Invent. Math. 91 (1988), no. 3, 423 – 464. , https://doi.org/10.1007/BF01388780 Marc Levine, Erratum to: ”Localization on singular varieties”, Invent. Math. 93 (1988), no. 3, 715 – 716. · Zbl 0646.14011 · doi:10.1007/BF01410206
[9] S. Mochizuki. Gersten’s conjecture for \( {K}_0\)-groups. www.math.uiuc.edu/K-theory/0842, 2007.
[10] Daniel Quillen, Higher algebraic \?-theory. I, Algebraic \?-theory, I: Higher \?-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85 – 147. Lecture Notes in Math., Vol. 341. · Zbl 0292.18004
[11] Marco Schlichting, Negative \?-theory of derived categories, Math. Z. 253 (2006), no. 1, 97 – 134. · Zbl 1090.19002 · doi:10.1007/s00209-005-0889-3
[12] R. W. Thomason and Thomas Trobaugh, Higher algebraic \?-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247 – 435. · Zbl 0731.14001 · doi:10.1007/978-0-8176-4576-2_10
[13] Ton Vorst, Localization of the \?-theory of polynomial extensions, Math. Ann. 244 (1979), no. 1, 33 – 53. With an appendix by Wilberd van der Kallen. · Zbl 0415.13005 · doi:10.1007/BF01420335
[14] Charles A. Weibel, \?-theory and analytic isomorphisms, Invent. Math. 61 (1980), no. 2, 177 – 197. · Zbl 0437.13009 · doi:10.1007/BF01390120
[15] Charles A. Weibel, Negative \?-theory of varieties with isolated singularities, Proceedings of the Luminy conference on algebraic \?-theory (Luminy, 1983), 1984, pp. 331 – 342. · Zbl 0551.14002 · doi:10.1016/0022-4049(84)90045-8
[16] Charles A. Weibel, A Brown-Gersten spectral sequence for the \?-theory of varieties with isolated singularities, Adv. in Math. 73 (1989), no. 2, 192 – 203. · Zbl 0696.18005 · doi:10.1016/0001-8708(89)90068-6
[17] Charles A. Weibel, A Quillen-type spectral sequence for the \?-theory of varieties with isolated singularities, \?-Theory 3 (1989), no. 3, 261 – 270. · Zbl 0714.14007 · doi:10.1007/BF00533372
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