𝐾-regularity, 𝑐𝑑ℎ-fibrant Hochschild homology, and a conjecture of Vorst

Cortiñas, G.; Haesemeyer, C.; Weibel, C.

doi:10.1090/S0894-0347-07-00571-1

$K$-regularity, $cdh$-fibrant Hochschild homology, and a conjecture of Vorst
HTML articles powered by AMS MathViewer

by G. Cortiñas, C. Haesemeyer and C. Weibel

J. Amer. Math. Soc. 21 (2008), 547-561

DOI: https://doi.org/10.1090/S0894-0347-07-00571-1

Published electronically: May 16, 2007

PDF | Request permission

Abstract:

In this paper we prove that for an affine scheme essentially of finite type over a field $F$ and of dimension $d$, $K_{d+1}$-regularity implies regularity, assuming that the characteristic of $F$ is zero. This verifies a conjecture of Vorst.

References

Théorie des topos et cohomologie étale des schémas. Tome 2

Hyman Bass and M. Pavaman Murthy, Grothendieck groups and Picard groups of abelian group rings, Ann. of Math. (2) 86 (1967), 16–73. MR 219592, DOI 10.2307/1970360
A. K. Bousfield and E. M. Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. MR 513569

Preprint

Joachim Cuntz and Daniel Quillen, Excision in bivariant periodic cyclic cohomology, Invent. Math. 127 (1997), no. 1, 67–98. MR 1423026, DOI 10.1007/s002220050115
Barry H. Dayton and Charles A. Weibel, $K$-theory of hyperplanes, Trans. Amer. Math. Soc. 257 (1980), no. 1, 119–141. MR 549158, DOI 10.1090/S0002-9947-1980-0549158-6
Thomas G. Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), no. 2, 187–215. MR 793184, DOI 10.1016/0040-9383(85)90055-2
Thomas G. Goodwillie, Relative algebraic $K$-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347–402. MR 855300, DOI 10.2307/1971283
Christian Haesemeyer, Descent properties of homotopy $K$-theory, Duke Math. J. 125 (2004), no. 3, 589–620. MR 2166754, DOI 10.1215/S0012-7094-04-12534-5
Christian Kassel, Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), no. 1, 195–216. MR 883882, DOI 10.1016/0021-8693(87)90086-X
Christian Kassel and Arne B. Sletsjøe, Base change, transitivity and Künneth formulas for the Quillen decomposition of Hochschild homology, Math. Scand. 70 (1992), no. 2, 186–192. MR 1189973, DOI 10.7146/math.scand.a-12395
Jean-Louis Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR 1217970, DOI 10.1007/978-3-662-21739-9
Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284
Ye. A. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic $K$-theory, Algebraic $K$-theory: connections with geometry and topology (Lake Louise, AB, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 279, Kluwer Acad. Publ., Dordrecht, 1989, pp. 241–342. MR 1045853
Andrei Suslin and Vladimir Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117–189. MR 1744945
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
Ton Vorst, Polynomial extensions and excision for $K_{1}$, Math. Ann. 244 (1979), no. 3, 193–204. MR 553251, DOI 10.1007/BF01420342
Charles A. Weibel, Homotopy algebraic $K$-theory, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 461–488. MR 991991, DOI 10.1090/conm/083/991991
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Charles Weibel, Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1655–1662. MR 1277141, DOI 10.1090/S0002-9939-96-02913-9
Charles Weibel, The Hodge filtration and cyclic homology, $K$-Theory 12 (1997), no. 2, 145–164. MR 1469140, DOI 10.1023/A:1007734302948
Charles Weibel, The negative $K$-theory of normal surfaces, Duke Math. J. 108 (2001), no. 1, 1–35. MR 1831819, DOI 10.1215/S0012-7094-01-10811-9