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
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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

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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-New York, 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