A negative answer to a question of Bass

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

doi:10.1090/S0002-9939-2010-10728-1

A negative answer to a question of Bass
HTML articles powered by AMS MathViewer

by G. Cortiñas, C. Haesemeyer, Mark E. Walker and C. Weibel PDF

Proc. Amer. Math. Soc. 139 (2011), 1187-1200 Request permission

Abstract:

We address Bass’ question, on whether $K_n(R)=K_n(R[t])$ implies $K_n(R)=K_n(R[t_1,t_2])$. In a companion paper, we establish a positive answer to this question when $R$ is of finite type over a field of infinite transcendence degree over the rationals. Here we provide an example of an isolated surface singularity over a number field for which the answer the Bass’ question is “no” when $n=0$.

References

Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
Hyman Bass, Some problems in “classical” algebraic $K$-theory, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 3–73. MR 0409606
G. Cortiñas, C. Haesemeyer, M. Schlichting, and C. Weibel, Cyclic homology, cdh-cohomology and negative $K$-theory, Ann. of Math. (2) 167 (2008), no. 2, 549–573. MR 2415380, DOI 10.4007/annals.2008.167.549
G. Cortiñas, C. Haesemeyer, Mark E. Walker, and C. Weibel, The $K$-theory of toric varieties, Trans. Amer. Math. Soc. 361 (2009), no. 6, 3325–3341. MR 2485429, DOI 10.1090/S0002-9947-08-04750-8
—, Bass’ $NK$ groups and $cdh$-fibrant Hochschild homology, preprint. Inventiones Math. 181 (2010), 421–448.
G. Cortiñas, C. Haesemeyer, and C. Weibel, $K$-regularity, $cdh$-fibrant Hochschild homology, and a conjecture of Vorst, J. Amer. Math. Soc. 21 (2008), no. 2, 547–561. MR 2373359, DOI 10.1090/S0894-0347-07-00571-1
Philippe Du Bois, Complexe de de Rham filtré d’une variété singulière, Bull. Soc. Math. France 109 (1981), no. 1, 41–81 (French). MR 613848
B. L. Feĭgin and B. L. Tsygan, Additive $K$-theory and crystalline cohomology, Funktsional. Anal. i Prilozhen. 19 (1985), no. 2, 52–62, 96 (Russian). MR 800920
Hans Grauert and Oswald Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292 (German). MR 302938, DOI 10.1007/BF01403182
G.-M. Greuel and G. Pfister, A singular 3.0 library for invariants of singularities, 2005, sing.lib.
G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3.0 — A computer algebra system for polynomial computations, 2005, http://www.singular.uni-kl.de.
Karsten Lebelt, Torsion äußerer Potenzen von Moduln der homologischen Dimension $1$, Math. Ann. 211 (1974), 183–197 (German). MR 371875, DOI 10.1007/BF01350711
Ben Lee, Local acyclic fibrations and the de Rham complex, Homology Homotopy Appl. 11 (2009), no. 1, 115–140. MR 2506129
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
Ruth Michler, Torsion of differentials of hypersurfaces with isolated singularities, J. Pure Appl. Algebra 104 (1995), no. 1, 81–88. MR 1359692, DOI 10.1016/0022-4049(94)00117-2
Ruth Ingrid Michler, Hodge-components of cyclic homology for affine quasi-homogeneous hypersurfaces, Astérisque 226 (1994), 10, 321–333. $K$-theory (Strasbourg, 1992). MR 1317123
Peter Orlik and Philip Wagreich, Isolated singularities of algebraic surfaces with C$^{\ast }$ action, Ann. of Math. (2) 93 (1971), 205–228. MR 284435, DOI 10.2307/1970772
Joseph H. M. Steenbrink, Du Bois invariants of isolated complete intersection singularities, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 5, 1367–1377 (English, with English and French summaries). MR 1600383
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
Carlo Traverso, Seminormality and Picard group, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 24 (1970), 585–595. MR 277542
Jonathan M. Wahl, A characterization of quasihomogeneous Gorenstein surface singularities, Compositio Math. 55 (1985), no. 3, 269–288. MR 799816
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