Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-25T00:12:13.264Z Has data issue: false hasContentIssue false
  • English
  • Français

On the K-Theory of the Coordinate Axes in the Plane

Published online by Cambridge University Press:  11 January 2016

Lars Hesselholt
Lars Hesselholt*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts, larsh@math.mit.edu, Nagoya University, Nagoya, Japan, larsh@math.nagoya-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k a regular noetherian p-algebra, let A = k[x, y]/(xy) be the coordinate ring of the coordinate axes in the affine k-plane, and let I = (x,y) be the ideal that defines the intersection point. We evaluate the relative K-groups Kq(A, I) completely in terms of the big de Rham-Witt groups of k. This generalizes a formula for K1(A, I) and K2(A, I) by Dennis and Krusemeyer.

Keywords

19G5019G5511G20
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 185 , 2007 , pp. 93 - 109
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[1] Beilinson, A. A., Higher regulators of modular curves, Applications of algebraic K-theory to Algebraic Geometry and Number Theory (Boulder, CO, 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 1–34.Google Scholar
[2] Bloch, S. and Esnault, H., An additive version of higher Chow groups, Ann. Sci. École Norm. Sup. 36 (2003), 463–477.Google Scholar
[3] Bökstedt, M., Hsiang, W.-C., and Madsen, I., The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 465–540.CrossRefGoogle Scholar
[4] Cortiñas, G., The obstruction to excision in K-theory and in cyclic homology, Invent. Math. 164 (2006), 143–173.Google Scholar
[5] Dennis, R. K. and Krusemeyer, M. I., K2(A[X, Y]/(XY)), a problem of Swan, and related computations, J. Pure Appl. Alg. 15 (1979), 125–148.CrossRefGoogle Scholar
[6] Geisser, T. and Hesselholt, L., Bi-relative algebraic K-theory and topological cyclic homology, Invent. Math. 166 (2006), 359–395.Google Scholar
[7] Geller, S., Reid, L., and Weibel, C. A., The cyclic homology and K-theory of curves, Bull. Amer. Math. Soc. 15 (1986), no. 2, 205–208.Google Scholar
[8] Geller, S., Reid, L., and Weibel, C. A., The cyclic homology and K-theory of curves, J. reine angew. Math. 393 (1989), 39–90.Google Scholar
[9] Guccione, J. A., Guccione, J. J., Redondo, M. J., and Villamayor, O. E., Hochschild and cyclic homology of hypersurfaces, Adv. Math. 95 (1992), 18–60.Google Scholar
[10] Hesselholt, L., On the p-typical curves in Quillen’s K-theory, Acta Math. 177 (1997), 1–53.CrossRefGoogle Scholar
[11] Hesselholt, L., K-theory of truncated polynomial algebras, Handbook of K-theory, Springer-Verlag, New York, 2005.Google Scholar
[12] Hesselholt, L. and Madsen, I., Cyclic polytopes and the K-theory of truncated polynomial algebras, Invent. Math. 130 (1997), 73–97.Google Scholar
[13] Hesselholt, L. and Madsen, I., On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), 29–102.Google Scholar
[14] Hesselholt, L. and Madsen, I., On the K-theory of nilpotent endomorphisms, Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math., vol. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 127–140.Google Scholar
[15] Hesselholt, L. and Madsen, I., On the de Rham-Witt complex in mixed characteristic, Ann. Sci. École Norm. Sup. 37 (2004), 1–43.CrossRefGoogle Scholar
[16] Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. 12 (1979), 501–661.CrossRefGoogle Scholar
[17] Lewis, L. G. and Mandell, M. A., Equivariant universal coefficient and Künneth spectral sequences, Proc. London Math. Soc. 92 (2006), 505–544.Google Scholar
[18] Loday, J. -L., Cyclic homology, Grundlehren der mathematischen Wissenschaften, vol. 301, Springer-Verlag, New York, 1992.Google Scholar
[19] Rülling, K., The generalized de Rham-Witt complex over a field is a complex of zero-cycles, J. Algebraic Geom. 16 (2007), 109–169.CrossRefGoogle Scholar
[20] Weibel, C. A., Mayer-Vietoris sequences and mod p K-theory, Algebraic K-theory, Part I (Oberwolfach, 1980), Lecture Notes in Math., vol. 966, Springer-Verlag, New York, 1982, pp. 390–407.Google Scholar