Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the $K$-theory and topological cyclic homology of smooth schemes over a discrete valuation ring


Authors: Thomas Geisser and Lars Hesselholt
Journal: Trans. Amer. Math. Soc. 358 (2006), 131-145
MSC (2000): Primary 11G25; Secondary 19F27
DOI: https://doi.org/10.1090/S0002-9947-04-03599-8
Published electronically: December 28, 2004
MathSciNet review: 2171226
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic $(0,p)$, the $p$-adic étale $K$-theory and $p$-adic topological cyclic homology agree.


References [Enhancements On Off] (What's this?)

  • 1. O. Gabber, $K$-theory of henselian local rings and henselian pairs, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59-70. MR 93c:19005
  • 2. T. Geisser and L. Hesselholt, Topological cyclic homology of schemes, $K$-theory (Seattle, 1997), Proc. Symp. Pure Math., vol. 67, 1999, pp. 41-87.MR 2001g:19003
  • 3. T. G. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, $K$-theory 4 (1990), 1-27. MR 92m:57027
  • 4. -, Calculus II: Analytic functors, $K$-theory 5 (1992), 295-332. MR 93i:55015
  • 5. L. Hesselholt, Stable topological cyclic homology is topological Hochschild homology, $K$-theory (Strasbourg, 1992), Asterisque, vol. 226, 1994, pp. 175-192. MR 96b:19004
  • 6. L. Hesselholt and I. Madsen, On the $K$-theory of local fields, Ann. of Math. 158 (2003), 1-113.
  • 7. J. P. May, Simplicial objects in algebraic topology. Reprint of the 1967 original, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 93m:55025
  • 8. R. McCarthy, Relative algebraic $K$-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222. MR 99e:19006
  • 9. I. A. Panin, On a theorem of Hurewicz and $K$-theory of complete discrete valuation rings, Math. USSR Izvestiya 29 (1987), 119-131. MR 88a:18021
  • 10. A. A. Suslin, Stability in algebraic $K$-theory, Algebraic $K$-theory (Proceedings, Oberwolfach, 1980), Lecture Notes in Mathematics, vol. 966, Springer-Verlag, 1982, pp. 304-333. MR 85d:18011
  • 11. -, On the $K$-theory of local fields, J. Pure Appl. Alg. 34 (1984), 304-318.MR 86d:18010
  • 12. R. W. Thomason, Algebraic $K$-theory and étale cohomology, Ann. Scient. École Norm. Sup. 13 (1985), 437-552. MR 87k:14016

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G25, 19F27

Retrieve articles in all journals with MSC (2000): 11G25, 19F27


Additional Information

Thomas Geisser
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: geisser@math.usc.edu

Lars Hesselholt
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: larsh@math.mit.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03599-8
Received by editor(s): August 15, 2002
Received by editor(s) in revised form: January 2, 2004
Published electronically: December 28, 2004
Additional Notes: Both authors were supported in part by the NSF and the Alfred P. Sloan Foundation. The first author received additional support from the JSPS
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society