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The de Rham-Witt complex and $p$-adic vanishing cycles

Authors: Thomas Geisser and Lars Hesselholt
Journal: J. Amer. Math. Soc. 19 (2006), 1-36
MSC (2000): Primary 11G25, 11S70; Secondary 14F30, 19D55
Published electronically: September 16, 2005
MathSciNet review: 2169041
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Abstract: We determine the structure of the reduction modulo $p$of the absolute de Rham-Witt complex of a smooth scheme over a discrete valuation ring of mixed characteristic $(0,p)$ with log-poles along the special fiber and show that the sub-sheaf fixed by the Frobenius map is isomorphic to the sheaf of $p$-adic vanishing cycles. We use this result together with the main results of op. cit. to evaluate the algebraic $K$-theory with finite coefficients of the quotient field of the henselian local ring at a generic point of the special fiber. The result affirms the Lichtenbaum-Quillen conjecture for this field.

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

  • 1. A. A. Beilinson, Letter to C. Soulé, January 11, 1982.
  • 2. -, Height pairing between algebraic cycles, $K$-theory, arithmetic and geometry (Moscow, 1984-1986), Lecture Notes in Math., vol. 1289, Springer-Verlag, 1987, pp. 1-25. MR 0923131 (89h:11027)
  • 3. S. Bloch and K. Kato, $p$-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107-152. MR 0849653 (87k:14018)
  • 4. M. Bökstedt, W.-C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic $K$-theory of spaces, Invent. Math. 111 (1993), 465-540. MR 1202133 (94g:55011)
  • 5. T. Geisser and L. Hesselholt, On the $K$-theory and topological cyclic homology of smooth schemes over a discrete valuation ring, Trans. Amer. Math. Soc. 358 (2006), 131-145.
  • 6. -, On the $K$-theory of complete regular local $\mathbb{F} _p$-algebras, Topology (to appear).
  • 7. -, Topological cyclic homology of schemes, $K$-theory (Seattle, 1997), Proc. Symp. Pure Math., vol. 67, 1999, pp. 41-87. MR 1743237 (2001g:19003)
  • 8. L. Hesselholt, The absolute and relative de Rham-Witt complexes, Compositio Math. 141 (2005), 1109-1127.
  • 9. L. Hesselholt, Algebraic $K$-theory and trace invariants, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 415-425. MR 1957052 (2004d:19005)
  • 10. L. Hesselholt and I. Madsen, On the $K$-theory of local fields, Ann. of Math. 158 (2003), 1-113. MR 1998478 (2004k:19003)
  • 11. -, On the de Rham-Witt complex in mixed characteristic, Ann. Sci. École Norm. Sup. 37 (2004), no. 1, 1-43. MR 2050204 (2005f:19005)
  • 12. O. Hyodo and K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Périodes $p$-adiques, Asterisque, vol. 223, 1994, pp. 221-268. MR 1293974 (95k:14034)
  • 13. L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Scient. Éc. Norm. Sup. (4) 12 (1979), 501-661. MR 0565469 (82d:14013)
  • 14. B. Kahn, Deux théorèmes de comparaison en cohomologie étale: applications, Duke Math. J. 69 (1993), 137-165. MR 1201695 (94g:14009)
  • 15. K. Kato, Galois cohomology of complete discrete valuation fields, Algebraic $K$-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math., vol. 967, Springer, Berlin-New York, 1982, pp. 215-238. MR 0689394 (84k:12006)
  • 16. -, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory, Proceedings of the JAMI Inaugural Conference (Baltimore, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191-224. MR 1463703 (99b:14020)
  • 17. E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. reine angew. Math. 44 (1852), 93-146.
  • 18. A. Langer and T. Zink, De Rham-Witt cohomology for a proper and smooth morphism, J. Inst. Math. Jussieu 3 (2004), 231-314. MR 2055710 (2005d:14027)
  • 19. S. Lichtenbaum, Values of zeta-functions at non-negative integers, Number theory, Lecture Notes in Math., vol. 1068, Springer-Verlag, 1983, pp. 127-138. MR 0756089
  • 20. R. McCarthy, Relative algebraic $K$-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222. MR 1607555 (99e:19006)
  • 21. K. Nielsen, On trace homology and algebraic $K$-theory of truncated polynomial algebras, Thesis, Aarhus University, 2001.
  • 22. I. A. Panin, On a theorem of Hurewicz and $K$-theory of complete discrete valuation rings, Math. USSR Izvestiya 29 (1987), 119-131. MR 0864175 (88a:18021)
  • 23. A. A. Suslin, On the $K$-theory of local fields, J. Pure Appl. Alg. 34 (1984), 304-318. MR 0772065 (86d:18010)
  • 24. T. Tsuji, $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233-411. MR 1705837 (2000m:14024)

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Additional Information

Thomas Geisser
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089

Lars Hesselholt
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Mathematics, Nagoya University, Nagoya, Japan

Keywords: de Rham-Witt complex, $p$-adic cohomology, algebraic $K$-theory
Received by editor(s): January 5, 2004
Published electronically: September 16, 2005
Additional Notes: A previous version of this paper was entitled On the $K$-theory of a henselian discrete valuation field with non-perfect residue field.
The authors were supported in part by grants from the National Science Foundation. The first author received additional support from the Japan Society for the Promotion of Science and the Alfred P. Sloan Foundation
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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