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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 | References | Similar Articles | Additional Information

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.

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  • 1. A. A. Beilinson, Letter to C. Soulé, January 11, 1982.
  • 2. A. A. Beĭlinson, Height pairing between algebraic cycles, 𝐾-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 1–25. MR 923131, 10.1007/BFb0078364
  • 3. Spencer Bloch and Kazuya Kato, 𝑝-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152. MR 849653
  • 4. M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic 𝐾-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539. MR 1202133, 10.1007/BF01231296
  • 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. Thomas Geisser and Lars Hesselholt, Topological cyclic homology of schemes, Algebraic 𝐾-theory (Seattle, WA, 1997) Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 41–87. MR 1743237, 10.1090/pspum/067/1743237
  • 8. L. Hesselholt, The absolute and relative de Rham-Witt complexes, Compositio Math. 141 (2005), 1109-1127.
  • 9. Lars 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
  • 10. Lars Hesselholt and Ib Madsen, On the 𝐾-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113. MR 1998478, 10.4007/annals.2003.158.1
  • 11. Lars Hesselholt and Ib Madsen, On the De Rham-Witt complex in mixed characteristic, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 1–43 (English, with English and French summaries). MR 2050204, 10.1016/j.ansens.2003.06.001
  • 12. Osamu Hyodo and Kazuya Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Astérisque 223 (1994), 221–268. Périodes 𝑝-adiques (Bures-sur-Yvette, 1988). MR 1293974
  • 13. Luc Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501–661 (French). MR 565469
  • 14. Bruno Kahn, Deux théorèmes de comparaison en cohomologie étale: applications, Duke Math. J. 69 (1993), no. 1, 137–165 (French). MR 1201695, 10.1215/S0012-7094-93-06907-4
  • 15. Kazuya 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 689394
  • 16. Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
  • 17. E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. reine angew. Math. 44 (1852), 93-146.
  • 18. Andreas Langer and Thomas Zink, De Rham-Witt cohomology for a proper and smooth morphism, J. Inst. Math. Jussieu 3 (2004), no. 2, 231–314. MR 2055710, 10.1017/S1474748004000088
  • 19. S. Lichtenbaum, Values of zeta-functions at nonnegative integers, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 127–138. MR 756089, 10.1007/BFb0099447
  • 20. Randy McCarthy, Relative algebraic 𝐾-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 197–222. MR 1607555, 10.1007/BF02392743
  • 21. K. Nielsen, On trace homology and algebraic $K$-theory of truncated polynomial algebras, Thesis, Aarhus University, 2001.
  • 22. I. A. Panin, The Hurewicz theorem and 𝐾-theory of complete discrete valuation rings, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 763–775, 878 (Russian). MR 864175
  • 23. Andrei A. Suslin, On the 𝐾-theory of local fields, Proceedings of the Luminy conference on algebraic 𝐾-theory (Luminy, 1983), 1984, pp. 301–318. MR 772065, 10.1016/0022-4049(84)90043-4
  • 24. Takeshi Tsuji, 𝑝-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), no. 2, 233–411. MR 1705837, 10.1007/s002220050330

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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.