Geodesic
The de Rham-Witt complex and 𝑝-adic vanishing cycles
Geisser, Thomas1 ; Hesselholt, Lars2
1 Department of Mathematics, University of Southern California, Los Angeles, California 90089
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Mathematics, Nagoya University, Nagoya, Japan
Journal of the American Mathematical Society, Tome 19 (2006) no. 1, pp. 1-36

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-05-00505-9

Geisser, Thomas 1 ; Hesselholt, Lars 2

1 Department of Mathematics, University of Southern California, Los Angeles, California 90089
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Mathematics, Nagoya University, Nagoya, Japan 
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Geisser, Thomas; Hesselholt, Lars. The de Rham-Witt complex and 𝑝-adic vanishing cycles. Journal of the American Mathematical Society, Tome 19 (2006) no. 1, pp. 1-36. doi: 10.1090/S0894-0347-05-00505-9

[1] Beä­Linson, A. A. Height pairing between algebraic cycles 1987 1 25

[2] Bloch, Spencer, Kato, Kazuya 𝑝-adic étale cohomology Inst. Hautes Études Sci. Publ. Math. 1986 107 152

[3] Bã¶Kstedt, M., Hsiang, W. C., Madsen, I. The cyclotomic trace and algebraic 𝐾-theory of spaces Invent. Math. 1993 465 539

[4] Geisser, Thomas, Hesselholt, Lars Topological cyclic homology of schemes 1999 41 87

[5] Hesselholt, Lars Algebraic 𝐾-theory and trace invariants 2002 415 425

[6] Hesselholt, Lars, Madsen, Ib On the 𝐾-theory of local fields Ann. of Math. (2) 2003 1 113

[7] Hesselholt, Lars, Madsen, Ib On the De Rham-Witt complex in mixed characteristic Ann. Sci. École Norm. Sup. (4) 2004 1 43

[8] Hyodo, Osamu, Kato, Kazuya Semi-stable reduction and crystalline cohomology with logarithmic poles Astérisque 1994 221 268

[9] Illusie, Luc Complexe de de Rham-Witt et cohomologie cristalline Ann. Sci. École Norm. Sup. (4) 1979 501 661

[10] Kahn, Bruno Deux théorèmes de comparaison en cohomologie étale: applications Duke Math. J. 1993 137 165

[11] Kato, Kazuya Galois cohomology of complete discrete valuation fields 1982 215 238

[12] Kato, Kazuya Logarithmic structures of Fontaine-Illusie 1989 191 224

[13] Langer, Andreas, Zink, Thomas De Rham-Witt cohomology for a proper and smooth morphism J. Inst. Math. Jussieu 2004 231 314

[14] Lichtenbaum, S. Values of zeta-functions at nonnegative integers 1984 127 138

[15] Mccarthy, Randy Relative algebraic 𝐾-theory and topological cyclic homology Acta Math. 1997 197 222

[16] Panin, I. A. The Hurewicz theorem and 𝐾-theory of complete discrete valuation rings Izv. Akad. Nauk SSSR Ser. Mat. 1986

[17] Suslin, Andrei A. On the 𝐾-theory of local fields J. Pure Appl. Algebra 1984 301 318

[18] Tsuji, Takeshi 𝑝-adic étale cohomology and crystalline cohomology in the semi-stable reduction case Invent. Math. 1999 233 411

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