The generalized de Rham-Witt complex over a field is a complex of zero-cycles
Author:
Kay Rülling
Journal:
J. Algebraic Geom. 16 (2007), 109-169
DOI:
https://doi.org/10.1090/S1056-3911-06-00446-2
Published electronically:
July 24, 2006
Erratum:
J. Algebraic Geom. 16 (2007), 793-795.
MathSciNet review:
2257322
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Abstract |
References |
Additional Information
Abstract: Bloch and Esnault defined additive higher Chow groups with modulus $m$ on the level of zero cycles over a field $k$ denoted by $\text {CH}^n((\mathbb {A}^1_k,(m+1)\{0\}),n-1)$, $n,m\ge 1$. Bloch and Esnault prove $\text {CH}^n((\mathbb {A}^1_k,2\{0\}),n-1)\cong \Omega ^{n-1}_{k/\mathbb {Z}}$. In this paper we generalize their result and prove that the additive Chow groups with higher modulus form a generalized Witt complex over $k$ and are as such isomorphic to the generalized de Rham-Witt complex of Bloch-Deligne-Hesselholt-Illusie-Madsen.
References
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References
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- A. Langer, T. Zink, De Rham-Witt cohomology for a proper and smooth morphism. J. Inst. Math. Jussieu 3, No. 2, 2004, 231-314. MR 2055710 (2005d:14027)
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- J.-P. Serre, Corps locaux. Deuxième édition. Hermann, Paris, 1968. MR 0354618 (50:7096)
- J.-P. Serre, Algebraic groups and class fields. Transl. of the French edition. Graduate Texts in Mathematics, 117. New York etc.: Springer-Verlag. ix, 1988. MR 0918564 (88i:14041)
- J. H. Silverman, The arithmetic of elliptic curves. Corrected reprint of the 1986 original. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1992. MR 1329092 (95m:11054)
- J. Tate, Residues of differentials on curves. Ann. Sci. Éc. Norm. Sup., IV. Sér. 1, No.1, 1968, 149-159. MR 0227171 (37:2756)
- B. Totaro, Milnor $K$-theory is the simplest part of algebraic $K$-theory. $K$-Theory 6, No. 2, 1992, 177-189. MR 1187705 (94d:19009)
- V. Voevodsky, A. Suslin, E. Friedlander, Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies. 143. Princeton, NJ: Princeton University Press, 2000. MR 1764197 (2001d:14026)
- E. Witt, Zyklische Körper und Algebren der Charakteristik $p$ vom Grad $p^n$. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik $p$. J. Reine Angew. Math. 176, 1936, 126-140.
Additional Information
Kay Rülling
Affiliation:
Universität Duisburg-Essen, Essen, FB6, Mathematik, 45117 Essen, Germany
Email:
kay.ruelling@uni-essen.de
Received by editor(s):
May 23, 2005
Published electronically:
July 24, 2006
Additional Notes:
The author was supported by the DFG Graduiertenkolleg 647.
