Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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
Full-text PDF

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 CH$ ^n((\mathbb{A}^1_k,(m+1)\{0\}),n-1)$, $ n,m\ge1$. Bloch and Esnault prove CH$ ^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 [Enhancements On Off] (What's this?)

  • [AnRo04] G. W. Anderson, F. P. Romo, Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an Artinian local ring. Commun. Algebra 32, No.1, 2004, 79-102. MR 2036223 (2005d:11099)
  • [BaTa73] H. Bass, J. Tate, The Milnor ring of a global field. Algebraic $ K$-theory, II: ``Classical" algebraic $ K$-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), pp. 349-446. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973. MR 0442061 (56:449)
  • [Be66] G.M. Bergman, Ring Schemes; the Witt Scheme. Lecture 26, in Lectures on Curves on an Algebraic Surface, Mumford, Ann. Math. Studies 59, Princeton University Press, 1966. MR 0209285 (35:187)
  • [Bl78] S. Bloch, Algebraic $ K$-Theory and Crystalline Cohomology. Pub. Math. I.H.E.S. 47, 1978, 187-268. MR 0488288 (81j:14011)
  • [Bl86] S. Bloch, Algebraic cycles and higher $ K$-theory. Adv. in Math. 61, No. 3, 1986, 267-304. MR 0852815 (88f:18010)
  • [BlEs03a] S. Bloch, H. Esnault, An additive version of higher Chow groups. Ann. Sci. École Norm. Sup. (4) 36, no. 3, 2003, 463-477. MR 1977826 (2004c:14035)
  • [BlEs03b] S. Bloch, E. Esnault, The additive dilogarithm. Doc. Math., J. Doc. Math. Extra Vol., 2003, 131-155. MR 2046597 (2005e:19006)
  • [Bo89] N. Bourbaki, Elements of mathematics. Commutative algebra. Chapters 1-7. Transl. from the French. 2nd printing. Berlin etc.: Springer-Verlag. x, 1989. MR 0979760 (90a:13001)
  • [Bo90] N. Bourbaki, Elements of mathematics. Algebra II. Chapters 4-7. Transl. from the French. Berlin etc.: Springer-Verlag., 1990. MR 1080964 (91h:00003)
  • [EGA IV] A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas IV. Pub. Math. I.H.E.S. 32, 1967. MR 0238860 (39:220)
  • [Fu84] W. Fulton, Intersection Theory. Springer-Verlag, Berlin, 1984. MR 0732620 (85k:14004)
  • [GeHe] T. Geisser, L. Hesselholt, On the $ K$-theory of complete regular local $ \mathbb{F}_p$-algebras, Topology, to appear.
  • [Ha77] R. Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [He04a] L. Hesselholt, Topological Hochschild homology and the de Rham-Witt complex for $ \mathbb{Z}_{(p)}$-algebras. Homotopy theory: Relations with algebraic geometry, group cohomology, and algebraic K-theory (Evanston, IL, 2002), Contemp. Math. 346, Amer. Math. Soc., Providence, RI, 2004, 253-259. MR 2066502 (2005c:19006)
  • [He04b] L. Hesselholt, The absolute and relative de Rham-Witt complexes. Compositio Math. 141 (2005), no. 5, 1109-1127. MR 2157132
  • [HeMa01] L. Hesselholt, I. Madsen, On the K-theory of nilpotent endomorphisms. Greenlees, J. P. C. (ed.) et al., Homotopy methods in algebraic topology. Proceedings of an AMS-IMS-SIAM joint summer research conference, University of Colorado, Boulder, CO, USA, June 20-24, 1999. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 271, 2001, 127-140. MR 1831350 (2002b:19003)
  • [HeMa04] L. Hesselholt, I. Madsen, On the de Rham-Witt complex in mixed characteristic. Ann. Scient. Éc. Norm. Sup., IV Ser. 37, 2004, 1-43. MR 2050204 (2005f:19005)
  • [Hu89] R. Hübl, Traces of differential forms and Hochschild homology. Lecture Notes in Mathematics 1368, 1989. MR 0995670 (92a:13010)
  • [Il79] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline. Ann. Scient. Éc. Norm. Sup., IV. Ser. 12, 1979, 501-661. MR 0565469 (82d:14013)
  • [Ka80] K. Kato, A generalization of local class field theory by using $ K$-groups. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 , No. 3, 1980, 603-683. MR 0603953 (83g:12020a)
  • [Ku86] E. Kunz, Kähler differentials. Advanced Lectures in Mathematics. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. VII, 1986. MR 0864975 (88e:14025)
  • [Ku64] E. Kunz, Arithmetische Anwendungen der Differentialalgebren. J. Reine Angew. Math. 214/215, 1964, 276-320. MR 0164957 (29:2248)
  • [LaZi04] 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)
  • [NeSu89] Y. P. Nesterenko, A. A. Suslin, Homology of the general linear group over a local ring, and Milnor's $ K$-theory. English translation. Math. USSR-Izv. 34, No. 1, 1990, 121-145. MR 0992981 (90a:20092)
  • [Ro04] F. P. Romo, A generalization of the Contou-Carrère symbol. Isr. J. Math. 141, 2004, 39-60. MR 2063024 (2005g:11115)
  • [Se68] J.-P. Serre, Corps locaux. Deuxième édition. Hermann, Paris, 1968. MR 0354618 (50:7096)
  • [Se88] 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)
  • [Si92] 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)
  • [Ta68] J. Tate, Residues of differentials on curves. Ann. Sci. Éc. Norm. Sup., IV. Sér. 1, No.1, 1968, 149-159. MR 0227171 (37:2756)
  • [To92] 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)
  • [VoSuFr00] 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)
  • [Wi36] 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

DOI: https://doi.org/10.1090/S1056-3911-06-00446-2
Received by editor(s): May 23, 2005
Published electronically: July 24, 2006
Additional Notes: The author was supported by the DFG Graduiertenkolleg 647.

American Mathematical Society