Hostname: page-component-5db58dd55d-xnzfm Total loading time: 0 Render date: 2026-06-07T04:19:36.679Z Has data issue: false hasContentIssue false
  • English
  • Français

An additive variant of Somekawa's K-groups and Kähler differentials

Published online by Cambridge University Press:  20 March 2014

Toshiro Hiranouchi
Toshiro Hiranouchi*
Affiliation:
Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan, hira@hiroshima–u.ac.jp
Get access

Abstract

We introduce a Milnor type K-group associated to commutative algebraic groups over a perfect field. It is an additive variant of Somekawa's K-group. We show that the K-group associated to the additive group and q multiplicative groups of a field is isomorphic to the space of absolute Kähler differentials of degree q of the field, thus giving us a geometric interpretation of the space of absolute Kähler differentials. We also show that the K-group associated to the additive group and Jacobian variety of a curve is isomorphic to the homology group of a certain complex.

Keywords

Milnor K-theoryChow groupsKähler differentials19D4514C15

Information

Type
Research Article
Information
Journal of K-Theory , Volume 13 , Issue 3 , June 2014 , pp. 481 - 516
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

1.Akhtar, R., Milnor K-theory of smooth varieties, K-Theory 32(3) (2004), 269–291.CrossRefGoogle Scholar
2.Bass, H. and Tate, J., 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), Springer, Berlin, 1973, pp. 349–446. Lecture Notes in Math. 342.Google Scholar
3.Bloch, S., Algebraic K-theory and classfield theory for arithmetic surfaces, Ann. of Math. 114 (1981), 229–266.CrossRefGoogle Scholar
4.Bloch, S. and Esnault, H., The additive dilogarithm, Doc. Math. (2003), Kazuya Kato's fiftieth birthday.Google Scholar
5.Bloch, S. and Esnault, H., An additive version of higher Chow groups, Ann. Sci. École Norm. Sup. (4) 36(3) (2003), 463–477.CrossRefGoogle Scholar
6.Bloch, S. and Kato, K., p-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. (1986), 107–152.Google Scholar
7.Fulton, W., Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete 3. A Series of Modern Surveys in Mathematics 2, Springer-Verlag, Berlin, 1998.Google Scholar
8.Hesselholt, L. and Madsen, I., On the K-theory of nilpotent endomorphisms, Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 127–140.Google Scholar
9.Hesselholt, L. and Madsen, I., On the De Rham-Witt complex in mixed characteristic, Ann. Sci. École Norm. Sup. (4) 37(1) (2004), 1–13.CrossRefGoogle Scholar
10.Hiranouchi, T., Finiteness of certain products of algebraic groups over a finite field, arXiv:1209.4457.Google Scholar
11.Illusie, L., Complexe de de Rham-Witt, Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. I, Astérisque 63, Soc. Math. France, Paris, 1979, pp. 83–112.Google Scholar
12.Ivorra, F. and Rülling, K., K-groups of reciprocity functors, arXiv:1209.1207.Google Scholar
13.Kahn, B., Nullité de certains groupes attachés aux variétés semi-abéliennes sur un corps fini; application, C. R. Acad. Sci. Paris Sér. I Math. 314(13) (1992), 1039–1042.Google Scholar
14.Kunz, E., Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986.CrossRefGoogle Scholar
15.Raskind, W. and Spiess, M., Milnor K-groups and zero-cycles on products of curves over p-adic fields, Compositio Math. 121 (2000), 1–33.CrossRefGoogle Scholar
16.Rülling, K., The generalized de Rham-Witt complex over afield is a complex of zero-cycles, J. Algebraic Geom. 16(1) (2007), 109–169.CrossRefGoogle Scholar
17.Saito, S., Class field theory for curves over local fields, J. Number Theory 21(1) (1985), 44–80.CrossRefGoogle Scholar
18.Serre, J.-P., Groupes algébriques et corps de classes, second ed., Publications de l'Institut Mathématique de l'Université de Nancago 7, Hermann, Paris, 1984, Actualités Scientifiques et Industrielles 1264.Google Scholar
19.Somekawa, M., On Milnor K-groups attached to semi-abelian varieties, K-Theory 4(2) (1990), 105–119.CrossRefGoogle Scholar