Proceedings of the Japan Academy, Series A, Mathematical Sciences

Remarks on zeta functions and K-theory over ${\mathbf F}_1$

Deitmar Anton

Full-text: Open access


We show that the notion of zeta functions over the field of one element $\F_1$, as given in special cases by Soulé, extends naturally to all $\F_1$-schemes as defined by the author in an earlier paper. We further give two constructions of K-theory for affine schemes or $\F_1$-rings, we show that these coincide in the group case, but not in general.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 82, Number 8 (2006), 141-146.

First available in Project Euclid: 6 November 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]
Secondary: 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27] 11S70: $K$-theory of local fields [See also 19Fxx] 14G15: Finite ground fields

Zeta function field of one element


Anton, Deitmar. Remarks on zeta functions and K-theory over ${\mathbf F}_1$. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 141--146. doi:10.3792/pjaa.82.141.

Export citation


  • A. Deitmar, Schemes over $\mathbf{F}_1$, in: Number fields and function fields –- Two parallel worlds (eds. G. van der Gee, B. J. J. Moonen, and R. Schoof), Prg. Math., vol.239, Birkhäuser, Boston, 2005.
  • A. Deitmar, Homological algebra over belian categories and cohomology of F1-schemes.
  • D. Grayson, Higher algebraic $K$-theory. II (after Daniel Quillen), in Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), pp.217–240. Lecture Notes in Math., 551, Springer, Berlin, 1976.
  • K., Kato, Toric singularities. Amer. J. Math. 116 (1994), no.5, 1073–1099.
  • B. Kurokawa, H. Ochiai and A. Wakayama, Absolute derivations and zeta functions. Doc. Math. 2003 (2003), Extra vol., 565–584.
  • N. Kurokawa, Zeta functions over $F\sb 1$. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no.10, 180–184 (2006).
  • Y. Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque No.228 (1995), 4, 121–163 (1995).
  • S. Priddy, On $\Omega \sp{\infty }S\sp{\infty }$ and the infinite symmetric group, in Algebraic topology (Proc. Sympos. Pure Math., Vol.XXII, Univ. Wisconsin, Madison, Wis., 1970), 217–220. Amer. Math. Soc., Providence, R.I., 1971.
  • D. Quillen, Higher algebraic $K$-theory. I, in Algebraic $K$-theory, I Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 85–147. Lecture Notes in Math., 341, Springer, Berlin, 1973.
  • C. Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (2004), no.1, 217–244, 312.
  • J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d'algèbre supérieure, Bruxelles du 19 au 22 décembre 1956, 261–289, Centre Belge de Recherches Aathématiques Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris.