Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-10-01T13:23:05.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Motives for perfect PAC fields with pro-cyclic Galois group

Published online by Cambridge University Press:  12 March 2014

Immanuel Halupczok
Immanuel Halupczok*
Affiliation:
Ecole Normale Supérieure, DMA, 45, Rue D'ulm, 75230 Paris Cedex 05, France, E-mail: immi@karimmi.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Denef and Loeser denned a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group.

In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

Keywords

Pseudo-finite fieldspseudo algebraically closed fieldsmotives
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 1036 - 1050
Copyright
Copyright © Association for Symbolic Logic 2008

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.)

References

REFERENCES

[1]Chatzidakis, Zoé and Hrushovski, Ehud, Perfect pseudo-algebraically closed fields are algebraically bounded, Journal of Algebra, vol. 271 (2004), no. 2, pp. 627–637.CrossRefGoogle Scholar
[2]Chatzidakis, Zoé, van den Dries, Lou, and Macintyre, Angus, Definable sets over finite field's, Journal für die Reine und Angewandte Mathematik, vol. 427 (1992), pp. 107–135.Google Scholar
[3]Rollin, Sebastian del Baño and Aznar, Vicente Navarro, On the motive of a quotient variety, Collectanea Mathematica, vol. 49 (1998), no. 2–3, pp. 203–226, dedicated to the memory of Fernando Serrano.Google Scholar
[4]Denef, Jan and Loeser, François, Definable sets, motives and p-adic integrals, Journal of the American Mathematical Society, vol. 14 (2001), no. 2, pp. 429–469, (electronic).CrossRefGoogle Scholar
[5]Denef, Jan and Loeser, François, On some rational generating series occurring in arithmetic geometry, Geometric Aspects of Dwork Theory, Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 509–526.CrossRefGoogle Scholar
[6]Fried, Michael D. and Moshe, Jarden, Field Arithmetic, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. [Results in Mathematics and Related Areas. 3rd Series.] A Series of Modern Surveys in Mathematics, vol. 11, Springer-Verlag, Berlin, 2005.CrossRefGoogle Scholar
[7]Halupczok, Immanuel, A measure for perfect pac fields with pro-cyclic Galois group, Journal of Algebra, (2007), no. 310, pp. 371–395.CrossRefGoogle Scholar
[8]Hrushovski, Ehud, Pseudo-finite fields and related structures, Model Theory and Applications, Quaderni di Matematica, vol. 11, Aracne, Rome, 2002, pp. 151–212.Google Scholar
[9]Nicaise, Johannes, Relative Motives and the Theory of Pseudo-finite Fields, International Mathematics Research Papers, vol. 2007, doi: 10.1093/imrp/rpm001.Google Scholar