Submission date: 18 April 2007. Revised version: 21 January 2008.

Abstract:

Denef and Loeser defined 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.

Mathematics Subject Classification: 03C10, 03C60, 03C98, 12L12, 12E30, 12F10, 14G15, 14G27

Keywords and phrases: pseudo-finite fields, pseudo algebraically closed fields, motives

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