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Grothendieck Rings of ℤ-Valued Fields

Published online by Cambridge University Press:  15 January 2014

Raf Cluckers  and
Deirdre Haskell
Raf Cluckers
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Heverlee, BelgiumE-mail:raf.cluckers@wis.kuleuven.ac.be
Deirdre Haskell
Affiliation:
Department of Mathematics and Statistics, Mcmaster University, 1280 Main St. West, Hamilton, Ontario, CanadaL8S 4K1, E-mail:haskell@math.mcmaster.ca
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Abstract

We prove the triviality of the Grothendieck ring of a ℤ-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K2 to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 2 , June 2001 , pp. 262 - 269
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1] Bélair, L., Types dans les corps valués munis d'applications coefficients, Illinois Journal of Mathematics, vol. 43 (1999), no. 2, pp. 410–425.CrossRefGoogle Scholar
[2] Denef, J., The diophantine problem for polynomial rings and fields of rational functions, Transactions of the American Mathematical Society, vol. 242 (1978), pp. 391–399.CrossRefGoogle Scholar
[3] Denef, J. and Loeser, F., Definable sets, motives and p-adic integrals, Journal of the American Mathematical Society, to appear, 45 pages.Google Scholar
[4] Denef, J. and Loeser, F., Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae, vol. 135 (1999), pp. 201–232.Google Scholar
[5] Kim, K. H. and Roush, F. W., Diophantine unsolvability over p-adic function fields, Journal of Algebra, vol. 176 (1995), no. 1, pp. 83–110.Google Scholar
[6] Krajíček, J., Uniform families of polynomial equations over a finite field and structures admitting an Euler characteristic of definable sets, Proc. LMS, vol. 81 (2000), pp. 257–284.Google Scholar
[7] Krajíček, J. and Scanlon, T., Combinatorics with definable sets: Euler characteristics and Grothendieck rings, this Bulletin, vol. 6 (2000), pp. 311–330.Google Scholar
[8] Pas, J., On the angular component map modulo p, The Journal of Symbolic Logic, vol. 55 (1990), pp. 1125–1129.Google Scholar
[9] Silvester, J. R., Introduction to algebraic K-theory, Mathematics Series, Chapman and Hall, 1981.Google Scholar