Grothendieck Rings of ℤ-Valued Fields

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 K1 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. At the Edinburgh meeting on the model theory of valued fields in May 1999, Luc Belair posed the question of whether there is a definable bijec? tion between the set of /?-adic integers and the set of /?-adic integers with one point removed. At the same meeting, Jan Denef asked what is the Grothendieck ring ofthe /?-adic numbers, as did Jan Krajicek independently in [K]. A general introduction to Grothendieck rings of logical structures was recently given in [KS] and in [DL2, par. 3.7]. Calculations of nontrivial Grothendieck rings and related topics such as motivic integration can be found in [DL] and [DL2]. The logical notion of the Grothendieck ring of a structure is analogous to that ofthe Grothendieck ring in the context of algebraic A^-theory and has analogous elementary properties (see [S]). Here we recall the definition. Definition 1. Let M be a structure and Vef(M) the set of definable subsets of Mn for every positive integer n. For any X, Y e Vef(M), write X = Y if and only if there is a definable bijection (an isomorphism) from X to Y. Let F be the free abelian group whose generators are isomorphism classes \X\ with X e Vef(M) (so [X\ = [Y\ ifand only if X ^ Y) and let E be the subgroup generated by all expressions [A"J + [ Y\ [X U Y\ [X n Y\ with X,Y e Vef[M). Then the Grothendieck group of M is the quotient group F/E. Write [X] for the image of X e Vef(M) in F/E. The Grothendieck group has a natural structure as a ring with multiplication induced by [X] ? [Y] = [X x Y] for X, Y e Vef[M). We call this ring the Grothendieck ring Kq(M) ofM. Received September 7, 2000; revised December 13, 2000. f Research supported by the Fund for Scientific Research, Flanders (F.W.O) Belgium. ft Research partially supported by the NSF. ? 2001, Association for Symbolic Logic 1079-8986/01 /0702-0005/S1.80