Model-theoretic K1 of free modules over PIDs
Abstract
Motivated by Krajicek and Scanlon's definition of the Grothendieck ring K0(M) of a first-order structure M, we introduce the definition of K-groups Kn(M) for n≥0 via Quillen's S-1S construction. We provide a recipe for the computation of K1(MR), where MR is a free module over a PID R, subject to the knowledge of the abelianizations of the general linear groups GLn(R). As a consequence, we provide explicit computations of K1(MR) when R belongs to a large class of Euclidean domains that includes fields with at least 3 elements and polynomial rings over fields with characteristic 0. We also show that the algebraic K1 of a PID R embeds into K1(RR).
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