Model-theoretic K1 of free modules over PIDs

Motivated by Krajiček and Scanlon's definition of the Grothendieck ring K 0 ( M ) of a first-order structure M , we introduce the definition of K -groups K n ( M ) for n ≥ 0 via Quillen's S − 1 S construction. We provide a recipe for the computation of K 1 ( M R ) , where M R is a free module over a PID R , subject to the knowledge of the abelianizations of the general linear groups G L n ( R ) . As a consequence, we provide explicit computations of K 1 ( M R ) 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 K 1 of a PID R embeds into K 1 ( R R ) .