Submission date: 18 July 2024

Abstract:

Motivated by Krajíček 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 GL_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).

Mathematics Subject Classification: 03C60, 19B99, 03C07, 19D23, 19B14

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Full text arXiv 2407.13624: pdf, ps.