2024/07/18 by Sourayan Banerjee, Banerjee, Sourayan, Amit Kuber +1 · 1 citation
Computer Science · #03C07 #03C60 #19B14 #19B99 #19D23 #FOS: Mathematics #K-Theory and Homology (math.KT) #Logic (math.LO) #Optimization and Search Problems
paper · pdf · doi:10.48550/arxiv.2407.13624
openalex publication_date 2024/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by Krajiček 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).