Model-theoretic $K_1$ of free modules over PIDs

TL;DR

This paper introduces a new approach to computing the algebraic K-theory group $K_1$ for free modules over PIDs, providing explicit calculations for various Euclidean domains and linking it to classical algebraic K-theory.

Contribution

It defines $K$-groups $K_n(M)$ for structures using Quillen's construction and computes $K_1(M_R)$ for free modules over PIDs, expanding understanding of algebraic K-theory in this context.

Findings

Explicit computation of $K_1(M_R)$ for Euclidean domains.

Embedding of algebraic $K_1$ of a PID into $K_1(R_R)$.

Provides a computational recipe based on $GL_n(R)$ abelianizations.

Abstract

Motivated by Kraji\v{c}ek 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 $n\geq0$ 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 $K_1(R_R)$.