Model-theoretic $K_1$ for modules over semisimple rings: (weak) Morita invariance
Sourayan Banerjee, Amit Kuber
TL;DR
The authors develop and apply model-theoretic K-theory for modules, using Quillen's $S^{-1}S$ on definable-bijection groupoids to define $K_n(M)$. They compute the model-theoretic $K_1$ for modules over division rings and matrix rings, obtaining explicit decompositions involving $GL_n(R)^{ab}$ and Dieudonné's determinants, and show a weak Morita invariance $K_1(R_R)\cong K_1((M_q(R))_{M_q(R)})$ for division rings $R$ and $q\ge1$ (provided $|M_q(R)|\neq 2$). The paper then extends the framework to semisimple rings, proving that $K_1$ commutes with finite products of modules and that the algebraic $K_1$ of a finite product of infinite matrix rings embeds into the model-theoretic $K_1$ of their right regular modules, linking model-theoretic and algebraic K-theory in this setting. Overall, the results clarify Morita-type invariance phenomena in model-theoretic $K$-theory and provide exact computations for key classes of modules. The approach relies on elimination of pp-formulas/imaginaries for von Neumann regular rings and Dieudonné's determinant results to describe $K_1$ explicitly.
Abstract
This paper is a sequel to a paper by the same authors, where they defined $K$-groups of model-theoretic structures, and computed $K_1$ of free modules over PIDs. In this paper, we compute $K_1$ of a right $M_q(R)$-module $M$, where $R$ is a division ring, $q\geq1$, and $|M_q(R)|\neq 2$. As a consequence, we obtain a (weak) Morita invariance $K_1(R_R)\cong K_1((M_q(R))_{M_q(R)})$ for all division rings $R$ and $q\geq 1$. Finally, we compute $K_1$ of a module over a semisimple ring by showing that the model-theoretic $K_1$ commutes with finite product of modules. We also show that the algebraic $K_1$ of a finite product of infinite matrix rings embeds into the model-theoretic $K_1$ of their right regular modules.