# Chevalley groups of polynomial rings over Dedekind domains

## TL;DR

This paper proves a stability result for Chevalley groups over polynomial rings with coefficients in Dedekind domains, extending classical results to a broader class of algebraic groups and rings.

## Contribution

It generalizes known stability theorems for special linear and symplectic groups to all simply connected Chevalley-Demazure groups over Dedekind domains.

## Key findings

- G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for Dedekind domains R and n>=1
- Extension of classical results to higher-dimensional regular rings and discrete Hodge algebras
- Provides corollaries for regular rings of higher dimension

## Abstract

Let R be a Dedekind domain, and let G be a simply connected Chevalley-Demazure group scheme of rank =>2. We prove that G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for any n=>1. This extends the corresponding results of A. Suslin and F. Grunewald, J. Mennicke, and L. Vaserstein for G=SL_n, Sp_2n. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.