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$K_1$-Stability of symplectic modules over monoid algebras (2504.19492v1)
Published 28 Apr 2025 in math.AC, math.KT, and math.RA
Abstract: Let $R$ be a regular ring of dimension $d$ and $L$ be a $c$-divisible monoid. If ${K}1{Sp}(R)$ is trivial and $k \geq d+2,$ then we prove that the symplectic group ${Sp}{2k}(R[L])$ is generated by elementary symplectic matrices over $R[L]$. When $d \leq 1$ or $R$ is a geometrically regular ring containing a field, then improved bounds have been established. We also discuss the linear case, extending the work of Gubeladze.