$K_1$-Stability of symplectic modules over monoid algebras

TL;DR

This work advances ${\rm K}_1{\rm Sp}$-stability for symplectic groups over monoid algebras by combining the geometry of polarized monoids with localization and patching techniques. It proves that for a regular ring $R$ of dimension $d$ with ${\rm K}_1{\rm Sp}(R)=0$ and a $c$-divisible monoid $L$, the unstable-to-stable map collapses for $k \ge d+2$, yielding ${\rm Sp}_{2k}(R[L])={\rm ESp}_{2k}(R[L])$, and shows ${\rm K}_1{\rm Sp}(R[L]) \simeq {\rm K}_1{\rm Sp}(R)$ (K1-sp invariance). The paper also establishes equalities between transvections and isometries, extends the stability framework to broader ring classes ${\mathcal R}'_n$, and extends unstable ${\rm K}_1$ phenomena from the linear to the symplectic setting, with improved bounds in low-dimensional or field-containing geometrically regular cases. These results extend stability phenomena to monoid algebras and illuminate excision-like behavior in ${\rm K}$-theory for such algebras.

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.

Theorems & Definitions (53)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 2.1
• Definition 2.2
• Example 3.1
• Theorem 3.2
• Corollary 3.3
• Example 3.4
• Example 3.5