Improved injective stability for relative $\mathrm{K_1Sp}$-groups

TL;DR

This paper extends key theorems in algebraic K-theory to relative settings, improving stability bounds for relative linear and symplectic K_1 groups over smooth affine algebras.

Contribution

It introduces a relative version of Vorst's theorem and the symplectic elementary Witt group, enhancing stability bounds in algebraic K-theory.

Findings

Proved a relative version of Vorst's theorem for invertible and elementary matrices.

Introduced a relative symplectic elementary Witt group fitting into a periodicity sequence.

Improved injective stability bounds for relative K_1 groups over various fields.

Abstract

We prove a relative version of Vorst's theorem concerning the equality of the group of all invertible matrices and the group of all elementary matrices over $R[X]$ with respect to an ideal $I\subset R$ such that $R/I$ is regular, where $R$ is a regular $k$-spot. We then introduce a relative version of the symplectic elementary Witt group and show that it fits into a relative version of the Karoubi periodicity sequence. Combining these results, we improve the existing injective stability bounds for relative linear and symplectic $\mathrm{K_1}$-groups of smooth affine algebras over various base fields.