Improved injective stability for relative ${\mathrm{K}}_1\mathrm{S}\mathrm{p}$-groups

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 $R\mathrm{/}I$ 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.