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arXiv · 2604.09476

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

Sourjya Banerjee·Kuntal Chakraborty
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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.

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Sourjya Banerjee, Kuntal Chakraborty. 2026-04-10. Improved injective stability for relative $\mathrm{K_1Sp}$-groups. https://arxiv.org/abs/2604.09476

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