Improved injective stability for relative $\mathrm{K_1Sp}$-groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918438908198912 |
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| author | Banerjee, Sourjya Chakraborty, Kuntal |
| author_facet | Banerjee, Sourjya Chakraborty, Kuntal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09476 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improved injective stability for relative $\mathrm{K_1Sp}$-groups Banerjee, Sourjya Chakraborty, Kuntal K-Theory and Homology Commutative Algebra 19B14, 19C20, 19G12 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. |
| title | Improved injective stability for relative $\mathrm{K_1Sp}$-groups |
| topic | K-Theory and Homology Commutative Algebra 19B14, 19C20, 19G12 |
| url | https://arxiv.org/abs/2604.09476 |