arXiv subjects

Kuntal Chakraborty

Publications and source records attributed to Kuntal Chakraborty.

4 recordsLinked to original sources

arXiv2026

Improved injective stability for relative $\mathrm{K_1Sp}$-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 $R/I$ 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. As an application, we give a necessary and sufficient condition for the freeness of stably free modules over smooth real $4$-folds with empty real locus.

math.KT↗

arXiv2023

Results on $\mathrm{K}_1$ of general quadratic groups

In the first part of this article we discuss the relative cases of Quillen-Suslin's local-global principle for the general quadratic (Bak's unitary) groups, and its applications for the (relative) stable and unstable $\mathrm{K}_1$-groups. The second part is dedicated to the graded version of the local-global principle for the general quadratic groups and its application to deduce a result for Bass' nil groups.

math.KT↗

arXiv2021

A note on relative Vaserstein symbol

In an unpublished work of Fasel-Rao-Swan the notion of the relative Witt group $W_E(R,I)$ is defined. In this article we will give the details of this construction. Then we studied the injectivity of the relative Vaserstein symbol $V_{R,I}: Um_3(R,I)/E_3(R,I)\rightarrow W_E(R,I)$. We established injectivity of this symbol if $R$ is an affine non-singular algebra of dimension $3$ over a perfect $C_1$-field and $I$ is a local complete intersection ideal of $R$. It is believed that for a $3$-dimensional affine algebra non-singularity is not necessary for establishing injectivity of the Vaserstein symbol . At the end of the article we will give an example of a singular $3$-dimensional algebra over a perfect $C_1$-field for which the Vaserstein symbol is injective.

math.KT↗

arXiv2021

NK_1 of Bak's unitary group over Graded Rings

For an associative ring $R$ with identity, we study the absence of $k$-torsion in NK_1GQ(R); Bass nil-groups for the general quadratic or Bak's unitary groups. By using a graded version of Quillen--Suslin theory we deduce an analog for the graded rings.

math.KT↗