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Mrinal Kanti Das

Publications and source records attributed to Mrinal Kanti Das.

9 recordsLinked to original sources
arXiv2025

Splitting criteria for projective modules over polynomial algebras

This article investigates the splitting problem for finitely generated projective modules $P$ over affine algebras over algebraically closed fields and their polynomial extensions. We then address an open question due to M. Roitman on monic inversion principle for projective modules and prove it in the affirmative for finitely generated rings. For affine algebras over $\overline{\mathbb{F}}_p$, we prove a monic inversion principle for ideals. We also exhibit some applications.

math.AC↗
arXiv2023

On a question of Nori: obstructions, improvements, and applications

This article concerns a question asked by M. V. Nori on homotopy of sections of Projective modules defined on the polynomial algebra over a smooth affine domain $R$. While this question has an affirmative answer, it is known that the assertion does not hold if: (1) $\dim(R)=2$; or (2) $d\geq 3$ but $R$ is not smooth. We first prove that an affirmative answer can be given for $\dim(R)=2$ when $R$ is an $\bar{\mathbb{F}}_p$-algebra. Next, for $d\geq 3$ we find the precise obstruction for the failure in the singular case. Further, we improve a result of Mandal (related to Nori's question) in the case when the ring $A$ is an affine $\bar{\mathbb{F}}_p$-algebra of dimension $d$. We apply this improvement to define the $n$-th Euler class group $E^n(A)$, where $2n\ge d+2.$ Moreover, if $A$ is smooth, we associate to a unimodular row $v$ of length $n+1$ its Euler class $e(v)\in E^n(A)$ and show that the corresponding stably free module, say, $P(v)$ has a unimodular element if and only if $e(v)$ vanishes in $E^n(A)$.

math.AC↗
arXiv2021

Euler class groups and motivic stable cohomotopy

We study maps from a smooth scheme to a motivic sphere in the Morel-Voevodsky ${\mathbb A}^1$-homotopy category, i.e., motivic cohomotopy sets. Following Borsuk, we show that, in the presence of suitable hypotheses on the dimension of the source, motivic cohomotopy sets can be equipped with functorial abelian group structures. We then explore links between motivic cohomotopy groups, Euler class groups à la Nori-Bhatwadekar-Sridharan and Chow-Witt groups. We show that, again under suitable hypotheses on the base field $k$, if $X$ is a smooth affine $k$-variety of dimension $d$, then the Euler class group of codimension $d$ cycles coincides with the codimension $d$ Chow-Witt group; the identification proceeds by comparing both groups with a suitable motivic cohomotopy group. As a byproduct, we describe the Chow group of zero cycles on a smooth affine $k$-scheme as the quotient of the free abelian group on zero cycles by the subgroup generated by reduced complete intersection ideals; this answers a question of S. Bhatwadekar and R. Sridharan.

math.AG↗
arXiv2019

${\mathbb P}^1$-gluing for local complete intersections

We prove an analogue of the Affine Horrocks' Theorem for local complete intersection ideals of height $n$ in $R[T]$, where $R$ is a regular domain of dimension $d$, which is essentially of finite type over an infinite perfect field of characteristic unequal to $2$, and $2n\geq d+3$.

math.AC↗
arXiv2018

Euler cycles and Mennicke symbols

Let $R$ be a smooth affine domain of dimension $d\geq 2$ over an infinite perfect field $k$. We establish a morphism from the Euler class group $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$, the group of elementary orbits of unimodular rows.

math.AC↗
arXiv2017

On two conjectures of Murthy

This article concerns two conjectures of M. P. Murthy. For Murthy's conjecture on complete intersections, the major breakthrough has still been the result proved by Mohan Kumar in 1978. In this article we improve "Mohan Kumar's bound" when the base field is $\overline{\mathbb F}_p$, and illustrate some applications of our result. Murthy's other conjecture is on a "splitting problem", which is roughly about finding the precise obstruction for a projective $R$-module $P$ of rank $\text{dim}(R)-1$ to split off a free summand of rank one, where $R$ is a smooth affine algebra over an algebraically closed field $k$. Asok-Fasel achieved the initial breakthrough, by settling it for $3$-folds and $4$-folds when $char(k)\neq 2$. For $k=\overline{\mathbb F}_p$ ($p\neq 2$) and $\text{dim}(R)\geq 5$ we define an obstruction group and an obstruction class for $P$ (whose determinant is trivial). As application we obtain: $P$ splits if and only if it maps onto a complete intersection ideal of height $\text{dim}(R)-1$.

math.AC↗
arXiv2017

Projective modules over affine threefolds: a simpler case

Let $p\neq 2$, and let $R$ be a smooth affine algebra of dimension $3$ over $\overline{F}_p$ and $P, Q$ be projective $R$-modules of rank $2$, each with trivial determinant. We prove: $P$ is isomorphic to $Q$ if and only if there is an ideal $J\subset R$ of height $2$ such that both $P$ and $Q$ map onto $J$.

math.AC↗
arXiv2017

From Euler class groups to Mennicke symbols and a monic inversion principle

Let $R$ be a regular domain of dimension $d\geq 2$ which is essentially of finite type over an infinite perfect field $k$. We compare the Euler class group $E^d(R)$ with the van der Kallen group $Um_{d+1}(R)/E_{d+1}(R)$. In the case $2R=R$, we define a map from $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$ and study it in intricate details. As application, this map enables us to carry out some interesting computations on real varieties, using some very basic arguments. The formalism required to carry out the above investigation also provides us a requisite tool to show that the monic inversion principle holds for the Euler class groups.

math.AC↗