arXiv subjects

Md. Ali Zinna

Publications and source records attributed to Md. Ali Zinna.

7 recordsLinked to original sources

arXiv2025

On set-theoretic complete intersections for smooth curves in three-dimensional affine schemes

We prove that every local complete intersection curve in $Spec(A)$, where $A$ is a commutative Noetherian ring of dimension three, is a set-theoretic complete intersection. An analogous result is established for local complete intersection surfaces when $A$ is a four-dimensional affine algebra over the algebraic closure of a finite field of $p$ elements. Furthermore, we show that any local complete intersection curve (respectively, surface) in $Spec(A)$, where $A$ has dimension three (respectively, four), having trivial conormal bundle is, in fact, a complete intersection.

math.AC↗

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

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↗

arXiv2016

Efficient generation of ideals in a discrete Hodge algebra

Let $R$ be a commutative Noetherian ring and $D$ be a discrete Hodge algebra over $R$ of dimension $d>\text{dim}(R)$. Then we show that (i) the top Euler class group $E^d(D)$ of $D$ is trivial. (ii) if $d>\text{dim}(R)+1$, then $(d-1)$-st Euler class group $E^{d-1}(D)$ of $D$ is trivial.

math.AC↗

arXiv2016

Unimodular Elements in Projective Modules and an analogue of a result of Mandal

Let $R$ be a Noetherian commutative ring of dimension $n$, $A=R[X_1,\cdots,X_m]$ be a polynomial ring over $R$ and $P$ be a projective $A[T]$-module of rank $n$. Assume that $P/TP$ and $P_f$ both contain a unimodular element for some monic polynomial $f(T)\in A[T]$. Then $P$ has a unimodular element.

math.AC↗

arXiv2016

Existence of unimodular elements in a projective module

Let $R$ be an affine algebra over an algebraically closed field of characteristic $0$ with dim$(R)=n$. Let $P$ be a projective $A=R[T_1,\cdots,T_k]$-module of rank $n$ with determinant $L$. Suppose $I$ is an ideal of $A$ of height $n$ such that there are two surjections $α:P\to\!\!\!\to I$ and $ϕ:L\oplus A^{n-1} \to\!\!\!\to I$. Assume that either (a) $k=1$ and $n\geq 3$ or (b) $k$ is arbitrary but $n\geq 4$ is even. Then $P$ has a unimodular element.

math.AC↗