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Husney Parvez Sarwar

Publications and source records attributed to Husney Parvez Sarwar.

12 recordsLinked to original sources
arXiv2026

Cancellation and splitting of Symplectic modules in the critical range and Euler class group

In this paper, we discuss the cancellation and splitting of the symplectic modules. The symplectic cancellation result presented here can be thought of as an analog of the Projective module cancellation result of Fasel. The symplectic splitting is similar to Murthy's splitting theorem. To prove the cancellation and splitting, we carefully analyze the Postnikov towers in the $\mathbb{A}^1$-homotopy category. Then we prove the vanishing of top cohomology with coefficients in some homotopy sheaf. As another application of the vanishing results, we answer partially a question of Mrinal Das about the isomorphism of $(d-1)$-th Euler class group and $(d-1)$-th Chow group, where $d$ is the dimension of the underlying smooth affine variety.

math.AG↗
arXiv2025

Subintegrality and ideal class groups of monoid algebras

$(1)$ Let $M\subset N$ be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension $A[M]\subset A[N]$, the two notions, subintegral and weakly subintegral coincide provided $\mathbb{Z}\subset A$. $(2)$ Let $A \subset B$ be an extension of commutative rings and $M\subset N$ an extension of commutative cancellative torsion-free positive monoids. Let $I$ be a radical ideal in $N$. Then $\frac{A[M]}{(I\cap M)A[M]}$ is subintegrally closed in $\frac{B[N]}{IB[N]}$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $\frac{A[M]}{(I\cap M)A[M]}$-submodules of $\frac{B[N]}{IB[N]}$.

math.AC↗
arXiv2025

Projective modules over Rees-like algebras and its monoid extensions

Let $A$ be a Rees-like algebra of dimension $d$ and $N$ a commutative partially cancellative torsion-free seminormal monoid. We prove the following results. \begin{enumerate} \item Let $P$ be a finitely generated projective $A$-module of $\rank\geq d$. Then $(i)$ $P$ has a unimodular element; $(ii)$ The action of $\EL(A\oplus P)$ on $\Um(A\oplus P)$ is transitive. \item Let $P$ be a finitely generated projective $A[N]$-module of $\rank~r$. Then $(i)$ $P$ has a unimodular element for $r\geq\max\{3,d\}$; $(ii)$ The action of $\EL(A[N]\oplus P)$ on $\Um(A[N]\oplus P)$ is transitive for $r\geq\max\{2,d\}$. \end{enumerate} These improve the classical results of Serre \cite{Se58} and Bass \cite{Ba64}.

math.AC↗
arXiv2023

Efficient generation, unimodular element in a geometric subring of a polynomial ring

Let $R$ be a commutative Noetherian ring of dimension $d$. First, we define the "geometric subring" $A$ of a polynomial ring $R[T]$ of dimension $d+1$ (the definition of geometric subring is more general, see (1.2)). Then we prove that every locally complete intersection ideal of height $d+1$ is a complete intersection ideal. Thus improving the general bound of Mohan Kumar \cite{NMK78} for an arbitrary ring of dimension $d+1$. Afterward, we deduce that every finitely generated projective $A$-module of rank $d+1$ splits off a free summand of rank one. This improves the general bound of Serre \cite{Serre58} for an arbitrary ring. Finally, applications are given to a set-theoretic generation of an ideal in the geometric ring $A$ and its polynomial extension $A[X]$.

math.AC↗
arXiv2021

The third homology of symplectic groups and algebraic K-theory

We improve the homology stability range for the 3rd integral homology of symplectic groups over commutative local rings with infinite residue field. As an application, we show that for local commutative rings containing an infinite field of characteristic not 2 the symbol map from Milnor-Witt K-theory to higher Grothendieck-Witt groups is an isomorphism in degrees 2 and 3.

math.KT↗
arXiv2019

Negative $K$-theory and Chow group of monoid algebras

We show, for a finitely generated partially cancellative torsion-free commutative monoid $M$, that $K_i(R) \cong K_i(R[M])$ whenever $i \le -d$ and $R$ is a quasi-excellent $\Q$-algebra of Krull dimension $d \ge 1$. In particular, $K_i(R[M]) = 0$ for $i < -d$. This is a generalization of Weibel's $K$-dimension conjecture to monoid algebras. We show that this generalization fails for $X[M]$ if $X$ is not an affine scheme. We also show that the Levine-Weibel Chow group of 0-cycles $\CH^{LW}_0(k[M])$ vanishes for any finitely generated commutative partially cancellative monoid $M$ if $k$ is an algebraically closed field.

math.AG↗
arXiv2018

Stability results for projective modules over Rees algebras

We provide a class of commutative Noetherian domains $R$ of dimension $d$ such that every finitely generated projective $R$-module $P$ of rank $d$ splits off a free summand of rank one. On this class, we also show that $P$ is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.

math.AC↗
arXiv2017

$K$-theory of monoid algebras and a question of Gubeladze

We show that for any commutative noetherian regular ring $R$ containing $\Q$, the map $K_1(R) \to K_1(\frac{R[x_1, \cdots , x_4]}{(x_1x_2 - x_3x_4)})$ is an isomorphism. This answers a question of Gubeladze. We also compute the higher $K$-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher $K$-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.

math.AG↗
arXiv2016

Serre Dimension of Monoid Algebras

Let $R$ be a commutative Noetherian ring of dimension $d$, $M$ a commutative cancellative torsion-free monoid of rank $r$ and $P$ a finitely generated projective $R[M]$-module of rank $t$. $(1)$ Assume $M$ is $Φ$-simplicial seminormal. $(i)$ If $M\in \CC(Φ)$, then {\it Serre dim} $R[M]\leq d$. $(ii)$ If $r\leq 3$, then {\it Serre dim} $R[int(M)]\leq d$. $(2)$ If $M\subset \BZ_+^2$ is a normal monoid of rank $2$, then {\it Serre dim} $R[M]\leq d$. $(3)$ Assume $M$ is $c$-divisible, $d=1$ and $t\geq 3$. Then $P\cong \wedge^t P\op R[M]^{t-1}$. $(4)$ Assume $R$ is a uni-branched affine algebra over an algebraically closed field and $d=1$. Then $P\cong \wedge^t P\op R[M]^{t-1}$.

math.AC↗
arXiv2015

Ideal class groups of monoid algebras

Let $A\subset B$ be an extension of commutative reduced rings and $M\subset N$ an extension of positive commutative cancellative torsion-free monoids. We prove that $A$ is subintegrally closed in $B$ and $M$ is subintegrally closed in $N$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $A[M]$-submodules of $B[N]$. In case $M=N$, we prove the same without the assumption that the ring extension is reduced.

math.AC↗
arXiv2014

Serre dimension and Euler class group of overrings of polynomial rings

Let R be a commutative Noetherian ring of dimension d and B=R[X_1,\ldots,X_m,Y_1^{\pm 1},\ldots,Y_n^{\pm 1}] a Laurent polynomial ring over R. If A=B[Y,f^{-1}] for some f\in R[Y], then we prove the following results: (i) If f is a monic polynomial, then Serre dimension of A is \leq d. In case n=0, this result is due to Bhatwadekar, without the condition that f is a monic polynomial. (ii) The p-th Euler class group E^p(A) of A, defined by Bhatwadekar and Raja Sridharan, is trivial for p\geq max \{d+1, \dim A -p+3\}. In case m=n=0, this result is due to Mandal-Parker.

math.AC↗