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Manoj K. Keshari

Publications and source records attributed to Manoj K. Keshari.

13 recordsLinked to original sources
arXiv2026

Applications of Swan's Bertini to unimodular rows

Let $R$ be an affine algebra of dimension $d\geq 4$ over a perfect field $k$ of char $\neq 2$ and $I$ be an ideal of $R$. Then - Um$_{d+1}(R,I)/{\rm E}_{d+1}(R,I)$ has nice group structure if $c.d._2(k)\leq 2$. - Um$_d(R,I)/{\rm E}_d(R,I)$ has nice group structure if $k$ is algebraically closed of char $k\neq 2,3$ and either (i) $k = \overline{\mathbb{F}}_{p}$ or (ii) $R$ is normal. - $MS_{d+1}(R)$ is uniquely divisible prime to characteristic of $k$ if $R$ is reduced and $k$ is infinite with $c.d.(k)\leq 1$.

math.KT↗
arXiv2025

Descent problem for certificate of non-negativity on semi-algebraic sets

Let $F$ be a subfield of $\mathbb R$ and let $K$ be a basic closed semi-algebraic set in $\mathbb R$ with $\partial K\subset F$. Let $\mathcal N$ be the natural choice of generators of $K$. We show that if $f\in F[x]$ is $\geq 0$ on $K$, then $f$ can be written as $$f=\sum_{e\in\{0,1\}^s } a_eσ_e g^e $$ where $a_e\in F_{\geq 0}$, $σ_e\in \sum F[x]^2$ and $g^{e}=g_1^{e_1} \cdots g_s^{e_s}$. In other words, the preordering $T_{\mathcal N}$ of $F[x]$ is saturated. In case $F=\mathbb R$, this result is due to Kuhlmann and Marshall. As an application, we prove that if $K$ is compact, then $M_{\mathcal N}=T_{\mathcal N}=Pos(K)$. In other words, the quadratic module $M_{\mathcal N}$ of $F[x]$ is saturated.

math.AG↗
arXiv2022

On a question of Moshe Roitman and Euler class of stably free module

Let $A$ be a ring of dimension $d$ containing an infinite field $k$, $T_1,\ldots,T_r$ be variables over $A$ and $P$ be a projective $A[T_1,\ldots,T_r]$-module of rank $n$. Assume one of the following conditions hold. (1) $2n\geq d+3$ and $P$ is extended from $A$. (2) $2n\geq d+2$, $A$ is an affine $\overline {\mathbb F}_p$-algebra and $P$ is extended from $A$. (3) $2n\geq d+3$ and singular locus of $Spec(A)$ is a closed set $V(\mathcal J)$ with ht $\mathcal J\geq d-n+2$. Assume $Um(P_f)\neq \varnothing$ for some monic polynomial $f(T_r)\in A[T_1,\ldots,T_r]$. Then $Um(P)\neq \varnothing$.

math.AC↗
arXiv2021

Nice group structure on the elementary orbit space of unimodular rows

(1) If $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ with $p>3$, then the group structure on ${\rm Um}_d(R)/{\rm E}_d(R)$ is nice. (2) If $R$ is a commutative noetherian ring of dimension $d\geq 2$ such that ${\rm E}_{d+1}(R)$ acts transitively on ${\rm Um}_{d+1}(R),$ then the group structure on ${\rm Um}_{d+1}(R[X])/{\rm E}_{d+1}(R[X])$ is nice.

math.AC↗
arXiv2021

Unimodular rows over monoid extensions of overrings of polynomial rings

Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative cancellative torsion-free seminormal monoid. Then (1) Let $A$ be a ring of type $R[d,m,n]$ and $P$ be a projective $A[M]$-module of rank $r \geq max\{2,d+1\}$. Then the action of $E(A[M] \oplus P)$ on $Um(A[M] \oplus P)$ is transitive and (2) Assume $(R, m, K)$ is a regular local ring containing a field $k$ such that either $char$ $k=0$ or $ char$ $k = p$ and $tr$-$deg$ $K/\mathbb{F}_p \geq 1$. Let $A$ be a ring of type $R[d,m,n]^*$ and $f\in R$ be a regular parameter. Then all finitely generated projective modules over $A[M],$ $A[M]_f$ and $A[M] \otimes_R R(T)$ are free. When $M$ is free both results are due to Keshari and Lokhande.

math.AC↗
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↗
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

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↗
arXiv2013

Cancellation of projective modules over non-Noetherian rings

Let R be a ring of dimension d and A be one of R[Y] or R[Y,Y^{-1}]. If P is a projective A-module of rank \geq d+1 satisfying some condition, then we show that E(A\oplus P) acts transitively on Um(A\oplus P). When P is free, this result is due to Yengui (when A=R[Y]) and Abedelfatah (when A=R[Y,Y^{-1}]).

math.AC↗
arXiv2012

A note on rigidity and triangulability of a derivation

Let A be a $\mathfrak Q$-domain, K=frac(A), B=A^{[n]} and D\in \lnd_A(B). Assume rank D= rank D_K=r, where D_K is the extension of D to K^{[n]}. Then we show that (i) If D_K is rigid, then D is rigid. (ii) Assume n=3, r=2 and B=A[X,Y,Z] with DX=0. Then D is triangulable over A if and only if D is triangulable over A[X]. In case A is a field, this result is due to Daigle.

math.AC↗
arXiv2011

A note on cancellation of projective modules

Let $A$ be a ring of dimension $d$. Assume that for every finite extension ring $R$ of $A$, E_{d+1}(R) acts transitively on Um_{d+1}(R). Then we prove that E(A\oplus P) acts transitively on Um(A\oplus P), for any projective A-module P of rank d. As a consequence of this, we generalise some results of Gubeladze.

math.AC↗
arXiv2010

Projective modules over overrings of polynomial rings

Let A be a commutative Noetherian ring of dimension d and let P be a projective R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac {1}{f_1\ldots f_m}]-module of rank r\geq max {2,dim A+1, where f_i\in A[Y_i]. Then (i) \EL^1(R\op P) acts transitively on Um(R\oplus P). In particular, P is cancellative. (ii) If A is an affine algebra over a field, then P has a unimodular element. (iii) The natural map Φ_r : GL_r(R)/EL^1_r(R) \ra K_1(R) is surjective. (iv) Assume f_i is a monic polynomial. Then Φ_{r+1} is an isomorphism. In the case of Laurent polynomial ring (i.e. f_i=Y_i), (i) is due to Lindel, (ii) is due to Bhatwadekar, Lindel and Rao and (iii, iv) is due to Suslin.

math.AC↗