Manoj Kumar Keshari
Let A be an affine algebra over the field of real numbers of dimension d. Let f \in A be an element not belonging to any real maximal ideal of A. Let P be a projective A-module of rank \geq d-1. Let (a,p) \in A_f \oplus P_f be a unimodular element. Then the projective A_f module Q=A_f \oplus P_f/(a,p)A_f is extended from A.
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In this paper, we find out a lower bound of the de Rham depth of affine cone of a projective space.
We study the projective closures of three important families of affine monomial curves in dimension $4$, namely the Backelin curve, the Bresinsky curve and the Arslan curve, in order to explore possible connections between syzygies and the arithmetic Cohen-Macaulay property.
In this paper our aim is twofold. First, we introduce the notion of star gluing of numerical semigroups and show that arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure are preserved under this gluing operation. We then give a condition on Gröbner basis of the defining ideal of an affine monomial curve which ensures that the Betti sequence of the affine curve is the same as the Betti sequence of its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct some numerical semigroups, using a g…
Splitting of projective modules over some affine algebras
本文针对特定仿射代数上的光滑仿射代数 A，定义稳定自由模的欧拉类群，分析对应射影模中 unimodular 元素的存在条件，是对 Banerjee 和 Das 相关工作的改进。
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 $J$.
We generalize the notion of symmetric semigroups, pseudo symmetric semigroups, and row factorization matrices for pseudo Frobenius elements of numerical semigroups to the case of semigroups with maximal projective dimension (MPD semigroups).
Our aim in this paper is to compute the Poincaré series of the derivation module of the projective closure of certain affine monomial curves.
This expository note delves into the theory of projective modules parallel to the one developed for injective modules by Matlis. Given a perfect ring $R$, we present a characterization of indecomposable projective $R$-modules and describe a one-to-one correspondence between the projective indecomposable $R$-modules and the simple $R$-modules.
Projective Closure of Semigroup Algebras
本文研究$\mathbb{N}^{d}$中单纯仿射半群的射影闭包，用Gröbner基刻画其Cohen-Macaulay性质、给出非Cohen-Macaulay情形下Buchsbaum性质的判据，还提出$k$-提升概念并研究其与原半群的关系。
Let A be a ring of dimension d and let P be a projective A-module of rank d. We prove that if for every finite extension R of A, R^d is cancellative, then P is cancellative. This gives an alternate proof of Bhatwadekar's result: every projective module of rank d over an affine algebra of dimension d over a C_1-field of characteristic 0 is cancellative. Let P be a projective module of rank d-1 over an affine agebra of dimension d over an algebraically closed field. Then, it is not known if P is cancellative. We prove some results in this direction also.
Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on ${\mathfrak{X}}$ when viewed as a projective limit of projective integral schemes having function field contained in $F$, and we characterize the locally ringed subspaces of ${\mathfrak{X}}$ that are affine schemes.
Splitting criteria for projective modules over polynomial algebras
本文研究了投射模在多项式代数上的分裂问题，并证明了monic逆原则在有限生成环上的成立，同时探讨了相关理想的应用。
Affine semigroups of maximal projective dimension-II
该研究聚焦极大投射维数仿射半群，给出单纯半群为Buchsbaum的充要条件、几类半群的刻画，证明部分半群满足扩展Wilf猜想，还发现Betti型可不受嵌入维数有界约束，并探索满足Arf性质的子幺半群。
Let $k$ be an arbitrary field, the purpose of this work is to provide families of positive integers $\mathcal{A} = \{d_1,\ldots,d_n\}$ such that either the toric ideal $I_{\mathcal A}$ of the affine monomial curve $\mathcal C = \{(t^{d_1},\ldots,\,t^{d_n}) \ | \ t \in k\} \subset \mathbb{A}_k^n$ or the toric ideal $I_{\mathcal A^{\star}}$ of its projective closure ${\mathcal C^{\star}} \subset \mathbb{P}_k^n$ is a complete intersection. More precisely, we characterize the complete intersection property for $I_{\mathcal A}$ and for $I_{\mathcal A^{\star}}$ when: (a) $\mathcal{A}$ is a generaliz…
Let $R$ be a ring. An $R$-module $M$ is said to be a weak $w$-projective module if ${\rm Ext}_R^1(M,N)=0$ for all $N \in \mathcal{P}_{w}^{\dagger_\infty}$ (see, \cite{FLQ}). In this paper, we introduce and study some properties of weak $w$-projective modules. And we use these modules to characterize some classical rings, for example, we will prove that a ring $R$ is a $DW$-ring if and only if every weak $w$-projective is projective, $R$ is a Von Neumann regular ring if and only if every FP-projective is weak $w$-projective if and only if every finitely presented $R$-module is weak $w$-projecti…
Let R be an affine algebra of dimension n \geq 3 over an algebraically closed field k. Suppose char k =0 or char k =p \geq n. Let g,f_1,...,f_r be a R-regular sequence and A=R[f_1/g,...,f_r/g]. Let P be a projective A-module of rank n-1 which is extended from R. Let (a,p) \in (A \op P) be a unimodular element and Q=A\op P/(a,p)A. Then, Q is extended from R. A similar result for affine algebras over reals are also proved.
On complete intersection projective closures of monomial curves
研究四维仿射空间中单项式曲线射影闭包的完全交性质，给出其定义理想为完全交的刻画，还探讨其是否算术科恩 - 麦考利。
Let $I$ be an ideal of the polynomial ring $A[x]=A[x_1,...,x_n]$ over the commutative, noetherian ring $A$. Geometrically $I$ defines a family of affine schemes over $\Spec(A)$: For $\p\in\Spec(A)$, the fibre over $\p$ is the closed subscheme of affine space over the residue field $k(\p)$, which is determined by the extension of $I$ under the canonical map $σ_\p:A[x]\to k(\p)[x]$. If $I$ is homogeneous there is an analogous projective setting, but again the ideal defining the fibre is $\sigI$. For a chosen term order this ideal has a unique reduced Gröbner basis which is known to contain consi…
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.
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