arXiv subjects

Kumari Saloni

Publications and source records attributed to Kumari Saloni.

10 recordsLinked to original sources

arXiv2026

Multiplicity Versus Buchsbaumness of the special fiber cone

Let $(A,\mathfrak m)$ be a Noetherian local ring of dimension $d>0$ with infinite residue field and $I$ an $\mathfrak{m}$-primary ideal. Let $\mathcal I$ be an $I$-good filtration. We study an equality of Hilbert coefficients, first given by Elias and Valla, versus passage of Buchsbaum property from the local ring to the blow-up algebras. Suppose $e_1(\mathcal I)-e_1(Q)=2e_0(\mathcal I)-2\ell(A/I_1)-\ell(I_1/(I_2+Q))$ where $Q\subseteq I$, a minimal reduction of $\mathcal I$, is a standard parameter ideal. Under some mild conditions, we prove that if $A$ is Buchsbaum (generalized Cohen-Macaulay respectively), then the associated graded ring $G(\mathcal I)$ is Buchsbaum (generalized Cohen-Macaulay respectively). Our results settle a question of Corso in general for an $I$-good filtration. Further, let $f_0(I)= e_1(I)-e_0(I)-e_1(Q)+\ell(A/I)+μ(I)-d+1$ and $e_1(I)-e_1(Q)=2e_0(I)-2\ell(A/I)-\ell(I/(I^2+Q))$. We prove, under mild conditions, that (1) if $A$ is generalized Cohen-Macaulay, then the special fiber ring $F_{\mathfrak{m}}(I)$ is generalized Cohen-Macaulay; In addition, if depth of $A$ is positive, then depth of $F_{\mathfrak {m}}(I)$ is same as depth of $A$ and (2) if $A$ is Buchsbaum and depth A$\geq d-1$, then $F_{\mathfrak{m}}(I)$ is Buchsbaum and the $I$-invariant of $F_{\mathfrak{m}}(I)$ is same as that of $A$.

math.AC↗

arXiv2025

Special fibers of coordinate sections of Hankel Matrices

We investigate the special fibers associated with certain coordinate sections of Hankel determinantal ideals. We provide explicit descriptions of their defining equations, showing that these equations admit a natural matrix structure. In particular, we prove that they are Cohen-Macaulay and cannot, in general, be minimally generated only by quadrics and cubics. Instead, we show that the degrees of their minimal generators grow with the size of the minors involved. In one case, we also prove that the Rees algebra is of fiber type. Additionally, we compute algebraic invariants of these special fibers. Our results partially build on and extend the work of Ramkumar and Sammartano on $2$-determinantal ideals and answer some of the questions posed by Cunha, Mostafazadehfard, Ramos, and Simis in earlier work.

math.AC↗

arXiv2025

Buchsbaumness, Macaulayfication and Castelnuovo-Mumford regularity of monomial curves

Projective monomial curves are associated with rings generated by monomials of equal degree in two variables. In this paper, we give an infinite class of non-smooth, non Cohen-Macaulay $k$-Buchsbaum projective monomial curves for any $k\geq 1$ and find the monomial generators for the respective Macaulayfication. More generally, we demonstrate a method to find the Macaulayfication of a $k$-Buchsbaum monomial curve for any $k\geq 1$. We also discuss Castelnuovo-Mumford regularity of certain curves in terms of $k$-Buchsbaumness.

math.AC↗

arXiv2024

Bounding reduction number and the Hilbert coefficients of filtration

Let $(A,\m)$ be a Cohen-Macaulay local ring of dimension $d\geq 3$, $I$ an $\m$-primary ideal and $\mathcal{I}=\{I_n\}_{n\geq 0}$ an $I$-admissible filtration. We establish bounds for the third Hilbert coefficient: (i) $e_3(\mathcal{I})\leq e_2(\mathcal{I})(e_2(\mathcal{I})-1)$ and (ii) $e_3(I)\leq e_2(I)(e_2(I)-e_1(I)+e_0(I)-\ell(A/I))$ if $I$ is an integrally closed ideal. Further, assume the respective boundary cases along with the vanishing of $e_i(\mathcal{I})$ for $4\leq i\leq d$. Then we show that the associated graded ring of the Ratliff-Rush filtration of $\mathcal{I}$ is almost Cohen-Macaulay, Rossi's bound for the reduction number $r_J(I)$ of $I$ holds true and the reduction number of Ratliff-Rush filtration of $\mathcal{I}$ is bounded above by $r_J(\I).$ In addition, if $\wt{I^{r_J(I)}}=I^{r_J(I)}$, then we prove that $\reg G_I(A)=r_J(I)$ and a bound on the stability index of Ratliff-Rush filtration is obtained. We also do a parallel discussion on the \textquotedblleft good behaviour of the Ratliff-Rush filtration with respect to superficial sequence''.

math.AC↗

arXiv2023

Bounds for the reduction number of primary ideal in dimension three

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an $\mathfrak{m}$-primary ideal of $R$. Let $r_J(I)$ be the reduction number of $I$ with respect to a minimal reduction $J$ of $I$. Suppose depth $G(I)\geq d-3$. We prove that $r_J(I)\leq e_1(I)-e_0(I)+λ(R/I)+1+(e_2(I)-1)e_2(I)-e_3(I)$, where $e_i(I)$ are Hilbert coefficients. Suppose $d=3$ and depth $G(I^t)>0$ for some $t\geq 1$. Then we prove that $r_J(I)\leq e_1(I)-e_0(I)+λ(R/I)+t$.

math.AC↗

arXiv2023

Ratliff-Rush filtration, Hilbert coefficients and the reduction number of integrally closed ideals

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an integrally closed $\mathfrak{m}$-primary ideal. We establish bounds for the third Hilbert coefficient $e_3(I)$ in terms of the lower Hilbert coefficients $e_i(I),~0\leq i\leq 2$ and the reduction number of $I$. When $d=3$, the boundary cases of these bounds characterize certain properties of the Ratliff-Rush filtration of $I$. These properties, though weaker than depth $G(I)\geq 1$, guarantees that Rossi's bound for reduction number $r_J(I)$ holds in dimension three. In that context, we prove that if $depth G(I)\geq d-3$, then $r_J(I)\leq e_1(I)-e_0(I)+\ell(R/I)+1+e_2(I)(e_2(I)-e_1(I)+e_0(I)-\ell(R/I))-e_3(I).$ We also discuss the signature of the fourth Hilbert coefficient $e_4(I).$

math.AC↗

arXiv2020

Buchsbaumness of the associated graded rings of filtration

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and $I$ an $\mathcal{I}$-primary ideal of $A$. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring $A$ to be passed onto the associated graded ring of filtration. Let $\mathcal{I}$ denote an $I$-good filtration. We prove that if $A$ is Buchsbaum and the I-invariant, $I(A)$ and $I(G(\mathcal{I}))$, coincide then the associated graded ring $G(\mathcal{I})$ is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions for Hilbert coefficients.

math.AC↗

arXiv2017

Bounding Hilbert coefficients of parameter ideals

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\geq d-1$. Let $Q$ be a parameter ideal of $R$. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient $e_i(Q)$ under certain assumptions on the depth of associated graded ring $G(Q)$. For $2\leq i\leq d $, we show that (1) $e_i(Q)\leq 0$ provided depth $G(Q)\geq d-2$ and (2) $e_i(Q)\geq -λ_R(H_{\mathfrak{m}}^{d-1}(R))$ provided depth $G(Q)\geq d-1$. It is proved that $e_3(Q)\leq 0$. Further, we obtain a necessary condition for the vanishing of the last coefficient $e_d(Q)$. As a consequence, we characterize the vanishing of $e_2(Q)$. Our results generalize \cite[Theorem 3.2]{goto-ozeki} and \cite[Corollary 4.5]{Lori}.

math.AC↗

arXiv2017

On the finiteness of the set of Hilbert coefficients

Let $(R,m)$ be a Noetherian local ring of dimension $d$ and $K,Q$ be $m$-primary ideals in $R.$ In this paper we study the finiteness properties of the sets $Λ_i^K(R):=\{g_i^K(Q): Q$ is a parameter ideal of $R\},$ where $g_i^K(Q)$ denotes the Hilbert coefficients of $Q$ with respect to $K,$ for $1 \leq i \leq d.$ We prove that $Λ_i^K(R)$ is finite for all $1\leq i \leq d$ if and only if $R$ is generalized Cohen-Macaulay. Moreover, we show that if $R$ is unmixed then finiteness of the set $Λ_1^K(R)$ suffices to conclude that $R$ is generalized Cohen-Macaulay. We obtain partial results for $R$ to be Buchsbaum in terms of $|Λ_i^K(R)|=1.$ We also obtain a criterion for the set $Δ^K(R):=\{g_1^K(I): I$ is an m-primary ideal of $R\}$ to be finite, generalizing preceding results.

math.AC↗

arXiv2016

On Hilbert coefficients of parameter ideals and Cohen-Macaulayness

Let $(R, \mathfrak m)$ be an unmixed Noetherian local ring, Q a parameter ideal and $K$ an $\mathfrak m$-primary ideal of $R$ containing $Q$. We give a necessary and sufficient condition for $R$ to be Cohen-Macaulay in terms of $g_0(Q)$ and $g_1(Q)$, the Hilbert coefficients of $Q$ with respect to $K$. As a consequence, we obtain a result of Ghezzi et al. which settles the negativity conjecture of W. V. Vasconcelos [15] in unmixed local rings.

math.AC↗