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Parangama Sarkar

Publications and source records attributed to Parangama Sarkar.

15 recordsLinked to original sources
arXiv2026

Epsilon multiplicity, multiplicity=volume formula and analytic spread of family of ideals

In an analytically unramified local ring $(R,\mathfrak m)$ of dimension $d\geq 1$, for a filtration of ideals $\mathfrak {I}=\{I_m\}_{m\in\mathbb N}$ satisfying $\mathfrak A(r)$ condition and for any $\mathfrak m$-primary ideal $K$, it is shown in $[18]$ that the epsilon multiplicity of the weakly graded family of ideals $\{(I_m:K)\}_{m\in\mathbb N}$ exists as a limit and it is bounded above by the epsilon multiplicity of $\mathfrak I$, $ε(\mathfrak I)$. In this article, we first show that $ε(\mathfrak I)$ coincides with the epsilon multiplicity of $\{(I_m:K)\}_{m\in\mathbb N}$ and this leads to the following: $(a)$ an expression for $ε(\mathfrak I)$ as a limit of the epsilon multiplicities of other graded families of ideals and $(b)$ a multiplicity=volume formula for the epsilon multiplicity of an ideal $I$ in $R$. In the final part of the article, we investigate the maximality of the analytic spread of filtrations of ideals.

math.AC↗
arXiv2025

$\operatorname{v}$-numbers of integral closure filtrations of monomial ideals

In this article, we investigate the $\operatorname{v}$-numbers of powers of monomial ideals and their integral closures in a polynomial ring $S$. We provide an alternative proof for determining the $\operatorname{v}$-numbers of powers of complete intersection monomial ideals. Furthermore, we analyze the $\operatorname{v}$-numbers associated to integral closure filtrations of irreducible monomial ideals and explore their relationship with the Castelnuovo-Mumford regularity of these ideals. Consequently, we obtain that for all $n\geq 1$, $\operatorname{reg}(S/\overline{I^n})=\operatorname{v}(\overline{I^n})=nα(I)-1$ where $I$ is an equigenerated irreducible monomial ideal. Finally, we give an upper bound for $\operatorname{v}$-numbers associated to the integral closure filtrations of complete intersection monomial ideals and explicitly compute these $\operatorname{v}$-numbers in certain cases. As a consequence, we show that for any integer $a\geq 1$, there exists a height two equigenerated complete intersection monomial ideal $I$ such that $\operatorname{reg}(S/\overline{I^n})-\operatorname{v}(\overline{I^n})=a-1$ for all $n\geq 1$. Moreover, we establish that for complete intersection monomial ideals, the $\operatorname{v}$-numbers of powers of ideals can be arbitrarily larger than the $\operatorname{v}$-numbers of integral closures of their powers.

math.AC↗
arXiv2025

Multiplicities of weakly graded families of ideals

In this article, we extend the notion of multiplicity for weakly graded families of ideals which are bounded below linearly. In particular, we show that the limit $e_W(\mathfrak{I}):=\lim\limits_{n\to\infty}d!\frac{\ell_R(R/I_n)}{n^d}$ exists where $\mathfrak I=\{I_n\}$ is a bounded below linearly weakly graded families of ideals in a Noetherian local ring $(R,\mathfrak m)$ of dimension $d\geq 1$ with $\dim(N(\hat{R}))<d$. Furthermore, we prove that ``volume=multiplicity" formula and Minkowski inequality hold for such families of ideals. We explore some properties of $e_W(\mathfrak J)$ for weakly graded families of ideals of the form $\mathfrak J=\{(I_n:K)\}$ where $\{I_n\}$ is an $\mathfrak m$-primary graded family of ideals. We provide a necessary and sufficient condition for the equality in Minkowski inequality for the weakly graded family of ideals of the form $\mathfrak J=\{(I_n:K)\}$ where $\{I_n\}$ is a bounded filtration. Moreover, we generalize a result of Rees characterizing the inclusion of ideals with the same multiplicities for the above families of ideals. Finally, we investigate the asymptotic behaviour of the length function $\ell_R(H_{\mathfrak m}^0(R/(I_n:K)))$ where $\{I_n\}$ is a filtration of ideals (not necessarily $\mathfrak m$-primary).

math.AC↗
arXiv2024

$v$-numbers of symbolic power filtrations

We study the asymptotic behaviour of $v$-number and local $v$-numbers of Noetherian generalized symbolic power filtrations $\mathcal I=\{I_n\}$ in a Noetherian $\mathbb N$-graded domain and show that they are quasi-linear type. We provide sufficient conditions for the existence of the limits $\lim\limits_{n\to\infty}\frac{v(I_n)}{n}$ and $\lim\limits_{n\to\infty}\frac{v_\mathfrak p(I_n)}{n}$ for all $\mathfrak p\in\overline A(\mathcal I)$. We explicitly compute local $v$-numbers and $v$-numbers of symbolic powers of cover ideals of complete bipartite graphs, complete graphs, cycles, $K_m^s$ and compare them with their Castelnuovo-Mumford regularity. We give an example of a bipartite graph $\mathcal H$ that is not a complete bipartite graph and $v(J(\mathcal H))>bight(I(\mathcal H))-1$. This answers a question in [25, Question 3.12]. We show that for both connected bipartite graphs and connected non-bipartite graphs, the difference between the regularity and the $v$-number of the cover ideals can be arbitrarily large. This strengthens and gives an alternative proof of[25,Theorem 3.10]. We provide a counterexample to a conjecture [12, Conjecture 5.4] due to A. Ficarra and E. Sgroi.

math.AC↗
arXiv2023

On a generalized Auslander-Reiten conjecture

It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten condition (ARC). In this paper we explore (SAC) along with the several canonical change of rings $R \to S$. First, we prove the equivalence of (SAC) for $R$ and $R/xR$, where $x$ is a non-zerodivisor on $R$, and the equivalence of (SAC) and (SACC) for rings with positive depth, where (SACC) is the symmetric Auslander condition for modules with constant rank. The latter assertion affirmatively answers a question posed by Celikbas and Takahashi. Secondly, for a ring homomorphism $R \to S$, we prove that if $S$ satisfies (SAC) (resp. (ARC)), then $R$ also satisfies (SAC) (resp. (ARC)) if the flat dimension of $S$ over $R$ is finite. We also prove that (SAC) for $R$ implies that (SAC) for $S$ when $R$ is Gorenstein and $S=R/Q^\ell$, where $Q$ is generated by a regular sequence of $R$ and the length of the sequence is at least $\ell$. This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to determinantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC). A relation between (SAC) and an invariant related to the finitistic extension degree is also explored.

math.AC↗
arXiv2023

Epsilon multiplicity and analytic spread of filtrations

We extend the epsilon multiplicity of ideals defined by Ulrich and Validashti to epsilon multiplicity of filtrations, and show that under mild assumptions this multiplicity exists as a limit. We show that in rather general rings, the epsilon multiplicity of a Q-divisorial filtration is positive if and only if the analytic spread of the filtration is maximal (equal to the dimension of the ring). The condition that filtrations $\mathcal J\subset \mathcal I$ have the same epsilon multiplicity is considered, and we find conditions ensuring that the filtrations have the same integral closure.

math.AC↗
arXiv2022

Analytic Spread of Filtrations and Symbolic Algebras

In this paper we define and explore the analytic spread $\ell(\mathcal I)$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal $I$, we have the classical bounds $\mbox{ht}(I)\le\ell(I)\le \dim R$. The upper bound $\ell(\mathcal I)\le \dim R$ is true for filtrations $\mathcal I$, but the lower bound is not true for all filtrations. We show that for the filtration $\mathcal I$ of symbolic powers of a height two prime ideal $\mathfrak p$ in a regular local ring of dimension three (a space curve singularity), so that $\mbox{ht}(\mathcal I) =2$ and $\dim R=3$, we have that $0\le \ell(\mathcal I)\le 2$ and all values of 0,1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if $\ell(\mathfrak p^{(n)})=3$ for all symbolic powers of $\mathfrak p$ and if and only if $\ell(\mathcal I_a)=3$ for all truncations $\mathcal I_a$ of $\mathcal I$.

math.AC↗
arXiv2021

Rees' theorem for filtrations, multiplicity function and reduction criteria

Let $J\subset I$ be ideals in a formally equidimensional local ring with $λ(I/J)<\infty.$ Rees proved that for all $n\gg0$, $λ(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$ if and only if deg $P(I/J)(X)\leq$ dim $R-1.$ We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg $P(I/J)$ achieves its maximal degree. On the other hand, for ideals $J\subset I$ in a formally equidimensional local ring, we consider the multiplicity function $e(I^n/J^n)$ which is a polynomial in $n$ for all large $n.$ We explicitly determine the deg $e(I^n/J^n)$ in some special cases. For an ideal $J$ of analytic deviation one, we give characterization of reductions in terms of deg $e(I^n/J^n)$ under some additional conditions.

math.AC↗
arXiv2021

Multiplicities and Mixed Multiplicities of arbitrary Filtrations

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of $m$-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of $r$ filtrations on an analytically unramified local ring $R$ come from the coefficients of a suitable homogeneous polynomial in $r$ variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of $m$-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.

math.AC↗
arXiv2020

Frobenius Betti numbers and syzygies of finite length modules

Let $(R,\mathfrak m)$ be a local (Noetherian) ring of dimension $d$ and $M$ a finite length $R$-module with free resolution $G_\bullet$. De Stefani, Huneke, and Núñez-Betancourt explored two questions about the properties of resolutions of $M$. First, in characteristic $p>0$, what vanishing conditions on the Frobenius Betti numbers, $β_i^F(M, R) : = \lim_{e \to \infty} λ(H_i(F^e(G_\bullet)))/p^{ed}$, force pd$_R M < \infty$. Second, if pd$_R M = \infty $, does this force $d+2$nd or higher syzygies of $M$ to have infinite length. For the first question, they showed, under rather restrictive hypotheses, that $d+1$ consecutive vanishing Frobenius Betti numbers forces pd$_R M < \infty$. And when $d=1$ and $R$ is CM then one vanishing Frobenius Betti number suffices. Using properties of stably phantom homology, we show that these results hold in general, i.e., $d+1$ consecutive vanishing Frobenius Betti numbers force pd$_R M < \infty$, and, under the hypothesis that $R$ is CM, $d$ consecutive vanishing Frobenius Betti numbers suffice. For the second question, they obtain very interesting results when $d=1$. In particular, no third syzygy of $M$ can have finite length. Their main tool is, if $d=1$, to show, if the syzygy has a finite length, then it is an alternating sum of lengths of Tors. We are able to prove this fact for rings of arbitrary dimension, which allows us to show that if $d=2$, no third syzygy of $M$ can be finite length! We also are able to show that the question has a positive answer if the dimension of the socle of $H^0_{\mathfrak m}(R)$ is large relative to the rest of the module, generalizing the case of Buchsbaum rings.

math.AC↗
arXiv2019

Mixed Multiplicities of Filtrations

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp.

math.AC↗
arXiv2016

Multigraded regularity, reduction vectors and postulation vectors

We relate the set of complete reduction vectors of a $\mathbb{Z}^s$-graded admissible filtration of ideals $\mathcal{F}$ with the set of multigraded regularities of $G(\mathcal{F}).$ We prove reg$(G(\mathcal{F}))$=reg$(\mathcal{R}(\mathcal{F})).$ We establish a relation between the sets of complete reduction vectors of $\mathcal{F}$ and postulation vectors of $\mathcal{F}$ under some cohomological conditions.

math.AC↗
arXiv2016

Variations on the Grothendieck-Serre Formula for Hilbert functions and their applications

In this expository paper we present proofs of Grothendieck-Serre Formula for multi-graded algebras and Rees algebras for admissible multi-graded filtrations. As applications, we derive formulas of Sally for postulation number of admissible filtrations and Hilbert coefficients. We also discuss a partial solution of Itoh's conjecture by Kummini and Masuti. We present an alternate proof of Huneke-Ooishi Theorem and a generalisation for multi-graded filtrations.

math.AC↗
arXiv2016

Postulation and reduction vectors of multigraded filtrations of ideals

We study relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal F= \{\mathcal F (\underline n)\}_{\underline n\in\mathbb Z^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalisation of the Kirby-Mehran Complex. An analysis of its homology leads to an analogue of Huneke's Fundamental Lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal F$ and reduction vectors with respect to complete reductions of $\mathcal F.$

math.AC↗