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Sarasij Maitra

Publications and source records attributed to Sarasij Maitra.

14 recordsLinked to original sources
arXiv2026

Trace ideals of syzygies

We study the behavior of trace ideals under taking syzygies. In particular, when $R$ is numerical semigroup ring and $I$ is a homogeneous ideal in $R$, we obtain an upper estimate for $\operatorname{tr}_R(Ω^1_R(I))$, and we show this estimate is sharp when $I$ is the conductor ideal $\mathfrak{c}_R$ of $R$. Using this result, we characterize the numerical semigroup rings for which $\operatorname{tr}_R(M) \subseteq \operatorname{tr}_R(Ω^1_R(M))$ holds for every finitely generated $R$-module $M$. In a similar vein, we characterize numerical semigroup rings for which $\operatorname{tr}_R(Ω^1_R(\mathfrak{c}_R))=\mathfrak{c}_R$.

math.AC↗
arXiv2025

Optimal Parametrizations and Valuations

This article discusses a way for uniquely setting up the valuations for the minimal generators of the maximal ideal of a one dimensional complete reduced and irreducible local algebra over an algebraically closed field, when treated as a subring of its integral closure. Our observations are a generalization of the more well-studied case of a numerical semigroup ring. These results provide completion to some missing arguments in certain proofs present in the existing literature, including some results concerning a long-standing conjecture of R. Berger.

math.AC↗
arXiv2025

Trace does not preserve Reflexivity

In this note, we address a question raised in a recent work by Dao-Maitra-Sridhar, regarding the preservation of reflexivity under taking trace. We answer this question negatively. We also study a few cases where the question has a positive answer in a one dimensional, analytically unramified Cohen-Macaulay local ring.

math.AC↗
arXiv2025

Unstable elements in cohomology and a question of Lescot

In his work on the Bass series of syzygy modules of modules over a commutative noetherian local ring $R$, Lescot introduces a numerical invariant, denoted $σ(R)$, and asks whether it is finite for any $R$. He proves that this is so when $R$ is Gorenstein or Golod. In the present work many new classes of rings $R$ for which $σ(R)$ is finite are identified. The new insight is that $σ(R)$ is related to the natural map from the usual cohomology of the module to its stable cohomology, which permits the use of multiplicative structures to study the question of finiteness of $σ(R)$.

math.AC↗
arXiv2024

A family of simplicial resolutions which are DG-algebras

Each monomial ideal over a polynomial ring admits a free resolution which has the structure of a DG-algebra, namely, the Taylor resolution. A pivot resolution of a monomial ideal, which we introduce, is a resolution that is always shorter than the Taylor resolution (unless the Taylor resolution is as short as possible) but still retains a DG-algebra structure. We study the basic properties of this family of resolutions including a characterization of when the construction is minimal. Following the work of Sobieska, we use the explicit nature of pivot resolutions to give formulae for the Eisenbud-Shamash construction of a free resolution of a given monomial ideal over complete intersections.

math.AC↗
arXiv2024

Annihilators of (co)homology and their influence on the trace Ideal

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, and suppose $R$ is Cohen-Macaulay with canonical module $ω_R$. We develop new tools for analyzing questions involving annihilators of several homologically defined objects. Using these, we study a generalization introduced by Dao-Kobayashi-Takahashi of the famous Tachikawa conjecture, asking in particular whether the vanishing of $\mathfrak{m} \operatorname{Ext}_R^i(ω_R,R)$ should force the trace ideal of $ω_R$ to contain $\mathfrak{m}$, i.e., for $R$ to be nearly Gorenstein. We show this question has an affirmative answer for numerical semigroup rings of minimal multiplicity, but that the answer is negative in general. Our proofs involve a technical analysis of homogeneous ideals in a numerical semigroup ring, and exploit the behavior of Ulrich modules in this setting.

math.AC↗
arXiv2024

Two Criteria For Quasihomogeneity

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. The ring $R$ is said to be quasihomogeneous if there exists a surjection $Ω_R\twoheadrightarrow \mathfrak{m}$ where $Ω_R$ denotes the module of differentials. We present two characterizations of quasihomogeneity of $R$ in the situation when $R$ is a domain: the first one on the valuation semigroup of $R$ and the other on the trace ideal of the module $Ω_R$.

math.AC↗
arXiv2023

Extremal behavior of reduced type of one dimensional rings

Let $R$ be a domain that is a complete local $\mathbb{k}$ algebra in dimension one. In an effort to address the Berger's conjecture, a crucial invariant reduced type $s(R)$ was introduced by Huneke et. al. In this article, we study this invariant and its max/min values separately and relate it to the valuation semigroup of $R$. We justify the need to study $s(R)$ in the context of numerical semigroup rings and consequently investigate the occurrence of the extreme values of $s(R)$ for the Gorenstein, almost Gorenstein, and far-flung Gorenstein complete numerical semigroup rings. Finally, we study the finiteness of the category $\text{CM}(R)$ of maximal Cohen Macaulay modules and the category $\text{Ref}(R)$ of reflexive modules for rings which are of maximal/minimal reduced type and provide many classifications.

math.AC↗
arXiv2022

Valuations and Nonzero Torsion in Module of Differentials

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $Ω_R$ is torsion free. When $R$ is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of $R$. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of $R$ and orders of units of the integral closure of $R$. We also prove the conjecture in the case when certain monomials are missing from the monomial support of the defining ideal of $R$. These monomials are based on the smallest power of $\mathfrak{m}_R$ that is contained within the conductor ideal. This also generalizes a previous result of Cortiñas, Geller and Weibel.

math.AC↗
arXiv2022

Partial Trace Ideals, Torsion and Canonical Module

For any finitely generated module $M$ with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant $h(M)$ was introduced and studied in the author's previous work "Partial Trace Ideals and Berger's Conjecture". We establish a bound on it which helps capture information about the torsion submodule of $M$ when $M$ has rank one and it also generalizes the discussion in the mentioned previous article. We further study bounds and properties of $h(M)$ in the case when $M$ is the canonical module $ω_R$. This in turn helps in answering a question of S. Greco and then provide some classifications. Most of the results in this article are based on the results presented in the author's doctoral dissertation "Partial Trace Ideals, The Conductor and Berger's Conjecture".

math.AC↗
arXiv2022

Partial Trace Ideals And Berger's Conjecture

For any finitely generated module $M$ with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant $\operatorname{h}(M)$ based on a partial trace ideal of $M$. We study its properties and explore relations between this invariant and the colength of the conductor. Finally we apply this to the universally finite module of differentials $Ω_{R/k}$, where $R$ is a complete $k$-algebra with $k$ any perfect field, to study a long-standing conjecture due to R. W. Berger.

math.AC↗
arXiv2022

Torsion in Differentials and Berger's Conjecture

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials $Ω_R$ is a torsion-free $R$-module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499-510, 1990) and Cortiñas et al. (Math Z 228:569-588, 1998).This is obtained by constructing a new subring $S$ of $\operatorname{Hom}_R(\mathfrak{m},\mathfrak{m})$ and constructing enough torsion in $Ω_S$, enabling us to pull back a nontrivial torsion to $Ω_R$.

math.AC↗
arXiv2021

RandomPoints package for Macaulay2

We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly to identify rational and geometric points in a variety over a finite field. We provide tools to estimate the dimension of a variety. We also present methods to obtain non-vanishing minors of a given size in a given matrix, by evaluating the matrix at a point.

math.AG↗