arXiv subjects

Souvik Dey

Publications and source records attributed to Souvik Dey.

49 records · Page 3Linked to original sources

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

Vanishing of (co)homology of Burch and related submodules

We introduce the notion of Burch submodules and weakly $\mathfrak m$-full submodules of modules over local rings and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test property. We also show that over a local ring $(R,\mathfrak m)$ a submodule $M$ of a finitely generated $R$-module $X$, such that either $M=\mathfrak m X$ or $M(\subseteq \mathfrak m X$) is weakly $\mathfrak m$-full in $X$, is 1-Tor rigid and a test module provided that $X$ is faithful (and $X/M$ has finite length when $M$ is weakly $\mathfrak m$-full). As an application, we give a new class of rings such that a conjecture of Huneke and Wiegand is affirmative over them.

math.AC↗

arXiv2023

Complexity and rigidity of Ulrich modules, and some applications

We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can be used as test modules, which detect finite homological dimensions of modules. We prove that Ulrich modules over CM local rings have maximal complexity and curvature. Various new characterizations of local rings are provided in terms of Ulrich modules. We show that every Ulrich module of dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not $s$-Tor-rigid in general (where $s\geq 1$). Over a deformation of a CM local ring of minimal multiplicity, we also study Tor rigidity.

math.AC↗

arXiv2022

Finite Birational extension with stable conductor

Let $S$ be a module finite birational extension of a $1$-dimensional local Cohen--Macaulay ring $R$. When is the conductor of $S$ in $R$ a stable ideal? If $R$ is also generically Gorenstein, then we show that the conductor of $S$ in $R$ is a stable ideal, and $S$ is a reflexive $R$-module if and only if $Ω\operatorname{CM}(S)=\operatorname{CM}(S)\cap Ω\operatorname{CM}(R)$.

math.AC↗

arXiv2022

Some characterizations of local rings via reducing dimensions

In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings via reducing homological dimensions. For example, we extend a classical result of Auslander and Bridger, and prove that a local ring is Gorenstein if and only if each finitely generated module over it has finite reducing Gorenstein dimension. Along the way, we prove various connections between complexity and reducing projective dimension of modules.

math.AC↗

arXiv2022

Trace ideal and annihilator of Ext and Tor of regular fractional ideals, and some applications

Given a commutative Noetherian ring $R$ with total ring of fractions $Q(R)$, and a finitely generated $R$-submodule $M$ of $Q(R)$, we prove an equality between trace ideal, and certain annihilator of Ext and Tor of $M$. As a consequence, we answer in one-dimensional local analytically unramified case, a question raised by the present author and R. Takahashi. As another application, we give an alternative proof of a recent result of Ö. Esentepe that for one-dimensional analytically unramified Gorenstein local rings, the cohomology annihilator of Iyengar and Takahashi coincides with the conductor ideal.

math.AC↗

arXiv2022

On the reducing projective dimension of the residue field

In this paper we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya-Celikbas and Araya-Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension, and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

math.AC↗

arXiv2022

Some new invariants of Noetherian local rings related to square of the maximal ideal

We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider the minimum among the number of generators of ideals $I$ such that either $I^2 = \mathfrak m^2$ or $I \supseteq \mathfrak m^2$ holds. We investigate subsequently the case that $R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by an ideal $I$ generated by homogeneous quadratic forms, and we compute these invariants. We devote specific attention to the case that $R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by the edge ideal of a finite simple graph $G.$

math.AC↗

arXiv2021

Comparisons between Annihilators of Tor and Ext

In this paper, we compare annihilators of Tor and Ext modules of finitely generated modules over a commutative noetherian ring. For local Cohen--Macaulay rings, one of our results refines a theorem of Dao and Takahashi.

math.AC↗

arXiv2021

On the Subcategories of n-Torsionfree Modules and Related Modules

Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we study n-torsionfree modules in the sense of Auslander and Bridger, by comparing them with n-syzygy modules, and modules satisfying Serre's condition (S_n). We mainly investigate closedness properties of the full subcategories of mod R consisting of those modules.

math.AC↗

arXiv2018

Commutative rings over which the support of any module is the collection of prime ideals containing the annihilator

The support of any module over a commutative ring is defined as the collection of all prime ideals of the ring at which the localization of the module is non-zero. For finitely generated modules, the support is the collection of all prime ideals containing the annihilator of the module. In this article, we raise the natural question that over which commutative rings, the support of every module is the collection of all the prime ideals of its annihilator. We completely classify such rings, and in the process it also comes out that it is enough to require that only for countably generated modules, the support is the collection of all prime ideals containing the annihilator of the module.

math.AC↗

arXiv2018

Commutative rings with every non-maximal ideal finitely generated

In commutative ring theory, there is a theorem of Cohen which states that if in a commutative ring all prime ideals are finitely generated then every ideal is finitely generated. However, it is known that having only maximal ideals finitely generated doesn't imply all ideals be finitely generated. In his article we ask the question that what happens if we assume all non-maximal ideals are finitely generated, and we answer the question by showing that indeed then all maximal ideals are also finitely generated i.e. the ring becomes Noetherian.

math.AC↗

arXiv2017

Classification of Rings which admits some special type of Polynomial functions

We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can be given by a polynomial with coefficients in $R$, can we say $R$ is a finite field ? We show that the answer is yes, and that in fact it is enough to only require that all bijections be given by polynomials. If we allow our rings to have unity, we show that the property that all characteristic functions can be given by polynomials actually characterizes finite fields and if we moreover allow our rings to be commutative, then to characterize finite fields, it is enough that some special characteristic function be given by a polynomial (with coefficients even in some extension ring). Motivated by this, we determine all commutative rings with unity which admits a characteristic function which can be given by some polynomial with coefficients in the ring.

math.AC↗