A note on the edge ideals of Ferrers graphs

Koszul blowup algebras associated to three-dimensional Ferrers diagrams

We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees algebra and the toric ring. We show that the toric ring is a Koszul Cohen--Macaulay normal domain, while the Rees algebra is Koszul and the defining ideal is of fiber type.

Ferrers图的覆盖理想幂的投射维数

Projective dimension of powers of cover ideal of Ferrers graphs

本文通过分析Ferrers图对应的覆盖理想J_λ的幂，证明了其投射维数等于min{t, d(λ)}+1，其中d(λ)是λ的不同部分数。

Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that …

Unimodality of partitions with distinct parts inside Ferrers shapes

We investigate the rank-generating function $F_λ$ of the poset of partitions contained inside a given shifted Ferrers shape $λ$. When $λ$ has four parts, we show that $F_λ$ is unimodal when $λ=\langle n,n-1,n-2,n-3 \rangle$, for any $n\ge 4$, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form $λ=\langle n,n-t,n-2t,n-3t \rangle$, for any given $t\ge 2$ and $n$ large enough with respect to $t$. When $λ$ has $b\le 3$ parts, we show that our rank-generating functions $F_λ$ are all unimodal. However, the situation remains mostly obscure for $b\ge 5$. In gen…

Enumeration of concave integer partitions

An integer partition λof n corresponds, via its Ferrers diagram, to an artinian monomial ideal I of colength n in the polynomial ring on two variables. If the partition λcorresponds to an integrally closed ideal we call λconcave. We study generating functions for the number of concave partitions, unrestricted or with at most r parts.

Depth and Singular Varieties of Exterior Edge Ideals

Edge ideals of finite simple graphs are well-studied over polynomial rings. In this paper, we initiate the study of edge ideals over exterior algebras, specifically focusing on the depth and singular varieties of such ideals. We prove an upper bound on the depth of the edge ideal associated to a general graph and a more refined bound for bipartite graphs, and we show that both are tight. We also compute the depth of several large families of graphs including cycles, complete multipartite graphs, spider graphs, and Ferrers graphs. Finally, we focus on the effect whiskering a graph has on the de…

Algebraic properties of edge ideals via combinatorial topology

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edges ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We app…

Linear strands of edge ideals of multipartite uniform clutters

We construct the first linear strand of the minimal free resolutions of edge ideals of $d$-partite $d$-uniform clutters. We show that the first linear strand is supported on a relative simplicial complex. In the case that the edge ideals of such clutters have linear resolutions, we give an explicit and surprisingly simple description of their minimal free resolutions, generalizing the known resolutions for edge ideals of Ferrers graphs and hypergraphs and co-letterplace ideals. As an application, we show that the Lyubeznik numbers that appear on the last column of the Lyubeznik table of the co…

Notes on a minimal set of generators for the radical ideal defining the diagonal locus of $(\C^2)^n$

We develop several techniques for the study of the radical ideal $I$ defining the diagonal locus of $(\C^2)^n$. Using these techniques, we give combinatorial construction of generators for $I$ of certain bi-degrees.

Simplicial ideals, 2-linear ideals and arithmetical rank

In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying the fiber cone of of a codimension two toric ideals. Let $I\subset S$ be an ideal defining a linearly joined arrangement of varieties: - We compute the depth, and the cohomological dimension. is the connectedness dimension. - We characterize sets of generators of $I$, and give an effective algorithm to find equations, as an application we compute arithmetical…

A Note on Co-Maximal Ideal Graph of Commutative Rings

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If $Γ(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

A note on stability properties of powers of polymatroidal ideals

Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring $R=K[x_1,...,x_n]$, where $K$ is a field. Let astab$(I)$ and dstab$(I)$ be the smallest integer $n$ for which Ass$(I^n)$ and depth$(I^n)$ stabilize, respectively. In this paper, we show that astab$(I)=1$ if and only if dstab$(I)=1$. Moreover, we prove that if $d=3$, then ${\rm astab}(I)={\rm dstab}(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese type ideal of degree $d$, then ${\rm astab}(I)={\rm dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.

关于图的 toric 理想与 Knutson-Miller-Yong 分解的注记

A note on toric ideals of graphs and Knutson-Miller-Yong decompositions

该研究使用 Knutson-Miller-Yong 提出的 Gröbner 基技术，分析图性质与对应 toric 理想的代数性质的关联，恢复了图 toric 理想的高度公式，提出检测二分图删除的代数性质，并利用初始理想界定色数。

A note on the reducedness and Gröbner bases of Specht ideals

The Specht ideal of shape $λ$, where $λ$ is a partition, is the ideal generated by all Specht polynomials of shape $λ$. Haiman and Woo proved that these ideals are reduced and found their universal Gröbner bases. In this short note, we give a short proof for these results.

A note on minimal resolutions of vector-spread Borel ideals

We consider vector-spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincaré series. A characterization of the extremal Betti numbers of such a class of ideals is given. Finally, we classify all Cohen-Macaulay vector-spread Borel ideals.

A note on pairs of rings with same prime ideals

We study the ring extensions R \subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples are also discussed to strengthen the results.

A note on monomial ideals

We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes.

Monomization of Power Ideals and Parking Functions

In this note, we find a monomization of a certain power ideal associated to a directed graph. This power ideal has been studied in several settings. The combinatorial method described here extends earlier work of other, and will work on several other types of power ideals, as will appear in later work.

A note on Stanley's conjecture for monomial ideals

We prove that the Stanley's conjecture holds for monomial ideals $I\subset K[x_1,...,x_n]$ generated by at most $2n-1$ monomials, i.e. $sdepth(I)\geq depth(I)$.

A note on conductor ideals

Let $S$ be a commutative ring with identity and $R$ a unitary subring of $S$. An ideal $I$ of $S$ is called an $R$-conductor ideal of $S$ if $I=\{x\in S\mid xS\subseteq V\}$ for some intermediate ring $V$ of $R$ and $S$. In this note we present necessary and sufficient criterions for being an $R$-conductor ideal of $S$. We generalize several well known facts about them and present a simple approach to rediscover the results of both old and recent papers. We sketch the boundaries of our criterions by providing a few counterexamples.

A note on the generic initial ideal for complete intersections

We prove that the $d$-component of the generic initial ideal, with respect to the reverse lexicographic order, of an ideal generated by a regular sequence of homogeneous polynomials of degree $d$ is revlex in a particular, but important, case. Using this property, we compute the generic initial ideal for several complete intersection with strong Lefschetz property.

A note on Bernstein-Sato ideals

We define the Bernstein-Sato ideal associated to a tuple of ideals and we relate it to the jumping points of the corresponding mixed multiplier ideals.

A note on almost Cohen-Macaulay monomial ideals

Let $R = k[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $k$ and let $I$ be a monomial ideal of $R$. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost Cohen-Macaulay polymatroidal Veronese type and transversal polymatroidal ideals and furthermore we give some examples.

Note on linearly equivalent ideal topologies over Noetherian modules

Let $R$ be a commutative Noetherian ring, and let $N$ be a non-zero finitely generated $R$-module. In this paper, the main result asserts that for any $N$-proper ideal $\frak a$ of $R,$ the $\frak a$-symbolic topology on $N$ is linearly equivalent to the $\frak a$-adic topology on $N$ if and only if, for every $\frak p\in \Supp(N)$, $\Ass_{R_{\mathfrak {p} }}N_{\mathfrak {p}}$ consists of a single prime ideal and $\dim N\leq 1$.

A note on linear resolution and polymatroidal ideals

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n-1$; $(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,...,x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.

A short note on the multiplier ideals of monomial space curves

Thompson (2014) exhibits a formula for the multiplier ideal with multiplier lambda of a monomial curve C with ideal I as an intersection of a term coming from the I-adic valuation, the multiplier ideal of the term ideal of I, and terms coming from certain specified auxiliary valuations. This short note shows it suffices to consider only one auxiliary valuation. This improvement is achieved through a more intrinsic approach, reduction to the toric case.

Notes on the behavior of the Ratliff-Rush filtration

We establish new classes of Ratliff-Rush closed ideals and some pathological behavior of the Ratliff-Rush closure. In particular, Ratliff-Rush closure does not behave well under passage modulo superficial elements, taking powers of ideals, associated primes, leading term ideals, and the minimal number of generators. In contrast, the reduction number of the Ratliff-Rush filtration behaves better: it preserves some information on the reduction number of the ideal.

关于根式 E-理想与素 E-理想的注记

Note on radical and prime E-ideals

本文研究指数多项式环中的 E-理想，证明其关于素 E-理想不满足 Noether 性，并刻画指数根式理想。

A note on Stanley conjecture for monomial ideals

In this paper, we prove that if $I\subset S:=K[x_1,...,x_n]$ is a monomial ideal then $I$ and $S/I$ satisfy the Stanley conjecture when $I$ has a small number of generators, with respect to $\depth(S/I)$ and $\max\{|P|:\;P\in\Ass(S/I)\}$. In particular, if $I$ be a monomial almost complete intersection ideal in $S$, then Stanley's Conjecture holds for $S/I$ and $I$.

A note on Mather-Jacobian multiplier ideals

By using Mather-Jacobian multiplier ideals, we first prove a formula on comparing Grauert-Riemenschneider canonical sheaf with canonical sheaf of a variety over an algebraically closed field of characteristic zero. Then we turn to study Mather-Jacobian multiplier ideals on algebraic curve, in which case the definition of Mather-Jacobian multiplier ideal can be extended to a ground field of any characteristic. We show that Mather-Jacobian multiplier ideal on curves is essentially the same as an integrally closed ideal. Finally by comparing conductor ideal with Mather-Jacobian multiplier ideal, …

关于二维正则局部环上秩为2的不可分解整闭模的注记

Note on indecomposable integrally closed modules of rank 2 over two-dimensional regular local rings

本文刻画了二维正则局部环中可由秩2不可分解整闭模的最大子式给出的理想，明确了这类理想的结构特征。

A note on the multiplicity of determinantal ideals

Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous $k$-algebra, the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies. The goal of this paper is to establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field $k$ for $k$-algebras $k[x_1, ..., x_n]/I$ being $I$ a determinantal ideal of arbitrary codimension.

A note on the Eisenbud-Mazur Conjecture

The Eisenbud-Mazur conjecture states that given an equicharacteristic zero, regular local ring (R,\mathfrak{m}) and a prime ideal P\subset R, we have that P^{(2)}\subseteq mP. In this paper, we computationally prove that the conjecture holds in the special case of certain prime ideals in formal power series rings.

A note on the quintasymptotic prime ideals

Let $R$ denote a commutative Noetherian ring, $I$ an ideal of $R$, and let $S$ be a multiplicatively closed subset of $R$. In \cite{Ra1}, Ratliff showed that the sequence of sets ${\rm Ass}_RR/\bar{I}\subseteq {\rm Ass}_RR/\bar{I^2} \subseteq {\rm Ass}_R R/\bar{I^3}\subseteq \dots $ increases and eventually stabilizes to a set denoted $\bar{A^\ast}(I)$. In \cite{Mc2}, S. McAdam gave an interesting description of $\bar{A^\ast}(I)$ by making use of $R[It,t^{-1}]$, the Rees ring of $I$. In this paper, we give a second description of $\bar{A^\ast}(I)$ by making use of the Rees valuation rings of $…

Note on resonance varieties

We study the irreducibility of resonance varieties of graded rings over an exterior algebra E with particular attention to Orlik-Solomon algebras. We prove that for a stable monomial ideal in E the first resonance variety is irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane arrangement, then we show that its first resonance variety is irreducible if and only if the subideal of J generated by all degree 2 elements has a 2-linear resolution. As an application we characterize those hyperplane arrangements of rank less than or equal to 3 where J is componentwise linea…

A note on Chern coefficients and Cohen-Macaulay rings

In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen-Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.

A note on joint reductions and mixed multiplicities

Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-primary ideal, $I_1,...,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this paper, we interpret mixed multiplicities of $(I_1,..., I_s,J)$ with respect to $M$ as multiplicities of joint reductions of them. This generalizes the Rees's theorem on mixed multiplicity (Theorem 2.4). As an application we show that mixed multiplicities are also multiplicities of Rees's superficial sequences.

Note on bounds for multiplicities

Let S=K[x_1,...,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I is a graded ideal in S. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in of Herzog, Srinivasan and Gold. We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.

The radical of an n-absorbing ideal

In this note we show that in a commutative ring $R$ with unity, for any $n > 0$, if $I$ is an $n$-absorbing ideal of $R$, then $(\sqrt{I})^{n} \subseteq I$.

A note on the homology of differentials

We deal with the complete-intersection property of maximally differential ideals. Also, we connect the Gorenstein homology of derivations to the Gorenstein property of the base rings. These equipped with some applications.

Note on the star operations over polynomial rings

This paper studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in $R[X]$ is a $*$-maximal ideal and when a $*$-maximal ideal $Q$ of $R[X]$ is extended from $R$, that is, $Q=(Q\cap R)[X]$ with $Q\cap R\not =0$, for a given star operation of finite character $*$ on $R[X]$. We also answer negatively some questions raised by Anderson-Clarke by constructing a Prüfer domain $R$ for which the $v$-operation is not stable.

A short note on containment of cores

We show that cores of ideals do not preserve the inclusion.

On the conjecture of Vasconcelos for Artinian almost complete intersection monomial ideals

In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay.

Ideals defining Gorenstein rings are (almost) never products

This note proves that if S is an unramified regular local ring and I and J are proper ideals of height at least two, then S/IJ is never Gorenstein.

关于UFD的一个注记

A note on UFD

研究高度为一的理想为主理想的条件，通过局部上同调技术简化Auslander-Buchsbaum定理的证明，并解决Samuel问题，证明重数至多为3的局部环是超曲面，肯定Braun关于有限内射维数自反理想的猜想，简化Hartshorne的一个结果。

Koszul homology of codimension 3 Gorenstein ideals

In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.

Stanley depth of the path ideal associated to a line graph

We consider the path ideal associated to a line graph, we compute \texttt{sdepth} for its quotient ring and note that it is equal with its \texttt{depth}. In particular, it satisfies the Stanley inequality.

Notes on C-graded modules over an affine semigroup ring K[C]

Let $C \subset {\bf N}^d$ be an affine semigroup, and $R=K[C]$ its semigroup ring. This paper is a collection of various results on "$C$-graded" $R$-modules, especially, monomial ideals. For example, we show the following: If $R$ is normal and $I$ is a radical monomial ideal (i.e., $R/I$ is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of $R/I$ is a topological property of the "geometric realization" of the cell complex associated with $I$. Moreover, we can give a squarefree modules/constructible sheaves version of this result.

Notes on local cohomology and duality

We provide a formula (see Theorem 1.5) for the Matlis dual of the injective hull of $R/\mathfrak{p}$ where $\mathfrak p$ is a one dimensional prime ideal in a local complete Gorenstein domain $(R,\mathfrak{m})$. This is related to results of Enochs and Xu (see [4] and [3]). We prove a certain 'dual' version of the Hartshorne-Lichtenbaum vanishing (see Theorem 2.2). There is a generalization of local duality to cohomologically complete intersection ideals $I$ in the sense that for $I=\mathfrak{m}$ we get back the classical Local Duality Theorem. We determine the exact class of modules to which …

A note on some top local cohomology modules

Let $\fa$ be an ideal of a $d$-dimensional commutative Noetherian ring $R$. In this paper we give some information on some last non-zero local cohomology modules known as top local cohomology modules in particular, $H^{d-1}_{\fa}(R)$.

A Note on Bernstein-Sato Varieties for Tame Divisors and Arrangements

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standar…

关于乘积的正则性的一个注记

A note on the regularity of products

该数学研究采用Herzog方法构造M/IM的多重分次S-自由预解式，证明了一大类理想和模满足Reg(IM)≤Reg(I)+Reg(M)的正则性不等式，且当M为理想时在不涉及维数的生成元相对条件下该式也成立。

On the containment problem for fat points ideals

In this note we show that Harbourne's conjecture is true for symbolic powers of ideals of points, we check that the stable version of this conjecture is valid for ideals of very general points (resp. generic points) in $\mathbb P_{\mathbb K}^N$ (resp. $\mathbb P_{\mathbb K(\underline{z})}^N$). We also show that this conjecture and the Harbourne-Huneke conjecture are true for a class of ideals $I$ defining fat points obtained from line arrangements in $\mathbb P_{\mathbb K}^2$.

F-thresholds, integral closure and convexity

The purpose of this note is to revisit the results of arXiv:1407.4324 from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence of a peculiar polynomial allows to compute the F-jumping numbers of all the ideals formed by taking sums of products of the original ones. The note concludes with the suggestion of a possible source of examples falling in such a framework.