Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

The Lefschetz Property for Componentwise Linear Ideals and Gotzmann Ideals

Abstract

For standard graded Artinian K-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.     

On n-Absorbing Ideals of Commutative Rings

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x ₁…x _n+1 ∈ I for x ₁,…, x _n+1 ∈ R (resp., I ₁…I _n+1 ? I for ideals I ₁,…, I _n+1 of R), then there are n of the x _i 's (resp., n of the I _i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.     

Betti numbers of determinantal ideals

Let R=k[x₁,…,x_n] be a polynomial ring and let IR be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number β_i(R/I) can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of R/I as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras k[x₁,…,x_n]/I when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Globally generated vector bundles and the Weak Lefschetz property

Artinian ideals \({I\subset R: =k[x_1,..., x_n]}\) generated by m general forms of given degree have attracted a great deal of attention recently, and one of the main challenging problems is to determine its Hilbert function. The Hilbert function of R/I was conjectured by Fröberg, and it is well known that the Weak Lefschetz property imposes severe constraints on the possible Hilbert functions. In this short note, we will focus our attention on Artinian ideals \({I \subset R}\) generated by m general forms all of the same degree d and we analyze whether the Weak Lefschetz property is satisfied. More precisely, for m = n or n ≤ 4 the Weak Lefschetz property holds, and our goal will be to prove that for any integers n and d, the Weak Lefschetz property also holds provided m falls into the interval \({[\frac{1}{d+1} \alpha_{n,d}, \alpha_{n,d}]}\) where \({\alpha_{n,d}={n+d-1\choose d}}\) .     

The Integral Closure of Ideals in ℂ{x,y}

Abstract

We give some techniques to determine the ideal K _I generated by the monomials x ^{k ₁} y ^{k ₂} belonging to the integral closure ī of an ideal I ? ?{x, y}. We also give a sufficient condition for a weighted homogeneous ideal I ? ?{x, y} to satisfy the relation ī = I + K _I .     

Betti Numbers of Path Ideals of Trees

Let R = k[x₁,…, x_n], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by I_t(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms.     

Gotzmann Edge Ideals

Let P = 𝕜[x ₁,…, x _n ] be the polynomial ring in n variables. A homogeneous ideal I ? P generated in degree d is called Gotzmann if it has the smallest possible Hilbert function out of all homogeneous ideals with the same dimension in degree d. The edge ideal of a simple graph G on vertices x ₁,…, x _n is the quadratic square-free monomial ideal generated by all x _i x _j where {x _i , x _j } is an edge of G. The only edge ideals that are Gotzmann are those edge ideals corresponding to star graphs.     

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.     

Prime T-Ideals in Polynomial and Power Series Rings Over a Pseudo-Valuation Domain

Let R be an m-dimensional pseudo-valuation domain with residue field k, let V be the associated valuation domain with residue field K, and let k ₀ be the maximal separable extension of k in K. We compute the t-dimension of polynomial and power series rings over R. It is easy to see that t-dim R[x ₁,…, x _n ] = 2 if m = 1 and K is transcendental over k, but equals m otherwise, and that t-dim R[[x ₁,…, x _n ]] = ∞ if R is a nonSFT-ring. When R is an SFT-ring, we also show that: (1) t-dim R[[x]] = m; (2) t-dim R[[x ₁,…, x _n ]] = 2m ? 1, if n ≥ 2, K has finite exponent over k ₀, and [k ₀: k] < ∞; (3) t-dim R[[x ₁,…, x _n ]] = 2m, otherwise.     

Depth of Initial Ideals of Normal Edge Rings

Let G be a finite graph on the vertex set [d] = {1,…, d} with the edges e ₁,…, e _n and K[t] = K[t ₁,…, t _d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t ^e  = t _i t _j such that e = {i, j} is an edge of G. Let K[x] = K[x ₁,…, x _n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: K[x] → K[G] by setting π(x _i ) = t ^{e _i} for i = 1,…, n. The toric ideal I _G of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <_rev on K[x] and a lexicographic order <_lex on K[x] such that (i) K[G] is normal with Krull-dim K[G] = d, (ii) depth K[x]/in_<rev (I _G ) = f and K[x]/in_<lex (I _G ) is Cohen–Macaulay, where in_<rev (I _G ) (resp., in_<lex (I _G )) is the initial ideal of I _G with respect to <_rev (resp., <_lex) and where depth K[x]/in_<rev (I _G ) is the depth of K[x]/in_<rev (I _G ).     

Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I ²) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ_?(J) = ? (resp., φ₀(J) = 0, φ₂(J) = J ²), a φ_?-prime ideal (resp., φ₀-prime ideal, φ₂-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.     

First Nonlinear Syzygies of Ideals Associated to Graphs

Consider an ideal I ? K[x ₁,…, x _n ], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number β _{s, s+3}. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.     

The Hilbert–Kunz Function in Graded Dimension Two

Let R denote a two-dimensional normal standard-graded K-domain over the algebraic closure K of a finite field of characteristic p, and let I ? R denote a homogeneous R ₊-primary ideal. We prove that the Hilbert–Kunz function of I has the form ? (q) = e _HK (I)q ² + γ(q) with rational Hilbert–Kunz multiplicity e _HK (I) and an eventually periodic function γ(q).     

On Generalizations of Prime Ideals

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {?} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n ? 1, n) ? φ-prime if, whenever a ₁, a ₂, ?, a _n  ∈ R and a ₁ a ₂?a _n  ∈ I?φ(I), the product of (n ? 1) of the a _i 's is in I. In this article, we study (n ? 1, n) ? φ-prime ideals (n ≥ 2). A number of results concerning (n ? 1, n) ? φ-prime ideals and examples of (n ? 1, n) ? φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n ? 1, n) ? φ-prime, are characterized.     

On Pseudo-Almost Valuation Domains

Let R be an integral domain with quotient field K and integral closure R ^′. Anderson and Zafrullah called R an “almost valuation domain” if for every nonzero x ∈ K, there is a positive integer n such that either x ⁿ  ∈ R or x ^?n  ∈ R. In this article, we introduce a new closely related class of integral domains. We define a prime ideal P of R to be a “pseudo-strongly prime ideal” if, whenever x, y ∈ K and xyP ? P, then there is a positive integer m ≥ 1 such that either x ^m  ∈ R or y ^m P ? P. If each prime ideal of R is a pseudo-strongly prime ideal, then R is called a “pseudo-almost valuation domain” (PAVD). We show that the class of valuation domains, the class of pseudo-valuation domains, the class of almost valuation domains, and the class of almost pseudo-valuation domains are properly contained in the class of pseudo-almost valuation domains; also we show that the class of pseudo-almost valuation domains is properly contained in the class of quasilocal domains with linearly ordered prime ideals. Among the properties of PAVDs, we show that an integral domain R is a PAVD if and only if for every nonzero x ∈ K, there is a positive integer n ≥ 1 such that either x ⁿ  ∈ R or ax ^?n  ∈ R for every nonunit a ∈ R. We show that pseudo-almost valuation domains are precisely the pullbacks of almost valuation domains, we characterize pseudo-almost valuation domains of the form D + M, and we use this characterization to construct PAVDs that are not almost valuation domains. We show that if R is a Noetherian PAVD, then R has Krull dimension at most one and R ^′ is a valuation domain; we show that every overring of a PAVD R is a PAVD iff R ^′ is a valuation domain and every integral overring of R is a PAVD.