Francesc Planas-Vilanova
The purpose of this paper is to prove the following theorem of uniform Artin-Rees properties: Let $A$ be an excellent (in fact J-2) ring and let $N\subset M$ be two finitely generated $A$-modules such that ${\rm dim}(M/N)\leq 1$. Then there exists an integer $s\geq 1$ such that, for all integers $n\geq s$ and for all ideals $I$ of $A$, $I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)$.
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We interpret the Artin-Rees lemma and the Izumi theorem in term of Artin function and we obtain a stable version of the Artin-Rees lemma. We present different applications of these interpretations. First we show that the Artin function of $X_1X_2-X_3X_4$, as a polynomial in the ring of power series in more than three variables, is not bounded by an affine function. Then we prove that the Artin functions of a class of polynomials are bounded by affine functions and we use this to compute approximated integral closures of ideals.
In this note we give a simple proof of the fact that local rings of dimension one have the strong uniform Artin-Rees property. Moreover, we give two examples of rings of dimension two where the property fails.
Let X be a set of smooth points in P^2, and I = \oplus_{t >= d} I_t the defining ideal of X. In this paper, we give a set of defining equations for the Rees algebra R(I_{d+1}) of the ideal generated by I_{d+1}. This study give information to completely answer questions on the defining ideals of projective embeddings of the blowup of P^2 along the points in X, which has been studied by many authors. In this paper, we also study the asymptotic behaviour of the Rees algebras R(I_t) of the ideal generated by I_t, as t gets large.
The uniform Artin-Rees lemma has been proved by C. Huneke using algebraic methods. We give a new proof for this result in the analytic setting using residue calculus. We also have to introduce a type of product of complexes of vector bundles, which may be applicable in the solution of other division problems with respect to product ideals.
Let $(R,m)$ be a local Noetherian ring, let $M$ be a finitely generated $R$-module and let $(F_{\bullet},\partial_{\bullet})$ be a free resolution of $M$. We find a uniform bound $h$ such that the Artin-Rees containment $I^n F_i\cap Im \, \partial_{i+1} \subseteq I^{n-h} Im \, \partial_{i+1}$ holds for all integers $i\ge d$, for all integers $n\ge h$, and for all ideals $I$ of $R$. In fact, we show that a considerably stronger statement holds. The uniform bound $h$ holds for all ideals and all resolutions of $d$th syzygy modules. In order to prove our statements, we introduce the concept of Ko…
For an ideal I in a regular local ring (R,m)$ with residue class field K = R/m or a standard graded K-algebra R we show that for k >> 0 --> the Artin--Rees number of the syzygy modules of I^k as submodules of the free modules from a free resolution is constant, and thereby present the Artin-Rees number as a proper replacement of regularity in the local situation, --> the ring R/I^k is Golod, its Poincer{é}-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of work of Kodiyalam and Cut…
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers
本文引入 Rees 模的筛型与中等 Artin-Rees 数，建立两者间的紧密联系，并通过实例阐明其意义与相互关联，为度量 Rees 模复杂度提供新工具。
On Rees algebras of ideals and modules with weak residual conditions
本文研究了在弱生成条件下（仅至余维数 $s-1$ 成立）模的 Rees 环 $\mathcal{R}(E)$，利用逐次逼近法提供了定义该代数的理想生成集，并在 $\mathrm{rank} \\, E=1$ 时恢复并扩展了关于 grade 二完美理想的 Rees 代数的已知结果。
On Rees algebras of linearly presented ideals and modules
本文在更弱的局部生成元数目条件下，确定了线性呈现的完美高度二理想的 Rees 代数的定义理想，并利用 Bourbaki 理想将结果推广到投影维数为一的线性呈现模。
The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number
本文证明了基座度数为d且Sperner数至多为d+1的artinian Gorenstein代数均满足弱Lefschetz性质，且与余维数无关，并指出该界限在一般情况下是严格的。
Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four
本文针对余维数4、正则度不超过6的Artinian Gorenstein环，研究其极小自由分解与弱/强Lefschetz性质及Jordan型之间的关系。
We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.
On Rees algebras of 2-determinantal ideals
本文证明了2×n线性形式矩阵子式理想的Rees代数及其特殊纤维环具有Cohen-Macaulay和Koszul性质，通过分析Hilbert概形分层并结合Groebner退化揭示了其二次代数特征。
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of…
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen-Macaulayness and Castelnuovo-Mumford regularity.
We prove that the Hilbert functions of codimension four graded Gorenstein Artin algebras R/I are unimodal provided I has a minimal generator in degree less than five. It is an open question whether all Gorenstein h-vectors in codimension four are unimodal. In this paper, we prove that Hilbert functions of all artinian codimension four Gorenstein algebras starting with (1,4,10, 20, h_4,...), where h_4\leq 34 are unimodal. Combining this with the previously known results, we obtain that all Gorenstein h-vectors (1, h_1, h_2,h_3, h_4, ....) are unimodal if h_4\leq 3.
Uniformity in nonreduced rings via Noetherian operators
本文通过Noetherian微分算子证明Artin-Rees引理的微分版本，进而得到非约化环的若干均匀性结果。
On the weak Lefschetz property of Artinian Gorenstein algebras of codimension three in arbitrary characteristic
本文将特征零代数闭域上三维余维Artinian Gorenstein代数的弱Lefschetz性质结论推广到任意特征的无限域，同时证明了相关的公因子结论。
We find a sufficient condition that $\H$ is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function $\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$ cannot be level if $h_d\le 2d+3$, and that there exists a level O-sequence of codimension 3 of type $\H$ for $h_d \ge 2d+k$ for $k\ge 4$. Furthermore, we show that $\H$ is not level if $β_{1,d+2}(I^{\rm lex})=β_{2,d+2}(I^{\rm lex})$, and also prove that any codimension 3 Artinian graded algebra $A=R/I$ cannot be level if $β_{1,d+2}(\Gin(I))=β_{2,d+2}(\Gin(I))$. In this case, the H…
A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J \subset S $ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal $I \subset S$ is said to be $\mathfrak m$-full if $\mathfrak mI:y=I$ for some $y \in \mathfrak m$, where $\mathfrak m$ is the graded maximal ideal of $S$. It was proved by one of the authors that $\mathfrak m$-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals whic…
Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann Property for Graded Oriented Artinian Gorenstein Algebras in Codimension Two
该研究引入$i$-洛伦兹多项式，证明其为余维2有向阿廷戈伦斯坦代数的Macaulay对偶生成元，建立其与全非负Toeplitz矩阵的一一对应，推导相关推论并给出新的Toeplitz矩阵惠特尼型定理。
New families of Artinian Gorenstein algebras with the weak Lefschetz property
本文构造了任意余维数且具有二项式 Macaulay 对偶生成元的 Artinian Gorenstein 代数新族，并证明其满足弱或强 Lefschetz 性质，是相关姊妹篇的补充。
Artinian Gorenstein algebras with binomial Macaulay dual generator
本文系统研究具有二项式Macaulay对偶生成元的Artinian Gorenstein代数，在余维数3证明其满足强Lefschetz性质并给出完全交刻画，同时为任意余维数建立最优的弱Lefschetz性质充分条件。
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Apéry set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I…
Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence we describe a family of non-monomial complete intersections that alwa…
The goal of this paper is the fine structure of the ideals in the title, with emphasis on the properties of the associated Rees algebra and the special fiber. The watershed between the present approach and some of the previous work in the literature is that here one does not assume that the ideals in question satisfy the common generic properties. One exception is a recent work of N. P. H. Lan which inspired the present work. Here we recover and extend his work. We strongly focus on the behavior of the ideals of minors of the corresponding so-called Hilbert--Burch matrix and on conjugation fea…
We deal with a generalization of a Theorem of P. Gordan and M. Noether on hypersurfaces with vanishing (first) Hessian. We prove that for any given $N\geq 3$, $d \geq 3$ and $2\leq k < \frac{d}{2}$ there are infinitely many irreducible hypersurfaces $X = V(f)\subset \mathbb{P}^N$, of degree $\operatorname{deg}(f)=d$, not cones and such that their Hessian determinant of order $k$, $\operatorname{hess}^k_f$, vanishes identically. The vanishing of higher Hessians is closely related with the Strong (or Weak) Lefschetz property for standard graded Artinian Gorenstein algebra, as pointed out firstly…
Let A = bigoplus_{i >= 0} A_i be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element ell of degree 1 such that the multiplication times ell : A_i --> A_{i+1} has maximal rank, for every i, and A has the Strong Lefschetz property if times ell^d : A_i --> A_{i+d} has maximal rank for every i and d. The main results obtained in this paper are the following. 1) EVERY height three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank two vector bundles on P^2 and the Grauert-Mulich theorem.) 2) W…
Let $R=\oplus_{i\geq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\dim R_2\leq 3$ and that $K$ is algebraically closed of characteristic $\neq 2$. We show that $R$ is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.
In this paper we study the Hilbert scheme, Hilb(P), of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in P^4 and 3-folds in P^5, and the Hilbert scheme stratification H_{c} of constant cohomology. For every (X) in Hilb(P) we define a number δ_X in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + δ_X - \dim_{(X)} H_{c} and 1 + δ_X - \dim T_{c} are CI-biliaison invariants where T_{c} is the tangent space of H_{c} at (X). As a corollary we get a formula for the dimension of any generically smooth component of H…
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 the generic initial ideal $\mathrm{gin}(I)$ of $I$ under the reverse lexicographic order. Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of $\mathrm{gin}(I)$ are determined just by the 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 codimens…
Let $R = k[w, x_1,..., x_n]/I$ be a graded Gorenstein Artin algebra . Then $I = \ann F$ for some $F$ in the divided power algebra $k_{DP}[W, X_1,..., X_n]$. If $RI_2$ is a height one idealgenerated by $n$ quadrics, then $I_2 \subset (w)$ after a possible change of variables. Let $J = I \cap k[x_1,..., x_n]$. Then $μ(I) \le μ(J)+n+1$ and $I$ is said to be generic if $μ(I) = μ(J) + n+1$. In this article we prove necessary conditions, in terms of $F$, for an ideal to be generic. With some extra assumptions on the exponents of terms of $F$, we obtain a characterization for $I = \ann F$ to be gener…
Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $Γ. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $Γ$ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, th…
Our purpose is to study the cohomological properties of the Rees algebras of a class of ideals generated by quadrics. For all such ideals $I\subset R = K[x,y,z]$ we give the precise value of depth $R[It]$ and decide whether the corresponding rational maps are birational. In the case of dimension $d \geq 3$, when $K=\mathbb{R}$, we give structure theorems for all ideals of codimension $d$ minimally generated by ${{d+1}\choose{2}}-1$ quadrics. For arbitrary fields $K$, we prove a polarized version.
We study a number of conditions on the Hilbert function of a level artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP.
We introduce a method for studying the Lefschetz properties for $k[x,y]$-modules based on the Lindström-Gessel-Viennot Lemma. In particular, we prove that certain modules over Artinian Clements-Lindström rings in characteristic zero have the strong Lefschetz property. In particular, we show that every homogeneous idea in a Clements-Lindström ring of embedding dimension two has the strong Lefschetz property. As an application, we study the strong Lefschetz property of type two monomial ideals of codimension three.
On lengths of modules over certain Artinian complete intersections
本文研究一类余维数c≥2的Artinian完全交环上有限生成模的长度性质，证明存在仅依赖于环的整数整除满足支撑簇条件的模的长度，并应用于单项式完全交环上有界Betti数模的长度素因子分析。
The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
本文证明了一个广义Briançon-Skoda型包含关系，并由此推出伪有理奇点（如正则环）和Du Bois奇点下的完整Briançon-Skoda包含，同时应用于证明有限维拟优秀环的一致Artin-Rees定理和一致Briançon-Skoda定理。
Ext functors, support varieties and Hilbert polynomials over complete intersection rings
该研究在完全交环框架下，证明了Ext函子相关多项式型函数的渐近次数为常数，揭示其与理想、环及模的支撑簇的不变性关系，推导了正则性有界条件并给出合冲Artin-Rees性质的应用。
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