Juergen Herzog, Dorin Popescu, Ngo Viet Trung
In this paper we study homological properties of the Rees ring R of the graded maximal ideal of a standard graded k-algebra A. In particular we are interested the comparison of the depth and regularity of A and R.
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In this paper we consider Hibi rings and Rees rings attached to a poset. We classify the ideal lattices of posets whose Hibi relations are indispensable and the ideal lattices of posets whose Hibi relations form a quadratic Groebner basis with respect to the rank lexicographic order. Similar classifications are obtained for Rees rings of Hibi ideals.
In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) onto it, where M is the direct sum of powers of m. We compute a Groebner basis of the ideal defining R(M). It turns out that the normal domain R(M) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well.
The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.
We present algorithms for the computation of the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring of equigenerated $\mm$-primary ideals in two variables. Applying these algorithms, we find a counter-example to a conjecture of Eisenbud and Ulrich which states that these regularities are equal.
Let I be a divisorial ideal of a strongly F-regular ring R. Watanabe asked if the symbolic Rees algebra R_s(I) is Cohen-Macaulay whenever it is Noetherian. We develop the notion of multi-symbolic Rees algebras, and use this to show that R_s(I) is indeed Cohen-Macaulay whenever a certain auxiliary ring is finitely generated over R.
Almost Complete Intersections: Regularity of powers and Rees algebra
针对几乎完全交理想，给出满足特定条件的分次几乎完全交理想幂的正则度精确公式，研究相关里斯代数的极小双分次自由分解及对角线的Cohen-Macaulay性、Koszul性等性质。
Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.
In this article, we compute the regularity of Rees algebra of binomial edge ideals of closed graphs. We obtain a lower bound for the regularity of Rees algebra of binomial edge ideals. We also study some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. We obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.
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。
Regularity of normal Rees algebras of edge ideals of graphs
本研究针对图的边理想的正规Rees代数，分类了所有满足代数正规且正则性与图的匹配数相等的图，完善了交换代数与图论交叉领域的相关结论。
We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals of maximal minors have linear resolutions. Our main theoretical contribution consists of introducing two new classes of ideals named after Cartwright and Sturmfels, and proving that they are closed under multigraded hyperplane sections. The gins of the ideals that we study enjoy special properties.
In this paper we show that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology of the Rees algebra and the associated graded ring for the good filtrations case. We obtain relationships between the Castelnuovo-Mumford regularity of the fiber cone, associated graded ring, Rees algebra and reduction number for the good filtrations case.
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.
Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gröbner basis of its defining equations; interestingly, both of them are described in terms of the combinatorics of $G$. We apply these ideas to study the regularity of the powers of $I$. For any $s \ge \text{match}(G)+\lvert E(G) \rvert +1$ we prove that $\text{reg}(I^{s+1})=\text{reg}(I^s)+2$.
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…
Recently, Peeva and the second author constructed irreducible projective varieties with regularity much larger than their degree, yielding counterexamples to the Eisenbud-Goto Conjecture. Their construction involved two new ideas: Rees-like algebras and step-by-step homogenization. Yet, all of these varieties are singular and the nature of the geometry of these projective varieties was left open. The purpose of this paper is to study the singularities inherent in this process. We compute the codimension of the singular locus of an arbitrary Rees-like algebra over a polynomial ring. We then sho…
The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and for the Artin-Rees numbers.
We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the Ratliff-Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo-Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjectu…
This Note provides first a generalization of the stabilization result of Eisenbud and Ulrich for the regularity of powers of a m-primary ideal to the case of ideals that are not generated in a single degree. We then partially extend our previous results expressing this stabilization degree in term of the regularity of a specific graded strand of the Rees ring : The natural extension of the statement holds at least if the stabilization index or the regularity is greater than the number of variables. In any case a precise comparison is given.
Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbol…
This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded ring $R$ over a general base ring. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers of an homogeneous ideal $I$ of $R$ is eventually a linear function in $v$. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple characterization of …
Some classes of sequences of Linear Type
针对分次环与齐次理想的线性型问题，通过分析$y$-正则性、$d$-序列与线性型理想的关系，证明$d$-序列理想是线性型理想的小子集，确定一类$d$-序列任意幂次生成格罗布纳线性型理想，并指出特定代数中弱$d$-序列即$d$-序列。
Let $(A,\m)$ be a Noetherian local ring, let $M$ be a finitely generated \CM $A$-module of dimension $r \geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = \bigoplus_{n\geq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $\R$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extend…
We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing…
Hochster and Huneke showed that the property of F-regularity deforms for Gorenstein rings, i.e., if (R,m) is a Gorenstein local ring such that R/tR is F-regular for some nonzerodivisor t in m, then R is F-regular. This result was later extended to the case of Q-Gorenstein rings by Smith (for rings of characteristic zero) and Aberbach, Katzman, and MacCrimmon (for rings of positive characteristic). We investigate the deformation of strong F-regularity using an anti-canonical cover of R, i.e., a symbolic Rees algebra S = R + It + I^(2)t^2 + ..., where I is an inverse for the canonical module in …
Let S=K[x_1,..., x_n], let A,B be finitely generated graded S-modules, and let m=(x_1,...,x_n). We give bounds for the Castelnuovo-Mumford regularity of the local cohomology of Tor_i(A,B) under the assumption that the Krull dimension of Tor_1(A,B) is at most 1. We apply the results to syzygies, Groebner bases, products and powers of ideals, and to the relationship of the Rees and Symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first (roughly) (n-1)/2 steps of the resolution of I are linear, then I^…
Dao's question on the asymptotic behaviour of fullness
本文针对Dao提出的局部环中理想fullness性质渐近行为控制问题，利用约化数、Ratliff-Rush闭包和Rees代数正则性等工具，建立了相关不变量的上界及部分精确公式。
In this paper we give a generalization of a result of Herzog, Hibi, and Zheng providing an upper bound for regularity of powers of an ideal. As the main result of the paper, we give a simple criterion in terms of Rees algebra of a given ideal to show that high enough powers of this ideal have linear resolution. We apply the criterion to two important ideals $J,J_{1}$ for which we show that $J^{k},$ and $J_{1}^{k}$ have linear resolution if and only if $k\neq 2.$ The procedures we include in this work is encoded in computer algebra package CoCoA.
Let $A$ be the homogeneous coordinate ring of a rational normal scroll. The ring $A$ is equal to the quotient of a polynomial ring $S$ by the ideal generated by the two by two minors of a scroll matrix $ψ$ with two rows and $\ell$ catalecticant blocks. The class group of $A$ is cyclic, and is infinite provided $\ell$ is at least two. One generator of the class group is $[J]$, where $J$ is the ideal of $A$ generated by the entries of the first column of $ψ$. The positive powers of $J$ are well-understood, in the sense that the $n^{\text{th}}$ ordinary power, the $n^{th}$ symmetric power, and th…
Powers of generalized binomial edge ideals of path graphs
本文研究了路径图广义二项式边理想的幂，显式计算了其正则性与深度极限，证明普通幂与符号幂一致，并利用 Sagbi 基理论给出了 Rees 代数及特殊纤维环正则性的精确公式。
We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism which resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung which states that the normalized $a$-invariants and the Castelnuovo-Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic…
In this article we analyze the implicitization problem of the image of a rational map $ϕ: X --> P^n$, with $T$ a toric variety of dimension $n-1$ defined by its Cox ring $R$. Let $I:=(f_0,...,f_n)$ be $n+1$ homogeneous elements of $R$. We blow-up the base locus of $ϕ$, $V(I)$, and we approximate the Rees algebra $Rees_R(I)$ of this blow-up by the symmetric algebra $Sym_R(I)$. We provide under suitable assumptions, resolutions $\Z.$ for $Sym_R(I)$ graded by the torus-invariant divisor group of $X$, $Cl(X)$, such that the determinant of a graded strand, $\det((\Z.)_μ)$, gives a multiple of the i…
Asymptotic regularity of graded families of ideals
本研究证明齐次理想梯度族渐近正则性在多种条件下存在，给出组合解释，否定若干极限存在性，并构造显式 Gröbner 基。
Symbolic powers of the generic linkage of maximal minors
本文研究了最大余子式生成的理想J的符号幂与普通幂相等性，通过格罗布纳退化描述生成元的首项，并证明关联格雷德环的格雷索斯性质及正特征下的Rees代数F-理性性。
Let $(R,\mathfrak m)$ be a local ring and $I, J$ two arbitrary ideals of $R$. Let $\operatorname{gr}_J(R/I)$ denote the associated ring of $R/I$ with respect to $J$, which corresponds to the normal cone in geometry. The main result of this paper shows that if $I = (f_1,...,f_r)$, where $f_1,...,f_r$ is a $J$-filter regular sequence, there exists a number $N$ such that if $f_i' \equiv f_i \mod J^N$ and $I' = (f_1',...,f_r')$, then $\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')$. If $J$ is an $\mathfrak m$-primary ideal, this result implies a long standing conjecture of Srinivas and T…
The homological shift algebra of a monomial ideal
本文引入单项式理想的同调移位代数，证明线性幂单项式理想的该代数是Rees代数上有限生成双分次模且不变量渐近良好，确定了若干类理想使其同调移位代数对大$k$有线性分解，并证明线性幂理想的同调移位代数对大$k$是Golod的。
Dao numbers and the asymptotic behaviour of fullness
本文研究Noether局部环或标准分次代数中理想的Dao数，借助Rees代数模的Castelnuovo-Mumford正则度给出一般界，证明Koszul代数下的界，并对单项式理想给出组合界。
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