The Rees Algebra Package in Macaulay2

$F$-rationality of Rees algebras

In this paper, we study the $F$-rationality of the Rees algebra and the extended Rees algebra of $\mathfrak{m}$-primary ideals in excellent local rings $(R, \mathfrak{m})$ of prime characteristic. We partially answer some conjectures and questions raised by N. Hara, K.-i. Watanabe and K.-i. Yoshida (J. Algebra, pp.153--190, vol 247, 2002).

A formula for the multiplicity of the multi-graded extended Rees algebra

Our aim in this paper is to obtain a formula for the multiplicity of the maximal homogeneous ideal of the multi-graded extended Rees algebra. This formula generalizes the one obtained in [8] by Katz and Verma for the ordinary extended Rees algebra.

Cohen-Macaulayness of Rees Algebras of Modules

We provide the sufficient conditions for Rees algebras of modules to be Cohen-Macaulay, which has been proven in the case of Rees algebras of ideals by Johnson-Ulrich and Goto-Nakamura-Nishida. As it turns out the generalization from ideals to modules is not just a routine generalization, but requires a great deal of technical development. We use the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos to obtain the Cohen-Macaulayness of Rees Algebras of modules.

Rees algebra and special fiber ring of binomial edge ideals of closed 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.

The equations of Rees algebras of height three Gorenstein ideals in hypersurface rings

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.

Multi-symbolic Rees algebras and strong F-regularity

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.

Rees 代数的非 $F$-有理轨迹

The non-$F$-rational locus of Rees algebras

本文研究 Rees 代数的非 $F$-有理轨迹，通过刻画其参数测试子模给出该轨迹的描述。

Multi-Rees Algebras and Toric Dynamical Systems

This paper explores the relation between multi-Rees algebras and ideals that arise in the study of toric dynamical systems from the theory of chemical reaction networks.

Gröbner bases and equations of the multi-Rees algebras

In this paper we describe the equations defining the multi-Rees algebra $R[I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where $R$ is a Noetherian ring and the ideals are generated by subsets of a fixed weak regular sequence.

Groebner bases and regularity of Rees algebras

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.

Mixed multiplicities and the multiplicity of rees modules of reductions (published in j. Algebra appl.)

This paper shows that mixed multiplicities and the multiplicity of Rees modules of good filtrations and that of their reductions are the same. As an application of this result, we obtain interesting results on mixed multiplicities and the multiplicity of Rees modules.

Gorensteinness in Rees algebras of powers of parameter ideals

This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the $d$-th power of parameter ideals in certain Noetherian local rings of dimension $d\ge 2$. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra $\mathcal{R}(\mathfrak{q}^d)=\oplus_{i\ge 0}\mathfrak{q}^{di}$ is Gorenstein for every parameter ideal $\mathfrak{q}$ that is a reduction of the maximal ideal in a $d$-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.

Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebras

The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A,m) in terms of local cohomology modules of Rees algebras of I and J. The cohomology of a variation of the Kirby-Mehran complex for bigraded Rees algebras is studied which is used to characterize the Cohen-Macaulay property of bigraded Rees algebra of I and J fo…

On the symmetric and Rees algebras of certain determinantal ideals

The purpose of this paper is to give an elementary proof to the theorem due to Avramov on certain determinantal ideals of linear type.

A characterization of the Arf property for quadratic quotients of the Rees algebra

We provide a characterization of the Arf property in both the numerical duplication of a numerical semigroup and in a member of a family of quotients of the Rees algebra studied in arXiv:1403.4200 [math.AC]

Rees algebras of sparse determinantal ideals

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.

Matching numbers and the regularity of the Rees algebra of an edge ideal

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.

Rees Algebras of Truncations of Complete Intersections

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.

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

On the computation of Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring

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.

The Rees Algebra for Certain Monomial Curves

In this article, we find the equations defining the Rees algebra for certain Monomial Curves explicitly and use them to prove that the blowup scheme is not smooth. This proves a conjecture of Francia in affirmative, which says that a dimension one prime in a regular local ring is a complete intersection if it has a smooth blowup.

与滤链相关的拟Gorenstein扩张Rees代数

Quasi-Gorenstein extended Rees algebras associated with filtrations

本文研究诺特局部环上Hilbert滤链对应的扩张Rees代数的拟Gorenstein性质，利用局部上同调模的Matlis对偶的Cohen-Macaulay性给出变形条件，并借助局部上同调长度给出拟Gorenstein性质的刻画。

Rees algebras of ideals of star configurations

In this article we study the defining ideal of Rees algebras of ideals of star configurations. We characterize when these ideals are of linear type and provide sufficient conditions for them to be of fiber type. In the case of star configurations of height two, we give a full description of the defining ideal of the Rees algebra, by explicitly identifying a minimal generating set.

Local cohomology modules of bigraded Rees algebras

Formulas are obtained in terms of complete reductions for the bigraded components of local cohomology modules of bigraded Rees algebras of 0-dimensional ideals in 2-dimensional Cohen-Macaulay local rings. As a consequence, cohomological expressions for the coefficients of the Bhattacharya polynomial of such ideals are obtained.

Local cohomology of multi-Rees algebras, Joint reduction numbers and product of complete ideals

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of Reid-Roberts-Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

The almost Gorenstein Rees algebras over two-dimensional regular local rings

Let $(R,\mathfrak{m})$ be a two-dimensional regular local ring with infinite residue class field. Then the Rees algebra $\mathcal{R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring in the sense of Goto-Takahashi-Taniguchi for every $\mathfrak{m}$-primary integrally closed ideal $I$ in $R$.

Minimal generators of the defining ideal of the Rees Algebra associated to a rational plane parameterization with μ=2

We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated to the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

通过Rees代数研究的极良覆盖图

Very well-covered graphs via the Rees algebra

该研究针对Cohen-Macaulay极良覆盖图，利用与其覆盖理想关联的Rees代数，深入分析了这类图覆盖理想的代数性质，为相关代数图论研究提供了新的分析视角。

Rees algebras, Monomial Subrings and Linear Optimization Problems

In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut pr…

A family of quotients of the Rees algebra

A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.

关于圈的$t$-路理想的里斯代数

On the Rees algebras of $t$-path ideals of cycles

本文研究$n$-圈的$t$-路理想$I_t(C_n)$的里斯代数性质，重点分析该理想为纤维型的情形。

Cohen-Macaulay fiber cones and defining ideal of Rees algebras of modules

Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the Cohen-Macaulay property of fiber cones of modules and to study the defining ideal of Rees algebras. This is possible as long as the Rees algebra of a given module $E$ is a deformation of the Rees algebra of a generic Bourbaki ideal $I$ of $E$. Our main technical result provides a deformation condition that in fact extends the applicability of generic Bourbaki ideals to…

Sequentially Cohen-Macaulay Rees algebras

This paper studies the question of when the Rees algebras associated to arbitrary filtration of ideals are sequentially Cohen-Macaulay. Although this problem has been already investigated by N. T. Cuong, S. Goto and H. L. Truong, their situation is quite a bit of restricted, so we are eager to try the generalization of their results.

The depth of the Rees algebra of three general binary forms

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that certain large matrices whose entries are quadratic forms have maximal rank. One also conjectures a shorter result that implies the main theorem of the paper.

Extremal Rees Algebras

We study almost complete intersections ideals whose Rees algebras are extremal in the sense that some of their fundamental metrics---depth or relation type---have maximal or minimal values in the class. The focus is on those ideals that lead to almost Cohen--Macaulay algebras and our treatment is wholly concentrated on the nonlinear relations of the algebras. Several classes of such algebras are presented, some of a combinatorial origin. We offer a different prism to look at these questions with accompanying techniques. The main results are effective methods to calculate the invariants of thes…

The Rees Algebra of Parametric Curves via liftings

We study the defining equations of the Rees algebra of ideals arising from curve parametrizations in the plane and in rational normal scrolls, inspired by the work of Madsen and Kustin, Polini and Ulrich. The curves are related by work of Bernardi, Gimigliano and Ida, and we use this framework to relate the defining equations.

On the Cohen-Macaulay property of the Rees algebra of the module of differentials

Let $R$ be an algebra essentially of finite type over a field $k$ and let $Ω_k(R)$ be its module of Kähler differentials over $k$. If $R$ is a homogeneous complete intersection and $\mathrm{char}(k)=0$, we prove that $Ω_k(R)$ is of linear type whenever its Rees algebra is Cohen-Macaulay and locally at every homogeneous prime $\mathfrak{p}$ the embedding dimension of $R_{\mathfrak{p}}$ is at most twice its dimension.

单位区间行列式面理想的 Rees 代数

Rees Algebras of Unit Interval Determinantal Facet Ideals

本文利用 SAGBI 基技术，为单位区间行列式面理想的 Rees 代数与特殊纤维环构造 Gröbner 基，证明其纤维型、Koszul 性及正规 Cohen-Macaulay 与有理奇点性质。

符号Rees代数上投射模中幺模元素的存在性

Existence of unimodular element in a projective module over symbolic Rees algebras

本文证明在维数为d的符号Rees代数上，秩至少为d的有限生成投射模必含幺模元素，从而将Serre经典结果推广到这类不必诺特的代数。

Rees Algebras and Almost Linearly Presented Ideals

Consider a grade 2 perfect ideal $I$ in $R=k[x_1,\cdots,x_d]$ which is generated by forms of the same degree. Assume that the presentation matrix $φ$ is almost linear, that is, all but the last column of $φ$ consist of entries which are linear. For such ideals, we find explicit forms of the defining ideal of the Rees algebra $\mathcal{R}(I)$. We also introduce the notion of iterated Jacobian duals.

Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Let $I$ be an ideal whose symbolic Rees algebra is Noetherian. For $m \geq 1$, the $m$-th symbolic defect, sdefect$(I,m)$, of $I$ is defined to be the minimal number of generators of the module $\frac{I^{(m)}}{I^m}$. We prove that sdefect$(I,m)$ is eventually quasi-polynomial as a function in $m$. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra…

素理想的环及其Rees代数

Links of prime ideals and their Rees algebras

在以前的文件中，我们展示了有点令人惊讶的性质，大多数直接联系的素理想在Gorenstein环是equimultiple理想减少数$1$。这导致了建设大家庭的科恩-麦考利里斯代数。本文的第一个目标是将这个结果推广到任意的Cohen-Macaulay环。证明的方法改变了，因为人们不能如此严重地依赖于联系理论。然后，我们研究的Rees代数的结构，这些链接，更具体地说，我们描述了他们的典型模块足够详细，能够表征自我-链接素理想。在最后一节中，建立了此类理想类的重数估计。

New algebraic properties of quadratic quotients of the Rees algebra

We study some properties of a family of rings $R(I)_{a,b}$ that are obtained as quotients of the Rees algebra associated with a ring $R$ and an ideal $I$. In particular, we give a complete description of the spectrum of every member of the family and describe the localizations at a prime ideal. Consequently, we are able to characterize the Cohen-Macaulay and Gorenstein properties, generalizing known results stated in the local case. Moreover, we study when $R(I)_{a,b}$ is an integral domain, reduced, quasi-Gorenstein, or satisfies Serre's conditions.

Quasi-Gorensteinness of extended Rees algebras

Let $R$ be a Noetherian local ring and $I$ an $R$-ideal. It is well-known that if the associated graded ring $\gr_I(R)$ is Cohen-Macaulay (Gorenstein), then so is $R$, but the converse is not true in general. In this paper we investigate the Cohen-Macaulayness and Gorensteinness of the associated graded ring $\gr_I(R)$ under the hypothesis of the extended Rees algebra $R[It,t^{-1}]$ is quasi-Gorenstein or the associated graded ring $\gr_I(R)$ is a domain.

有限生成饱和多重Rees代数

Finitely generated saturated multi-Rees algebras

本文研究了饱和多重Rees代数的有限生成问题及相关长度函数的渐近行为，并在excellent local domains、二维local UFDs及多项式环商等设定下证明了有限生成性及长度函数的多项式或准多项式性质。

A theorem about Cremona maps and symbolic Rees algebras

This work is about the structure of the symbolic Rees algebra of the base ideal of a Cremona map. We give sufficient conditions under which this algebra has the "expected form" in some sense. The main theorem in this regard seemingly covers all previous results on the subject so far. The proof relies heavily on a criterion of birationality and the use of the so-called inversion factor of a Cremona map. One adds a pretty long selection of examples of plane and space Cremona maps tested against the conditions of the theorem, with special emphasis on Cohen--Macaulay base ideals.

Singularities of Rees-like Algebras

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…

几乎完全交：幂的正则度与里斯代数

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。

Regularity, Rees algebra and Betti numbers of certain cover ideals

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.

Multi-Rees algebras on principal ideal rings

When $R$ is a Noetherian ring and we have a family of ideals in which every ideal contains at least one nonzero divisor, then it is already known that the defining ideal of the multi-Rees algebra of these ideals is equal to a saturated ideal. In such a case to get the defining ideal of the multi-Rees algebra we only need to saturate the first syzygies of direct sum of this family of ideals. However, this fact is not true, when at least one of these ideals does not contain any nonzero divisor. In this paper we show that the defining ideal of the multi-Rees algebra of a family of ideals of a pol…

关于Rees代数与de Jonquières变换

On Rees algebras and de Jonquières transformations

本文研究高维de Jonquières变换的隐式化问题，通过转向映射图像，在驯顺Cremona支集下隐式化对应理想的Rees代数，还研究了广义情形的Rees环并解答了Ramos与Simis的一个猜想。

Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex

Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and ${\mathcal R}(\mathfrak{n})$ be the Rees algebra of the maximal homogeneous ideal $\mathfrak{n}$ of $R.$ We show that the generalized Hilbert-Kunz function $HK(s)=\ell({\mathcal R}(\mathfrak n)/(\mathfrak n, \mathfrak n t)^{[s]})$ is given by a polynomial for all large $s.$ We calculate it in many examples and also provide a Macaulay2 code for computing $HK(s).$

牛顿-奥库诺夫体、雷斯代数与格雷德家族的单调理想解析度

Newton-Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

本文利用牛顿-奥库诺夫体研究格雷德单调理想家族的雷斯代数诺特性及解析度的组合解释，并探讨符号雷斯代数的生成类型与韦罗内塞次数。

Residual intersections and modules with Cohen-Macaulay Rees algebra

In this paper, we consider a finite, torsion-free module $E$ over a Gorenstein local ring. We provide sufficient conditions for $E$ to be of linear type and for the Rees algebra $\mathcal{R}(E)$ of $E$ to be Cohen-Macaulay. Our results are obtained by constructing a generic Bourbaki $I$ ideal of $E$ and exploiting properties of the residual intersections of $I$.