Cohen-Macaulay and Gorenstein property of Rees algebras of non-singular equimultiple prime ideals

We give criteria for the Cohen-Macaulay and Gorenstein property of Rees algebras of height 2 non-singular equimultiple prime ideals in terms of explicite representations of the associated graded rings. As consequences, we show that in general, the Cohen-Macaulay resp. Gorenstein property of such Rees algebras does not imply the Cohen-Macaulay resp. Gorenstein property of the base ring and that these properties depend upon the characteristic. Dedicated to the memrory of Professor Lê Van Thiêm Professor Lê Van Thiêm was the first directorof Hanoi Institute of Mathematics     

On ideals with the Rees property

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 ${\mathfrak{m}I:y=I}$ 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 which have the Rees property but are not ${\mathfrak{m}}$ -full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.     

Regularity of Rees Algebras

Let B = k[x₁, ..., x_n] be a polynomial ring over a field k,and let A be a quotient ring of B by a homogeneous ideal J.Let m denote the maximal graded ideal of A. Then the Rees algebraR = A[m t] also has a presentation as a quotient ring of thepolynomial ring k[x₁, ..., x_n, y₁, ..., y_n] by a homogeneousideal J^*. For instance, if A = k[x₁, ..., x_n], then Rk[x₁,...,x_n,y₁,...,y_n]/(x_iy_j–x_jy_i|i, j=1,...,n). In this paper we want to compare the homological propertiesof the homogeneous ideals J and J^*.     

Rees Algebras of Modules

We study Rees algebras of modules within a fairly general framework.We introduce an approach through the notion of Bourbaki idealsthat allows the use of deformation theory. One can talk aboutthe (essentially unique) generic Bourbaki ideal I(E) of a moduleE which, in many situations, allows one to reduce the natureof the Rees algebra of E to that of its Bourbaki ideal I(E).Properties such as Cohen–Macaulayness, normality and beingof linear type are viewed from this perspective. The known numericalinvariants, such as the analytic spread, the reduction numberand the analytic deviation, of an ideal and its associated algebrasare considered in the case of modules. Corresponding notionsof complete intersection, almost complete intersection and equimultiplemodules are examined in some detail. Special consideration isgiven to certain modules which are fairly ubiquitous becauseinteresting vector bundles appear in this way. For these modulesone is able to estimate the reduction number and other invariantsin terms of the Buchsbaum–Rim multiplicity. 2000 MathematicsSubject Classification 13A30 (primary), 13H10, 13B21 (secondary)     

Gorenstein代数上的Hopf代数作用

令H是有限维Hopf代数，A是左H-模代数。本文证明了A是Gorenstein代数的充分必要条件。A^H也是Gorenstein代数的条件。它是Enochs EE,GarciaJJ和del RioA关于群作用相应的理论的推广，同时给出A/A^H是Frobenius扩张的条件。     

On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

On the Weak-Lefschetz property for Artinian Gorenstein algebras

We deal with the weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension 3. We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence \(H\) of codimension 3 we found several Gorenstein Betti sequences compatible with \(H\) which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.     

Gorenstein代数上的逼近扩张

本文引进了（极小）逼近扩张，证明了极小逼近扩张在Ｇｏｒｅｎｓｔｅｉｎ代数上的存在性和唯一性，并给出了极小逼近扩张的一个应用．     

Gorenstein Dimension and AS-Gorenstein Algebras

The purpose of this paper is to connect the notion of Gorenstein dimension with AS-Gorenstein algebras. In particular, we show that a noetherian connected graded algebra having a balanced dualizing complex is AS-Gorenstein if the balanced dualizing complex has finite Gorenstein dimension. As a preparation, we generalize the Auslander–Bridger formula to the class of noncommutative noetherian connected graded algebras having balanced dualizing complexes.     

Koszul Modules and Gorenstein Algebras

I first define Koszul modules, which are a generalization to arbitrary rank of complete intersections. After a study of some of their properties, it is proved that Gorenstein algebras of codimension one or two over a local or graded CM ring are Koszul modules, thus generalizing a well known statement for rank one modules. The general techniques used to describe Koszul modules are then used to obtain a structure theorem for Gorenstein algebras in codimension one and two, over a local or graded CM ring.     

On the Cohen-Macaulay property of diagonal subalgebras of the Rees algebra

We consider the blowing up of ℙ _k ^/n−1 along a closed subscheme defined by a homogeneous idealI ∪A=k[X ₁, …,X _n ] generated by forms of degree ≤d, and its projective embeddings by the linear systems corresponding to (I ^e ) _c , forc≥de+1. The homogeneous coordinate rings of these embeddings arek[(I ^e ) _c ]. One wants to study the Cohen-Macaulay property of these rings. We will prove that if the Rees algebraR _A (I) is Cohen-Macaulay, thenk[(I ^e ) _c ] are Cohen-Macaulay forc>>e>0, thus proving a conjecture stated by A. Conca, J. Herzog, N.V. Trung and G. Valla. Supported by a F.P.I. grant of Ministerio de Educación y Ciencia (Spain) This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

连通微分分次代数的Gorenstein性质

证明了由两个同调光滑的,Koszul,Gorenstein连通微分分次代数作张量得到的连通微分分次代数仍为同调光滑的,Koszul,Gorenstein连通微分分次代数;假设A是同凋光滑的连通微分分次代数使得H(A)是Koszul连通分次代数,则A是Gorenstein连通微分分次代数当且仅当H(A)是Gorenstein连通分次代数.     

Notes on Hilbert-Kunz Multiplicity of Rees Algebras

Abstract

In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese subrings.     

Isomorphism Classes of Certain Artinian Gorenstein Algebras

In this paper we attack the problem of the classification, up to analytic isomorphism, of Artinian Gorenstein local k-algebras with a given Hilbert Function. We solve the problem in the case the square of the maximal ideal is minimally generated by two elements and the socle degree is high enough.     

Rees Algebras of a Class of Graded Ideals

Mathematical Notes - The paper deals with the Rees algebra $$\mathcal{R}$$ of a graded ideal I of a standard graded algebra A generated by a subset of generators of the maximal graded ideal of A....     

Fitting ideals and the Gorenstein property

Let p be a prime number and G be a finite commutative group such that p ² does not divide the order of G. In this note we prove that for every finite module M over the group ring Z _p [G], the inequality #M  ￡  #Z_p[G]/Fit_Zp[G](M){\#M\,\leq\,\#{\bf Z}_{p}[G]/{{\rm Fit}}_{{\bf Z}_{p}[G]}(M)} holds. Here, Fit_Zp[G](M){\rm Fit}_{{\bf Z}_{p}[G]}(M) is the Z _p [G]-Fitting ideal of M.     

Koszul and Gorenstein Properties for Homogeneous Algebras

The Koszul property was generalized to homogeneous algebras of degree in [5], and related to -complexes. We show that if the -homogeneous algebra is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem to i.e., there is a Poincaré duality between Hochschild homology and cohomology of as for .