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

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.

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.

The Hilbert-Kunz function of some quadratic quotients of the Rees algebra

Given a commutative local ring $(R,\mathfrak m)$ and an ideal $I$ of $R$, a family of quotients of the Rees algebra $R[It]$ has been recently studied as a unified approach to the Nagata's idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When $R$ is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either $I$ is $\mathfrak{m}$-primary or $R$ is regular and F-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explor…

对偶性与Rees环和切代数的方程

Duality and the equations of Rees rings and tangent algebras

本文利用对称代数与其定义理想之间的对偶性，研究Noetherian环上投射维数为一的模的Rees环，并应用于由二次型定义的完全交环的Kähler微分模及其切代数。

极大阶 Pfaffian 的 Rees 代数及其对角子代数

Rees algebra of maximal order Pfaffians and its diagonal subalgebras

本文研究一般反对称矩阵极大阶 Pfaffian 理想的 Rees 代数，给出其显式定义方程并证明对角子代数均为 Koszul，同时分析稀疏三对角情形的线性型理想及其幂的线性商性质。

The bi-graded structure of Symmetric Algebras with applications to Rees rings

Consider a rational projective plane curve C parameterized by three homogeneous forms h1,h2,h3 of the same degree d in the polynomial ring R=k[x,y] over the field k. Extracting a common factor, we may harmlessly assume that the ideal I=(h1,h2,h3)R has height two. Let phi be a homogeneous minimal Hilbert-Burch matrix for the row vector [h1,h2,h3]. So, phi is a 3 by 2 matrix of homogeneous forms from R; the entries in column m have degree dm, with d1 \le d2 and d1+d2=d. The Rees algebra $cal R$ of I is the subring k[h1t,h2t,h3t] of the polynomial ring k[t]. The bi-projective spectrum of $cal R$ …