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

Minimal Hilbert-Kunz multiplicity

In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal Hilbert-Kunz multiplicity for strongly F-regular rings. Moreover, we calculate this invariant for quotient singularities and for the coordinate ring of the Segre embedding: $P^{r-1} \times P^{s-1} \hookrightarrow P^{rs-1}$, respectively.

Some extensions of Hilbert-Kunz multiplicity

Let $R$ be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do \emph{not} assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures g…

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。

Explicit Hilbert-Kunz functions of 2 x 2 determinantal rings

Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in $m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2 \times 2$ minors of the generic $m \times n$ matrix $[x_{i,j}]$. We give a closed formulation for the dimensions of the $k$-vector space $k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q))$ as $q$ varies over all positive integers, i.e., we give a closed form for the generalized Hilbert-Kunz function of the determinantal ring $k[X]/I_{2}[X]$. We also give a closed formulation of dimensions of related quoti…

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 $cal R$ …

Families of Gorenstein and almost Gorenstein rings

Starting with a commutative ring $R$ and an ideal $I$, it is possible to define a family of rings $R(I)_{a,b}$, with $a,b \in R$, as quotients of the Rees algebra $\oplus_{n \geq 0} I^nt^n$; among the rings appearing in this family we find Nagata's idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a,b$; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a,b$. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we …

时空图的因果边Rees代数

Causal Edge Rees Algebras for Spatiotemporal Graphs

本文提出因果边Rees代数，用于编码时空系统中连接性的演变，通过因果约束的时间过滤生成边理想序列，揭示连接性变化的因果结构，定义了时间桥模块并提出桥检测定理。

One-dimensional Gorenstein local rings with decreasing Hilbert function

In this paper we solve a problem posed by M.E. Rossi: {\it Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? } More precisely, for any integer $h>1$, $h \notin\{14+22k, \, 35+46k \ | \ k\in\mathbb{N} \}$, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level $h$; moreover we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing…

Dao数与满性的渐近行为

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。

晶格同调与积分闭合子模及Artin代数

Lattice homology of integrally closed submodules and Artin algebras

本文研究晶格同调在积分闭合子模和Artin代数中的应用，证明了当两个扩展离散估值实现实现同一子模时，晶格同调模块同构且欧拉特征等于商模的维数。