Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebras

The Rees Algebra of a monomial plane parametrization

We compute a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve. We describe explicitly both the bigraded Betti numbers and the maps of the resolution in terms of a generalized version of the Euclidean Algorithm. We also explore the relation between pencils of adjoints of the monomial plane curve and elements in a suitable piece of the defining ideal of the Rees Algebra.

An extension of Rees theorem and two interpretations of a vector in the joint reduction lattice

In \cite{rees} Rees gave a characterization for the normal joint reduction number zero of two $\m$-primary ideals in an analytically unramified Cohen-Macaulay local ring of dimension two. Rees' result is a generalization of Zariski's product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees' theorem for the ordinary powers of $\m$-primary ideals $I$ and $J$ in a Cohen-Macaulay local ring of dimension two. Following Rees' approach, we define the modified Koszul homology modules $M^1_{r,s}(a^k,b^k)$ for a joint reduction $(a,b)$ of $I$ …

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。

Local Cohomology of Multi-Rees Algebras with Applications to Joint Reductions and Complete Ideals

In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{ \overline{I^rJ^s}\}$ of complete ${\mathfrak m}$-primary ideals and vanishing of the second normal Hilbert coefficient $\overline{e}_2(IJ)$ where $R$ is a two-dimensional Cohen-Macaulay analytically unramified local ring with maximal ideal $\mathfrak m.$ This generalization is obtained as a consequence of a formula for the third local cohomology module of the extended Rees algebras of the $\mathbb Z^3$-graded filtration $\{\overline{I^rJ^sK^t}\}$ with su…

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。

Variations on the Grothendieck-Serre Formula for Hilbert functions and their applications

In this expository paper we present proofs of Grothendieck-Serre Formula for multi-graded algebras and Rees algebras for admissible multi-graded filtrations. As applications, we derive formulas of Sally for postulation number of admissible filtrations and Hilbert coefficients. We also discuss a partial solution of Itoh's conjecture by Kummini and Masuti. We present an alternate proof of Huneke-Ooishi Theorem and a generalisation for multi-graded filtrations.

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.

A $D$-module approach on the equations of the Rees algebra

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce information about the defining equations of the Rees algebra of $I$. Let $\mathcal{K}$ be the kernel of the canonical map $α: \text{Sym}(I) \rightarrow \text{Rees}(I)$ from the symmetric algebra of $I$ onto the Rees algebra of $I$. We prove that $\mathcal{K}$ can be described as the solution set of a system of differential equations, that the whole bigraded stru…

On the defining equations of Rees algebra of a height two perfect ideal using the theory of $D$-modules

Let $k$ be a field of characteristic zero, and $R=k[x_1, \ldots, x_d]$ with $d \geq 3$ be a polynomial ring in $d$ variables. Let $\m=(x_1, \ldots, x_d)$ be the homogeneous maximal ideal of $R$. Let $\mathcal{K}$ be the kernel of the canonical map $α: \sym(I) \rightarrow \R(I)$, where $\sym(I)$ (resp. $\R(I)$) denotes the symmetric algebra (resp. the Rees algebra) of an ideal $I$ in $R$. We study $\mathcal{K}$ when $I$ is a height two perfect ideal minimally generated by $d+1$ homogeneous elements of same degree and satisfies $G_d$, that is, the minimal number of generators of the ideal $I_{\m…

A duality theorem for generalized local cohomology

We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

Rees-like代数及其幺半群扩张上的投射模

Projective modules over Rees-like algebras and its monoid extensions

针对Rees-like代数$A$及其幺半群扩张$A[N]$，证明了不同秩条件下投射模的幺模元素存在性及特殊群作用的传递性，改进了Serre和Bass的经典结果。

Diagonal Subalgebras of Residual Intersections

Let ${\sf k}$ be a field, $S$ be a bigraded ${\sf k}$-algebra, and $S_Δ$ denote the diagonal subalgebra of $S$ corresponding to $Δ= \{ (cs,es) \; | \; s \in \mathbb{Z} \}$. It is know that the $S_Δ$ is Koszul for $c,e \gg 0$. In this article, we find bounds for $c,e$ for $S_Δ$ to be Koszul, when $S$ is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the …

在Lean4中对Auslander-Buchsbaum-Serre准则的正式化

Formalization of Auslander--Buchsbaum--Serre criterion in Lean4

本文在Lean4中形式化了Auslander-Buchsbaum-Serre准则，通过Ext函子定义深度，证明正则局部环的性质，并发展了Cohen-Macaulay模的理论。

关于Gorenstein环上极大Cohen-Macaulay模的稳定范畴（II）

On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II

本文研究Gorenstein局部环上极大Cohen-Macaulay模稳定范畴的三角等价，证明其保持环的维数、曲率、Serre条件和穿孔谱上的完全交性质，并构造扩展Rees代数稳定范畴到原稳定范畴的三角函子及其诱导等价。