scholarly journals On the associated graded module of an ideal generated by an unconditioned strong $d$-sequence

1999 ◽  
Vol 39 (4) ◽  
pp. 607-618 ◽  
Author(s):  
K. Khashyarmanesh ◽  
SH. Salarian ◽  
H. Zakeri
Keyword(s):  
Graded Module

scholarly journals Constructions of vector-valued modular forms of rank four and level one

2020 ◽  
Vol 16 (05) ◽  
pp. 1111-1152
Author(s):  
Cameron Franc ◽  
Geoffrey Mason
Keyword(s):  
Irreducible Representation ◽  
Differential Equations ◽  
Modular Forms ◽  
Modular Group ◽  
Tensor Products ◽  
Two Dimensional ◽  
Minimal Weight ◽  
Graded Module ◽  
Holomorphic Modular Forms ◽  
Vector Valued

This paper studies modular forms of rank four and level one. There are two possibilities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the [Formula: see text]-matrix. In the remaining sections we describe how to write down the corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately, the differential equations that arise can only be solved recursively in general. We conclude the paper by studying the cases of tensor products of two-dimensional representations, symmetric cubes of two-dimensional representations, and inductions of two-dimensional representations of the subgroup of the modular group of index two. In these cases, the differential equations satisfied by minimal weight forms can be solved exactly.

Annihilator of local cohomology of homogeneous parts of a graded module

2020 ◽  
pp. 2150092
Author(s):  
Tran Do Minh Chau ◽  
Nguyen Thi Kieu Nga ◽  
Le Thanh Nhan
Keyword(s):  
Asymptotic Stability ◽  
Local Ring ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module

Let [Formula: see text] be a homogeneous graded ring, where [Formula: see text] is a Noetherian local ring. Let [Formula: see text] be a finitely generated graded [Formula: see text]-module. For [Formula: see text] set [Formula: see text]. Denote by [Formula: see text] the set of all prime ideals of [Formula: see text] containing [Formula: see text]. For [Formula: see text], let [Formula: see text] be the set of all [Formula: see text] such that [Formula: see text] In this paper, we prove that the sets [Formula: see text] and [Formula: see text] do not depend on [Formula: see text] for [Formula: see text]. We show that the annihilators [Formula: see text], [Formula: see text] are eventually stable, where [Formula: see text] for [Formula: see text]. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of [Formula: see text].

scholarly journals Truncated Euler Systems over Imaginary Quadratic Fields

2009 ◽  
Vol 195 ◽  
pp. 97-111
Author(s):  
Soogil Seo
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Abelian Extension ◽  
Iwasawa Theory ◽  
Quadratic Fields ◽  
Galois Module ◽  
Imaginary Quadratic Fields ◽  
Euler Systems ◽  
Graded Module ◽  
Elliptic Units

AbstractLet K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group ClF of F is equal to the order of the quotient UF/ElF of the group of global units UF by the group of elliptic units ElF of F. We introduce a filtration on UF/ElF made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

scholarly journals GENERALIZED HILBERT–KUNZ FUNCTION IN GRADED DIMENSION 2

2016 ◽  
Vol 230 ◽  
pp. 1-17 ◽  
Author(s):  
HOLGER BRENNER ◽  
ALESSIO CAMINATA
Keyword(s):  
Rational Number ◽  
Bounded Function ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Algebraically Closed Field ◽  
Graded Module ◽  
Dimension 2

We prove that the generalized Hilbert–Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+\unicode[STIX]{x1D6FE}(q)$, with rational generalized Hilbert–Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\unicode[STIX]{x1D6FE}(q)$. Moreover, we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow +\infty$ of the generalized Hilbert–Kunz multiplicity $e_{gHK}^{R_{p}}(M_{p})$ over the fibers $R_{p}$ exists, and it is a rational number.

scholarly journals COMPARISON OF MULTIGRADED AND UNGRADED COUSIN COMPLEXES

2001 ◽  
Vol 44 (2) ◽  
pp. 365-378
Author(s):  
M. H. Dogani Aghcheghloo ◽  
R. Enshaei ◽  
S. Goto ◽  
R. Y. Sharp
Keyword(s):  
Noetherian Ring ◽  
Subject Classification ◽  
Sufficient Condition ◽  
Commutative Noetherian Ring ◽  
Additional Input ◽  
Graded Module ◽  
Cousin Complexes ◽  
Cousin Complex

AbstractThis paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a $\mathbb{Z}$-graded module $M$ over a $\mathbb{Z}$-graded commutative Noetherian ring $R$, the graded Cousin complex for $M$ introduced by Goto and Watanabe can be regarded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations considered in this paper are, firstly, to multigraded situations and, secondly, to Cousin complexes with respect to more general filtrations than the basic ones considered by Petzl and Sharp. New arguments are presented to provide a sufficient condition for the exactness of the quotient complex in this generality, as the arguments of Petzl and Sharp will not work for this situation without additional input.AMS 2000 Mathematics subject classification: Primary 13A02; 13E05; 13D25; 13D45

Castelnuovo–Mumford regularity and Degree of nilpotency

2007 ◽  
Vol 142 (3) ◽  
pp. 429-437 ◽  
Author(s):  
CAO HUY LINH
Keyword(s):  
Finite Number ◽  
Primary Ideal ◽  
Fixed Degree ◽  
Graded Module

AbstractIn this paper we show that the Castelnuovo–Mumford regularity of the associated graded module with respect to anm-primary idealIis effectively bounded by the degree of nilpotency ofI. From this it follows that there are only a finite number of Hilbert-Samuel functions for ideals with fixed degree of nilpotency.

Regularity,a*-Invariant and the Associated Graded Module to a Filtration

2012 ◽  
Vol 40 (10) ◽  
pp. 3745-3752
Author(s):  
Dancheng Lu ◽  
Tongsuo Wu
Keyword(s):  
Graded Module

INFINITE ORDER PARAMETRIC NORMAL FORM OF HOPF SINGULARITY

2008 ◽  
Vol 18 (11) ◽  
pp. 3393-3408 ◽  
Author(s):  
MAJID GAZOR ◽  
PEI YU
Keyword(s):  
Normal Form ◽  
State Space ◽  
Integral Domain ◽  
Infinite Order ◽  
State Parameter ◽  
Efficient Computation ◽  
Graded Ring ◽  
Codimension One ◽  
Lie Bracket ◽  
Graded Module

In this paper, we introduce a suitable algebraic structure for efficient computation of the parametric normal form of Hopf singularity based on a notion of formal decompositions. Our parametric state and time spaces are respectively graded parametric Lie algebra and graded ring. As a consequence, the parametric state space is also a graded module. Parameter space is observed as an integral domain as well as a vector space, while the near-identity parameter map acts on the parametric state space. The method of multiple Lie bracket is used to obtain an infinite order parametric normal form of codimension-one Hopf singularity. Filtration topology is revisited and proved that state, parameter and time (near-identity) maps are continuous. Furthermore, parametric normal form is a convergent process with respect to filtration topology. All the results presented in this paper are verified by using Maple.

scholarly journals A FILTRATION ON EQUIVARIANT BOREL–MOORE HOMOLOGY

2019 ◽  
Vol 7 ◽  
Author(s):  
ARAM BINGHAM ◽  
MAHIR BILEN CAN ◽  
YILDIRAY OZAN
Keyword(s):  
Fixed Point ◽  
Maximal Torus ◽  
Flag Varieties ◽  
Direct Sum ◽  
Equivariant Embedding ◽  
Graded Module ◽  
Homogeneous Variety

Let $G/H$ be a homogeneous variety and let $X$ be a $G$ -equivariant embedding of $G/H$ such that the number of $G$ -orbits in $X$ is finite. We show that the equivariant Borel–Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel–Moore homologies of the $G$ -orbits. If $T$ is a maximal torus of $G$ such that each $G$ -orbit has a $T$ -fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel–Moore homology of $X$ . We apply our findings to certain wonderful compactifications as well as to double flag varieties.

scholarly journals Hilbert polynomials of j-transforms

2016 ◽  
Vol 161 (2) ◽  
pp. 305-337
Author(s):  
SHIRO GOTO ◽  
JOOYOUN HONG ◽  
WOLMER V. VASCONCELOS
Keyword(s):  
Hilbert Functions ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Graded Module ◽  
Systems Of Parameters ◽  
Partial Systems

AbstractWe study transformations of finite modules over Noetherian local rings that attach to a module M a graded module H(x)(M) defined via partial systems of parameters x of M. Despite the generality of the process, which are called j-transforms, in numerous cases they have interesting cohomological properties. We focus on deriving the Hilbert functions of j-transforms and studying the significance of the vanishing of some of its coefficients.

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