Graded Ring Research Articles

Monomial curve families supporting Rossiʼs conjecture

In this article, we give a constructive method to form infinitely many families of monomial curves in affine 4-space with corresponding Gorenstein local rings in embedding dimension 4 supporting Rossiʼs conjecture. Starting with any monomial curve in affine 2-space, we obtain large families of Gorenstein local rings with embedding dimension 4, having non-decreasing Hilbert functions, although their associated graded rings are not Cohen–Macaulay.

The γ-filtration and the Rost invariant

Abstract Grothendieck studied two filtrations – the γ and topological filtrations – on the ring K 0 ( X ) $K_0(X)$ for a projective variety X, together with the associated graded rings. He gave a necessary and sufficient condition for the graded ring ⊕ i γ i / γ i + 1 $\oplus _i \gamma ^{i} / \gamma ^{i+1}$ to have non-zero torsion elements when X is the variety of Borel subgroups of a simply connected semisimple algebraic group G over an algebraically closed field. We sharpen his observation by calculating explicitly the torsion in γ 2 / γ 3 $\gamma ^{2} / \gamma ^3$ , and we do this under weaker hypotheses on G. We apply this result to describing the torsion in the group CH 3 ( X ) $\operatorname{CH}^3(X)$ of codimension-3 cycles on X, and providing an extension of the Rost invariant.

Improving electrical performance of ehv post station insulators under severe icing conditions using modified grading rings

This paper presents some mitigation solutions for improving the performance of EHV post station insulators under severe icing conditions using modified grading rings. For this purpose, numerical simulations using the finite element method (FEM), implemented by the commercial software Comsol Multiphysics, were performed, in parallel with experimental tests to study and evaluate the influence of the proposed solutions on ice-covered insulator performance during melting period. The solution proposed consists in adding a fine metallic and conductive mesh on the standard grading ring. In addition to improving the potential grading, the metallic mesh allows the grading ring to behave like a booster shed by created a large air gap under it. Experimental tests under severe icing conditions reveals that using two modified standard grading rings permits to improve maximum withstand voltage by 17% compared to the standard grading ring alone. Also, numerical simulations permits to demonstrate that increasing the diameter of the modified grading ring helps to improve both maximum withstand voltage and uniformity of potential distribution of the EHV post station insulator. The results obtained will also be useful to improve general knowledge on the use of grading rings for the EHV post station insulator under clean and severe icing conditions.

Optimized design of electric field grading systems in 115 kV non-ceramic insulators

In this work two stress grading options for 115 kV non-ceramic suspension insulators are analyzed by means of 2D and 3D simulations. Both options were optimized in order to compare the maximum reduction that can be obtained on the electric field at the surface of the insulator. In the first option the shape and permittivity value of the housing material next to the energized end were the optimized parameters. For the second option, the installation of a corona ring, the position and dimensions of the ring were optimized. Electric field simulations were performed with the finite element method (FEM) while for the optimization process different functions of the MATLAB optimization toolbox were used. The optimization was performed with 2D FEM simulations and the optimal designs were then used on 3D models, in order to verify that these designs remain as the best option. According to the results, it was found that the use of a corona ring in 115 kV lines produces the maximum reduction of electric field as long as it is installed in the optimal position. A modification of the housing profile next to the energized end will also produce a significant reduction on the maximum electric field on 115 kV non-ceramic insulators.

Apolarity, Hessian and Macaulay Polynomials

A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree δ can be realized as the apolar ring of a homogeneous polynomial g of degree δ in x 0, ⋅, x n . If R is the Jacobian ring of a smooth hypersurface f(x 0, ⋅, x n ) = 0, then δ is equal to the degree of the Hessian polynomial of f. In this article we investigate the relationship between g and the Hessian polynomial of f, and we provide a complete description for n = 1 and deg(f) ≤4 and for n = 2 and deg(f) ≤3.

On certain rings of differentiable type and finiteness properties of local cohomology

Let R be a commutative F-algebra, where F is a field of characteristic 0, satisfying the following conditions: R is equidimensional of dimension n, every residual field with respect to a maximal ideal is an algebraic extension of F, and DerF(R) is a finitely generated projective R-module of rank n such that Rm⊗RDerF(R)=DerF(Rm). We show that the associated graded ring of the ring of differential operators, D(R,F), is a commutative Noetherian regular with unity and pure graded dimension equal to 2dim(R). Moreover, we prove that D(R,F) has weak global dimension equal to dim(R) and that its Bernstein class is closed under localization at one element. Using these properties of D(R,F), we show that the set of associated primes of every local cohomology module, HIi(R), is finite. If (S,m,K) is a complete regular local ring of mixed characteristic p>0, we show that the localization of S at p, Sp, is such a ring. As a consequence, the set of associated primes of HIi(S) that does not contain p is finite. Moreover, we prove this finiteness property for a larger class of functors.

真空绝缘堆均压环的结构优化

真空绝缘堆均压环的结构优化

Полупростые градуированные кольца

Leo Tolstoy Tula State Pedagogical University, Russia, 300026, Tula, Lenina pr., 125, ibalaba@mail.ru, KrasnovaEN.ne@yandex.ruThegradedversionofWedderburn–Artintheoremisobtained.ItgivesdescriptionofsemisimpleG-gradedringforarbitrarygroupG.Homological classification of graded semisimple rings is given.Key words: graded rings, graded modules, semisimple rings.

Cost, Power Consumption and Performance Evaluation of Metro Networks

We provide models for evaluating the performance, cost and power consumption of different architectures suitable for a metropolitan area network (MAN). We then apply these models to compare today’s synchronous optical network/synchronous digital hierarchy metro rings with different alternatives envisaged for next-generation MAN: an Ethernet carrier grade ring, an optical hub-based architecture and an optical time-slotted wavelength division multiplexing (WDM) ring. Our results indicate that the optical architectures are likely to decrease power consumption by up to 75% when compared with present day MANs. Moreover, by allowing the capacity of each wavelength to be dynamically shared among all nodes, a transparent slotted WDM yields throughput performance that is practically equivalent to that of today’s electronic architectures, for equal capacity.

Even More Spectra: Tensor Triangular Comparison Maps via Graded Commutative 2-rings

We initiate the theory of graded commutative 2-rings, a categorification of graded commutative rings. The goal is to provide a systematic generalization of Paul Balmer’s comparison maps between the spectrum of tensor-triangulated categories and the Zariski spectra of their central rings. By applying our constructions, we compute the spectrum of the derived category of perfect complexes over any graded commutative ring, and we associate to every scheme with an ample family of line bundles an embedding into the spectrum of an associated graded commutative 2-ring.

Syzygies of differentials of forms

Given a standard graded polynomial ring R=k[x1,…,xn] over a field k of characteristic zero and a graded k-subalgebra A=k[f1,…,fm]⊂R, one relates the module ΩA/k of Kähler k-differentials of A to the transposed Jacobian module D⊂∑i=1nRdxi of the forms f1,…,fm by means of a Leibniz mapΩA/k→D whose kernel is the torsion of ΩA/k. Letting D denote the R-submodule generated by the (image of the) syzygy module of ΩA/k and Z the syzygy module of D, there is a natural inclusion D⊂Z coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality – in which case one says that the forms f1,…,fm are polarizable. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in R and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own.

The maximal denumerant of a numerical semigroup

Given a numerical semigroup S = <a_0, a_1, a_2,..., a_t> and n in S, we consider the factorization n = c_0 a_0 + c_1 a_1 + ... + c_t a_t where c_i >= 0. Such a factorization is maximal if c_0 + c_1 + ... + c_t is a maximum over all such factorizations of n. We provide an algorithm for computing the maximum number of maximal factorizations possible for an element in S, which is called the maximal denumerant of S. We also consider various cases that have connections to the Cohen-Macualay and Gorenstein properties of associated graded rings for which this algorithm simplifies.

On the derived category of a graded commutative noetherian ring

For any graded commutative noetherian ring, where the grading group is abelian and where commutativity is allowed to hold in a quite general sense, we establish an inclusion-preserving bijection between, on the one hand, the twist-closed localizing subcategories of the derived category, and, on the other hand, subsets of the homogeneous spectrum of prime ideals of the ring. We provide an application to weighted projective schemes.

When the associated graded ring of a semigroup ring is Complete Intersection

Let (R,m) be the semigroup ring associated to a numerical semigroup S. In this paper we study the property of its associated graded ring grm(R) to be Complete Intersection. In particular, we introduce and characterize β-rectangular and γ-rectangular Apéry sets, which will be the fundamental concepts of the paper and will provide, respectively, a sufficient condition and a characterization for grm(R) to be Complete Intersection. Then we use these notions to give four equivalent conditions for grm(R) in order to be Complete Intersection.

Characterizing of corona rings applied to composite insulators

In this paper, the electric field stress is investigated for a set of composite insulators. Three designs of such insulators, with different dimensions and shapes, are used to build a model of a composite system. The critical values of electric field are determined for 230kV and 400kV voltage classes. The field is calculated via a set of basic equations, reflecting the influence of several factors such as voltage, material and corona ring parameters on the insulator aging. The impact of corona rings on reducing electric field stress is determined for different voltages and insulator designs using the concept of efficiency. The ring parameters are optimized to determine their impact on reducing the field stress on the insulator. Therefore, a MATLAB-based code is set to cater for a wide range of parameters and to determine the optimal dimensions and locations of such rings. In all studied cases, the Finite Element techniques are used as a simulation tool for analysis, field contour plotting and stress monitoring.

The equations of Rees algebras of equimultiple ideals of deviation one

We describe the equations of the Rees algebra R ( I ) \mathbf {R}(I) of an equimultiple ideal I I of deviation one provided that I I has a reduction generated by a regular sequence x 1 , … , x s x_1,\ldots ,x_s such that the initial forms x 1 ∗ , … , x s − 1 ∗ x_1^*,\ldots ,x_{s-1}^* are a regular sequence in the associated graded ring. In particular, we prove that there is a single equation of maximum degree in a minimal generating set of the equations of R ( I ) \mathbf {R}(I) , which recovers some previous known results.

STRONG COHOMOLOGICAL RIGIDITY OF A PRODUCT OF PROJECTIVE SPACES

Abstract. We prove that for a toric manifold (respectively, a quasitoricmanifold) M, any graded ring isomorphism H ∗ (M) → H ∗ (Q mi=1 CP n i )can be realized by a diffeomorphism (respectively, a homeomoQrphism) mi=1 CP n i →M. 1. IntroductionThe cohomological rigidity problem for toric manifolds asks whether the inte-gral cohomology ring of a toric manifold determines its topological type or not.So far, there is no negative answer to the question but some positive results.In [2], the authors with M. Masuda show that if Mis a toric manifold whosecohomology ring is isomorphic to that ofQ mi=1 CP n i , a product of complexprojective spaces, then Mis actually diffeomorphic toQ mi=1 CP n i , which givesa positive result to the cohomological rigidity problem.On the other hand, one might aska strongerquestion as follows. Throughoutthis paper, H ∗ (X) denotes the integral cohomology ring of a topological spaceX.Problem 1.1. Let M and N be toric manifolds, and ϕ: H ∗ (N) → H ∗ (M)a graded ring isomorphism. Then, does there exist a homeomorphism or adiffeomorphism f: M→ Nsuch that f

Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen–Macaulay case

We use results of Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination of Betti diagrams of modules with a pure resolution. This implies the multiplicity conjecture of Herzog, Huneke, and Srinivasan for modules that are not necessarily Cohen-Macaulay and also implies a generalized version of these inequalities. We also give a combinatorial proof of the convexity of the simplicial fan spanned by pure diagrams.

The Leading Ideal of a Complete Intersection of Height Two, Part III

Let (S, 𝔫) be an s-dimensional regular local ring with s > 2, and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. As in [2, 3], we examine the leading form ideal I* of I in the associated graded ring G: = gr𝔫(S). Let μ G (I*) = n ≥ 3, and let {ξ1, ξ2,…, ξ n } be a minimal homogeneous system of generators of I* such that ξ1 = f* and ξ2 = g*, and c i : = deg ξ i ≤ deg ξ i+1: = c i+1 for each i ≤ n − 1. For m ≤ n, we say that K m : = (ξ1,…, ξ m )G is an ideal generated by part of a minimal homogeneous generating set of I*. Let D i : = GCD(ξ1,…, ξ i ) and d i = deg D i for i with 1 ≤ i ≤ m. Let K m be perfect with ht G K m = 2. We prove that the following are equivalent: 1. deg ξ i+1 = deg ξ i + (d i−1 − d i ) +1, for all i with 3 ≤ i ≤ m − 1; 2. deg ξ i+1 ≤ deg ξ i + (d i−1 − d i ) +1, for all i with 3 ≤ i ≤ m − 1. Furthermore, if these equivalent conditions hold, then K m = I*. Moreover, if e(G/K m ) = e(G/I*), we prove that K m = I*. We illustrate with several examples in the cases where K m is or is not perfect.

A formal proof of the projective Eisenbud–Evans–Storch theorem

We extend a constructive proof of the Eisenbud–Evans–Storch theorem, developed in a previous work by Coquand, Schuster, and Lombardi, from the affine to the projective case. The main tool is that of distributive lattices, which allows us to replace the classical topological arguments by more algebraic and constructive ones. Given a suitable graded ring, we work in the distributive lattice in which the prime filters correspond to the homogeneous prime ideals. The proof presented here is one of the first examples of concrete results obtained using this tool.