scholarly journals Standard Bases for fractional ideals of the local ring of an algebroid curve

2020 ◽  
Vol 551 ◽  
pp. 342-361 ◽  
Author(s):  
E. Carvalho ◽  
M.E. Hernandes
Keyword(s):  
Local Ring ◽  
Standard Bases ◽  
Algebroid Curve

Alternative algorithms for computing generic μ∗-sequences and local Euler obstructions of isolated hypersurface singularities

2019 ◽  
Vol 18 (08) ◽  
pp. 1950156
Author(s):  
Katsusuke Nabeshima ◽  
Shinichi Tajima
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Rational Functions ◽  
Isolated Singularity ◽  
Euler Obstruction ◽  
Standard Bases ◽  
Hypersurface Singularities ◽  
Previous Algorithm ◽  
Computation Speed

A new algorithm is introduced for computing [Formula: see text]-sequences of isolated hypersurface singularities. It is shown that the new algorithm results in better performance, compared to our previous algorithm that utilizes parametric local cohomology systems, in computation speed. Furthermore, it can be used to compute local Euler obstruction of a hypersurface with an isolated singularity. The key idea of the new algorithm is computing standard bases in a local ring over a field of rational functions.

scholarly journals Form rings and regular sequences

1978 ◽  
Vol 72 ◽  
pp. 93-101 ◽  
Author(s):  
Paolo Valabrega ◽  
Giuseppe Valla
Keyword(s):  
Local Ring ◽  
Singular Points ◽  
Point Of View ◽  
Algebraic Varieties ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Standard Bases ◽  
Regular Sequences ◽  
Initial Forms ◽  
The Ideal

Hironaka, in his paper [H1] on desingularization of algebraic varieties over a field of characteristic 0, to deal with singular points develops the algebraic apparatus of the associated graded ring, introducing standard bases of ideals, numerical characters ν* and τ* etc. Such a point of view involves a deep investigation of the ideal b* generated by the initial forms of the elements of an ideal A of a local ring, with respect to a certain ideal a.

On Generalized Regular Local Ring

2015 ◽  
Vol 3 (1) ◽  
pp. 145-152
Author(s):  
Zubayda Ibraheem ◽  
Naeema Shereef
Keyword(s):  
Local Ring ◽  
Regular Local Ring

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

On equimultiplicity

1982 ◽  
Vol 91 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Herrmann ◽  
U. Orbanz
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Analytic Spread

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

KUMMER COVERINGS AND SPECIALISATION

2021 ◽  
pp. 1-37
Author(s):  
Martin Olsson
Keyword(s):  
Local Ring ◽  
Fundamental Groups ◽  
Technical Result ◽  
Noetherian Local Ring ◽  
Log Schemes ◽  
Log Scheme

Abstract We prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

On *-clean non-commutative group rings

2016 ◽  
Vol 15 (08) ◽  
pp. 1650150 ◽  
Author(s):  
Hongdi Huang ◽  
Yuanlin Li ◽  
Gaohua Tang
Keyword(s):  
Finite Field ◽  
Local Ring ◽  
Group Algebra ◽  
Rational Numbers ◽  
Semisimple Group ◽  
Group Algebras ◽  
Group Rings ◽  
Dihedral Groups ◽  
Commutative Local Ring ◽  
Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

The probability that the multiplication of two ring elements is zero

2018 ◽  
Vol 17 (03) ◽  
pp. 1850054 ◽  
Author(s):  
M. A. Esmkhani ◽  
S. M. Jafarian Amiri
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Minimum Value ◽  
Finite Commutative Ring ◽  
Ring Elements

Let [Formula: see text] be a finite commutative ring. We denote by [Formula: see text] the probability that the multiplication of two randomly chosen elements of [Formula: see text] is zero. In this paper, we show that either [Formula: see text] or [Formula: see text] for any ring [Formula: see text] and determine all rings [Formula: see text] with [Formula: see text]. Also, we obtain the structures of rings [Formula: see text] having maximum or minimum value of [Formula: see text] among all rings with identity of the same size. We characterize all rings [Formula: see text] with identity such that [Formula: see text]. Finally, we compute [Formula: see text] where [Formula: see text] is a PIR local ring.

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