scholarly journals On Ratliff-Rush closure of modules

2020 ◽  
Vol 126 (2) ◽  
pp. 170-188
Author(s):  
Naoki Endo
Keyword(s):  
Integral Closure ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Projective Scheme

In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.

Implosive graphs: Square-free monomials on symbolic Rees algebras

2016 ◽  
Vol 16 (08) ◽  
pp. 1750145 ◽  
Author(s):  
A. Flores-Méndez ◽  
I. Gitler ◽  
E. Reyes
Keyword(s):  
Structural Properties ◽  
Monomial Ideal ◽  
Minimal System ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Vertex Set ◽  
Minimal System Of Generators

Let [Formula: see text] be the edge monomial ideal of a graph [Formula: see text], whose vertex set is [Formula: see text]. [Formula: see text] is implosive if the symbolic Rees algebra [Formula: see text] of [Formula: see text] has a minimal system of generators [Formula: see text] where [Formula: see text] are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under clique-sums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.

scholarly journals Fibers of rational maps and Rees algebras of their base ideals

2020 ◽  
Vol 129 (1B) ◽  
pp. 5-14
Author(s):  
Tran Quang Hoa ◽  
Ho Vu Ngoc Phuong
Keyword(s):  
Projective Space ◽  
Local Cohomology ◽  
Rational Maps ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Local Cohomology Modules ◽  
Dimensional Projective Space ◽  
Cohomology Modules

We consider a ratinonal map $\phi$ from m-dimensional projective space to n-dimensional projective space that is a parameterization of m-dimensional variety. Our main goal is to study the (m-1)-dimensional fibers of $\phi$ in relation with the m-th local cohomology modules of Rees algebra of its base ideal.

scholarly journals Splittings and Symbolic Powers of Square-free Monomial Ideals

2019 ◽  
Author(s):  
Jonathan Montaño ◽  
Luis Núñez-Betancourt
Keyword(s):  
Linear Optimization ◽  
Monomial Ideals ◽  
Sufficient Condition ◽  
Graded Algebra ◽  
Prime Characteristic ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Symbolic Powers ◽  
Frobenius Map

Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

The parametric blowing-up of two-dimensional local domains

1983 ◽  
Vol 94 (2) ◽  
pp. 217-228 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Integral Domain ◽  
Two Dimensional ◽  
Rees Algebra ◽  
Graded Ring ◽  
Proper Mapping ◽  
Projective Scheme ◽  
Blowing Up ◽  
Image Position

Let (A, M) be a local Noetherian integral domain of dimension two and X = Spec A. For an ideal IA the graded ring RA(I) = noIn denotes the Rees algebra of A with respect to I. The projective scheme Y = Proj RA(I) is called the blowing-up of X (resp. A) along I. Then there exists a proper mapping : YX. The preimage Z = 1(V(I)) is called the exceptional fibre. Note that induces an isomorphism

scholarly journals The Cohen-Macaulayness of the Rees algebras of local rings

1983 ◽  
Vol 89 ◽  
pp. 47-63 ◽  
Author(s):  
Shin Ikeda
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Noetherian Local Ring

Let (A, m, k) be a Noetherian local ring. We defineand call it the Rees algebra of A. Let X be an indeterminate over A, then R(A) can be identified with the A-subalgebra .

scholarly journals Topics on symbolic Rees algebras for space monomial curves

1991 ◽  
Vol 124 ◽  
pp. 99-132 ◽  
Author(s):  
Shiro Goto ◽  
Koji Nishida ◽  
Yasuhiro Shimoda
Keyword(s):  
Prime Ideals ◽  
Gorenstein Ring ◽  
Regular Local Ring ◽  
Rees Algebra ◽  
Power Series Ring ◽  
Series Ring ◽  
Rees Algebras ◽  
Monomial Curves ◽  
Formal Power Series Ring ◽  
Formal Power

Let A be a regular local ring of dim A = 3 and p a prime ideal in A of dim A/p = 1. We put Rs(p) = (here t denotes an indeterminate over A) and call it the symbolic Rees algebra of p. With this notation the authors [5, 6] investigated the condition under which the A-algebra Rs(p) is Cohen-Macaulay and gave a criterion for Rs(p) to be a Gorenstein ring in terms of the elements f and g in Huneke’s condition [11, Theorem 3.1] of Rs(p) being Noetherian. They furthermore explored the prime ideals p = p(n1, n2, n3) in the formal power series ring A = k[X, Y, Z] over a field k defining space monomial curves and Z = with GCD(n1, n2, nz) = 1 and proved that Rs(p) are Gorenstein rings for certain prime ideals p = p(n1 n2, n3). In the present research, similarly as in [5, 6], we are interested in the ring-theoretic properties of Rs(p) mainly for p = p(n1 n2) nz) and the results of [5, 6] will play key roles in this paper.

scholarly journals Systems with the integer rounding property in normal monomial subrings

2010 ◽  
Vol 82 (4) ◽  
pp. 801-811 ◽  
Author(s):  
Luis A. Dupont ◽  
Carlos Rentería-Márquez ◽  
Rafael H. Villarreal
Keyword(s):  
Graph Theory ◽  
Linear System ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Incidence Matrix ◽  
Edge Ideal ◽  
Rees Algebra ◽  
Canonical Module ◽  
Rees Algebras ◽  
Integer Rounding

Let C be a clutter and let A be its incidence matrix. If the linear system x > 0; x A < 1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.

A construction of Prüfer rings involving quotients of Rees algebras

2018 ◽  
Vol 17 (06) ◽  
pp. 1850098
Author(s):  
Carmelo Antonio Finocchiaro
Keyword(s):  
Rees Algebra ◽  
Rees Algebras ◽  
Fixed Ideal ◽  
Prüfer Rings

A family of quotients of a Rees algebra associated to a ring with respect to a fixed ideal was recently introduced by Barucci, D’Anna and Strazzanti. In this paper, we will classify rings of this family that satisfy certain Prüfer-like properties and, as a particular case, we will extend results obtained for amalgamated duplications and Nagata idealizations.

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