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.

Frobenius closure and prime characteristic singularities

2020 ◽  
Author(s):  
◽  
Kyle Logan Maddox
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Sufficient Condition ◽  
Zariski Topology ◽  
Prime Characteristic ◽  
Joint Work ◽  
University Of Missouri ◽  
Mild Conditions ◽  
Flat Maps ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] This dissertation outlines several results about prime characteristic singularities for which the nilpotent part under the induced Frobenius action on local cohomology is either finite colength or the entire module, collectively referred to here as nilpotent singularities. First, we establish a sufficient condition for the finiteness of the Frobenius test exponent for a local ring and apply it to conclude that nilpotent singularities have finite Frobenius test exponent. In joint work with Jennifer Kenkel, Thomas Polstra, and Austyn Simpson, we show that under mild conditions nilpotent singularities descend and ascend along faithfully flat maps. Consequently, we then prove that the loci of primes which are weakly F-nilpotent and F-nilpotent are open in the Zariski topology for rings which are either F-finite or essentially of fiiite type over an excellent local ring.

scholarly journals Stanley depth and symbolic powers of monomial ideals

2017 ◽  
Vol 120 (1) ◽  
pp. 5 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Limit Behavior ◽  
Stanley Depth ◽  
Symbolic Powers

The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities $\mathrm{sdepth} (S/I^{(ks)}) \leq \mathrm{sdepth} (S/I^{(s)})$ and $\mathrm{sdepth}(I^{(ks)}) \leq \mathrm{sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, $\mathrm{sdepth}(I^{(k+d)})\leq \mathrm{sdepth}(I^{{(k)}})$ and $\mathrm{sdepth}(S/I^{(k+d)})\leq \mathrm{sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley depth of symbolic powers of a squarefree monomial ideal. We also introduce a method for comparing the Stanley depth of factors of monomial ideals.

scholarly journals BERNSTEIN–SATO ROOTS FOR MONOMIAL IDEALS IN POSITIVE CHARACTERISTIC

2020 ◽  
pp. 1-10
Author(s):  
EAMON QUINLAN-GALLEGO
Keyword(s):  
Positive Characteristic ◽  
Monomial Ideals ◽  
Prime Characteristic ◽  
The Ideal

Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$ ) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic- $p$ notion of Bernstein–Sato root is reasonable.

scholarly journals ON THE CONJECTURE OF VASCONCELOS FOR ARTINIAN ALMOST COMPLETE INTERSECTION MONOMIAL IDEALS

2019 ◽  
pp. 1-15
Author(s):  
KUEI-NUAN LIN ◽  
YI-HUANG SHEN
Keyword(s):  
Complete Intersection ◽  
Monomial Ideal ◽  
Short Note ◽  
Monomial Ideals ◽  
Rees Algebra

In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.

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 Rees algebras of square-free monomial ideals

2015 ◽  
Vol 7 (1) ◽  
pp. 25-53 ◽  
Author(s):  
Louiza Fouli ◽  
Kuei-Nuan Lin
Keyword(s):  
Monomial Ideals ◽  
Rees Algebras
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