scholarly journals Symbolic powers of monomial ideals and vertex cover algebras

2007 ◽  
Vol 210 (1) ◽  
pp. 304-322 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Ngô Viêt Trung
Keyword(s):  
Vertex Cover ◽  
Monomial Ideals ◽  
Symbolic Powers

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 Symbolic powers of cover ideals of graphs and Koszul property

2021 ◽  
pp. 1-17
Author(s):  
Yan Gu ◽  
Huy Tài Hà ◽  
Joseph W. Skelton
Keyword(s):  
Vertex Cover ◽  
Cycle Cover ◽  
Symbolic Powers ◽  
Koszul Property

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

scholarly journals The Golod property for products and high symbolic powers of monomial ideals

2014 ◽  
Vol 400 ◽  
pp. 290-298 ◽  
Author(s):  
S.A. Seyed Fakhari ◽  
V. Welker
Keyword(s):  
Monomial Ideals ◽  
Symbolic Powers

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.

Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals

2010 ◽  
Vol 149 (2) ◽  
pp. 229-246 ◽  
Author(s):  
LÊ TUÂN HOA ◽  
TRÂN NAM TRUNG
Keyword(s):  
Asymptotic Behavior ◽  
Linear Function ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Symbolic Powers ◽  
Integral Closures

AbstractLet I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits $\lim_{n\to\infty} a_i(R/(\cap_j \overline{I_{1j}^n} + \cdots + \cap_j \overline{I_{pj}^n}))/n $ also exist.As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

scholarly journals Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

2018 ◽  
pp. 491-510
Author(s):  
Philippe Gimenez ◽  
José Martínez-Bernal ◽  
Aron Simis ◽  
Rafael H. Villarreal ◽  
Carlos E. Vivares
Keyword(s):  
Monomial Ideals ◽  
Symbolic Powers

scholarly journals On the Stanley Depth of Powers of Monomial Ideals

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 607 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Polynomial Ring ◽  
Upper Bound ◽  
Integral Closure ◽  
Monomial Ideals ◽  
The Other ◽  
Finitely Generated ◽  
Stanley Depth ◽  
Survey Article ◽  
Symbolic Powers ◽  
Other Hand

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.

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