scholarly journals Associated Primes of Powers of Monomial Ideals: A Survey

2020 ◽  
Vol 6 (1) ◽  
pp. 1
Author(s):  
Mehrdad Nasernejad
Keyword(s):  
Monomial Ideals ◽  
Associated Primes

scholarly journals Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

2011 ◽  
Vol 331 (1) ◽  
pp. 224-242 ◽  
Author(s):  
Christopher A. Francisco ◽  
Huy Tài Hà ◽  
Adam Van Tuyl
Keyword(s):  
Monomial Ideals ◽  
Perfect Graphs ◽  
Associated Primes

scholarly journals Associated primes of monomial ideals and odd holes in graphs

2010 ◽  
Vol 32 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Christopher A. Francisco ◽  
Huy Tài Hà ◽  
Adam Van Tuyl
Keyword(s):  
Monomial Ideals ◽  
Associated Primes

scholarly journals Limit behavior of the rational powers of monomial ideals

2021 ◽  
Author(s):  
James Lewis
Keyword(s):  
Local Cohomology ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Limit Behavior ◽  
Associated Primes ◽  
Powers Of Ideals ◽  
Symbolic Powers ◽  
Local Cohomology Modules ◽  
Cohomology Modules

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

scholarly journals ON A CLASS OF MONOMIAL IDEALS

2013 ◽  
Vol 87 (3) ◽  
pp. 514-526 ◽  
Author(s):  
KEIVAN BORNA ◽  
RAHELEH JAFARI
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Total Degree ◽  
Associated Primes ◽  
Minimal Set ◽  
Irreducible Decomposition ◽  
Maximal Height

AbstractLet $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$.

scholarly journals Monomial Ideals of Weighted Oriented Graphs

10.37236/8479 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Yuriko Pitones ◽  
Enrique Reyes ◽  
Jonathan Toledo
Keyword(s):  
Oriented Graph ◽  
Monomial Ideals ◽  
Associated Primes ◽  
Edge Ideal ◽  
Combinatorial Characterization ◽  
Oriented Graphs ◽  
Irreducible Decomposition ◽  
Underlying Graph

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D$ whose underlying graph is $G$. We determine the irredundant irreducible decomposition of $I$. Also, we characterize the associated primes and the unmixed property of $I$. Furthermore, we give a combinatorial characterization for the unmixed property of $I$, when $G$ is bipartite, $G$ is a graph with whiskers or $G$ is a cycle. Finally, we study the Cohen–Macaulay property of $I$.

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