Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications

	Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications
Journal of Algebra and Related Topics
مقاله 2، دوره 2، شماره 1، شهریور 2014، صفحه 15-25 اصل مقاله (157.27 K)
نوع مقاله: Research Paper
نویسنده
M. Nasernejad^*
University of Payame Noor
چکیده
Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. We say that $I$ satisfies the persistence property if $\mathrm{Ass}_R(R/I^k)\subseteq \mathrm{Ass}_R(R/I^{k+1})$ for all positive integers $k\geq 1$, which $\mathrm{Ass}_R(R/I)$ denotes the set of associated prime ideals of $I$. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring $R=K[x_1,\ldots,x_n]$ over field $K$ which are associated to unrooted trees such that if $G$ is a unrooted tree and $I_t(G)$ is the ideal generated by the paths of $G$ of length $t$, then $J_t(G):=I_t(G)^\vee$, where $I^\vee$ denotes the Alexander dual of $I$, satisfies the persistence property. We also present a class of graphs such that the path ideals generated by paths of length two satisfy the persistence property. We conclude this paper by giving a criterion for normally torsion-freeness of monomial ideals.
کلیدواژه‌ها
Monomial ideals؛ associated prime ideals؛ trees؛ paths

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