A family of monomial ideals with the persistence property

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers [Formula: see text] and [Formula: see text], we consider the monomial ideal [Formula: see text] generated by all monomials [Formula: see text], where [Formula: see text] is an independent set of vertices of the path graph [Formula: see text] of size [Formula: see text], which is indeed the facet ideal of the [Formula: see text]th skeleton of the independence complex of [Formula: see text]. We describe the set of associated primes of all powers of [Formula: see text] explicitly. It turns out that any such ideal [Formula: see text] has the persistence property. Moreover, the index of stability of [Formula: see text] and the stable set of associated prime ideals of [Formula: see text] are determined.

A finite result for the set of associated prime ideals of formal local cohomology modules

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] two ideals of [Formula: see text], and [Formula: see text] two finitely generated [Formula: see text]-modules. It is first shown that [Formula: see text] is a finite set. We also prove that except the maximal ideal [Formula: see text], the set [Formula: see text] is stable for large [Formula: see text], where we use [Formula: see text] to denote [Formula: see text]-module [Formula: see text] or [Formula: see text] and [Formula: see text] is the eventual value of [Formula: see text].

ON THE ASSOCIATED PRIMES OF LOCAL COHOMOLOGY

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of $H_{I}^{t}(R)$ is finite for any ideal $I$ and for any $t\geqslant 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set $\operatorname{Ass}H_{I}^{2}(R)$ is always finite.

Persistence property for some classes of monomial ideals of a polynomial ring

Let [Formula: see text] be a field and [Formula: see text] be a polynomial ring in the variables [Formula: see text]. In this paper, we introduce two classes of monomial ideals of [Formula: see text], which have the following properties: (i) The (strong) persistence property of associated prime ideals. (ii) There exists a strongly superficial element. (iii) Ratliff–Rush closed. Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.

Ratliff–Rush closures and linear growth of primary decompositions of ideals

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a nonzero finitely generated [Formula: see text]-module and [Formula: see text] an ideal of [Formula: see text]. First purpose of this paper is to show that the sequences [Formula: see text] and [Formula: see text], [Formula: see text] of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff [On the relevant transform and the relevant component of an ideal, J. Algebra 111 (1987) 507–519, Theorem 3.1]. In addition, a characterization concerning the set [Formula: see text] is included. A second purpose of this paper is to prove that [Formula: see text] has linear growth primary decompositions for Ratliff–Rush closures with respect to [Formula: see text], that is, there exists a positive integer [Formula: see text] such that for every positive integer [Formula: see text], there exists a minimal primary decomposition [Formula: see text] in [Formula: see text] with [Formula: see text], for all [Formula: see text].

Localization at countably infinitely many prime ideals and applications

In this paper we present a technical lemma about localization at countably infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.