Asymptotic behaviour of monomial ideals on regular sequences (Q877803)

Summary: Let \(R\) be a commutative noetherian ring, and let \({\mathbf x}= x_1,\dots, x_d\) be a regular \(R\)-sequence contained in the Jacobson radical of \(R\). An ideal \(I\) of \(R\) is said to be a monomial ideal with respect to \({\mathbf x}\) if it is generated by a set of monomials \(x^{e_1}_1\cdots x^{e_d}_d\). The monomial closure of \(I\), denoted by \(\widetilde I\), is defined to be the ideal generated by the set of all monomials \(m\) such that \(m^n\in I^n\) for some \(n\in\mathbb{N}\). It is shown that the sequences \(\text{Ass}_R R/\widetilde{I^n}\) and \(\text{Ass}_R\widetilde{I^n}/I^n\), \(n= 1,2,\dots\), of associated prime ideals are increasing and ultimately constant for large \(n\). In addition, some results about the monomial ideals and their integral closures are included.