On the uniform behaviour of the Frobenius closures of ideals
Volume 109 / 2007
Colloquium Mathematicum 109 (2007), 1-7
MSC: 13A35, 13A15, 13E05.
DOI: 10.4064/cm109-1-1
Abstract
Let ${\mathfrak a}$ be a proper ideal of a commutative Noetherian ring $R$ of prime characteristic $p$ and let $Q({\mathfrak a})$ be the smallest positive integer $m$ such that $({\mathfrak a} ^{\rm F})^{[p^m]} = {\mathfrak a} ^{[p^m]}$, where ${\mathfrak a} ^{\rm F}$ is the Frobenius closure of ${\mathfrak a}$. This paper is concerned with the question whether the set $ \{ Q({\mathfrak a}^{[p^m]}) : m \in {\mathbb N}_0 \}$ is bounded. We give an affirmative answer in the case that the ideal ${\mathfrak a}$ is generated by an u.s.$d$-sequence $c_1, \dots ,c_n$ for $R$ such that
(i) the map $R/\sum_{j=1}^n Rc_j\to R/\sum_{j=1}^n Rc_j^{2}$ induced by multiplication by $c_1 \dots c_n$ is an $R$-monomorphism;
(ii) for all ${\mathfrak p} \in \mathop{\rm ass}\nolimits (c_1^j, \dots ,c_n^j) $, $c_1/1,\dots ,c_n /1$ is a ${\mathfrak p} R_{{\mathfrak p}}$-filter regular sequence for $R_{{\mathfrak p}}$ for $j \in \{1, 2 \}$.
