On the uniform behaviour of the Frobenius closures of ideals

K. Khashyarmanesh¹

¹ Department of Mathematics Ferdowsi University of Mashhad P.O. Box 1159-91775, Mashhad, Iran and Institute for Studies in Theoretical Physics and Mathematics P.O. Box 19395-5746, Tehran, Iran

Colloquium Mathematicum, Tome 109 (2007) no. 1, pp. 1-7.

Résumé

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 $j \in \{1, 2 \}$.

Détail

DOI : 10.4064/cm109-1-1

Keywords: mathfrak proper ideal commutative noetherian ring prime characteristic mathfrak smallest positive integer mathfrak mathfrak where mathfrak frobenius closure mathfrak paper concerned question whether set mathfrak mathbb bounded affirmative answer the ideal mathfrak generated d sequence dots map sum sum induced multiplication dots r monomorphism mathfrak mathop ass nolimits dots dots mathfrak mathfrak filter regular sequence mathfrak