Stability properties of powers of ideals in regular local rings of small dimension (Q1708449)

Denote the smallest integer \(n\) for which Ass \(I^n\), Ass \(\overline{I}^n\) or depth \(I^n\) stabilizes by astab \(I\), \(\overline{\text{astab}}\;I\), and dstab \(I\) respectively. In this short note, the authors show for a regular local ring \(R\) containing a field of dimension 2 or less that \(\text{astab}\;I=\overline{\text{astab}}\;I= \text{dstab}\;I\). In contrast, if the dimension is 3, although \(\text{astab}\;I= \text{dstab}\;I\), the authors exhibit rings for which \(\text{astab}\;I \neq \overline{\text{astab}}\;I\). In dimension 4, now all three measures of stability for powers of \(I\) can differ and in particular \(\text{astab}\;I-\text{dstab}\;I\) can achieve any integer value.