Comparison of Stability Indices of Powers of Graded Ideals

Abstract

In this paper, we compare the index of ass-stability [Formula: see text] and the index of [Formula: see text]-stability [Formula: see text] of powers of a graded ideal [Formula: see text]. We prove that [Formula: see text] for any graded ideal [Formula: see text] in a 2-dimensional polynomial ring, and that [Formula: see text] can be any positive integer in this situation. Moreover, given any integers [Formula: see text], we construct a graded ideal [Formula: see text] in a [Formula: see text]-dimensional polynomial ring such that [Formula: see text].