Primary decompositions of three-generated ideals

Abstract

Let I be a 3-generated ideal of height 2 in a RLR ( R, m, k) of dimension d ≥ 3 and J be its unmixed part, i.e. the intersection of all the primary components of I of height 2. In this paper, we will deduce the result of Huneke and Ulrich that if J is Cohen-Macaulay, i.e. pd R R/ J = 2, then depth R/ I ≥ d − 3 from a general and more elementary setting. For the next case that pd R R/ J = 3, we show that for p ϵ Min( J/ I), p e n I: J and p e is not contained in any p-primary component of I for e < (h + 1)(h − 3) 2(h − 2) where h = ht p. Also, a negative answer is given to the question of Huneke whether depth R/ I ≥ depth R/ J − 1 if depth R/ J ≥ 2.