Abstract :
Let image be a local Cohen–Macaulay ring of dimension d. Let I be an image-primary ideal and let J be the ideal generated by a maximal superficial sequence for I. Under these assumptions Valabrega and Valla (Nogoya Math. J. 72 (1978) 93–101) proved that the associated graded ring G of I is Cohen–Macaulay if and only if In∩J=JIn−1 for every integer n. In this paper we consider the class of the image-primary ideals I such that, for some positive integer k, we have In∩J=JIn−1 for n≤k and λ(Ik+1/JIk)≤1. In this case G need not be Cohen–Macaulay. In Theorem 2.2. we show that G is Cohen–Macaulay unless the ideals we are considering are of maximal Cohen–Macaulay type. One can use the ideas of Russi and Valla (Comm. Algebra 24(13) (1996) 4249–4261; Pure Appl. Algebra 122 (1997) 293–311) to prove that, for the ideals we consider, the depth of G is at least d−1 and that its h-vector has no negative components. We characterize the possible Hilbert function of G. Our approach gives proof of an extended version of a conjecture of Sally (proved in Russi and Valla Comm. Algebra 24(13) (1996) 4249–4261)) and independently in Wang (J. Algebra 190 (1997) 226–240) in the case image). Several results proved in Huckaba (Comm. Algebra, to appear), Russi and Valla (Nogoya Math. J. 110 (1988) 81–111; Comm. Algebra 24(13) (1996) 4249–4261; J. Pure Appl. Algebra 122 (1997) 293–311) and Sally (J. Algebra 83 (1983) 325–333) are unified and generalized.
