Primary ideals with good associated graded ring

AbstractLet (A,M) be a local Cohen–Macaulay ring of dimension d. Let I be an M-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 M-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 I=M). 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