ON THE FIRST GENERALIZED HILBERT COEFFICIENT AND DEPTH OF ASSOCIATED GRADED RINGS

摘要

Let (R, m) be a d-dimensional Cohen-Macaulay local ring with infinite residue field. Let I be an ideal of R that has analytic spread `(I) = d, satisfies the Gd condition, the weak Artin-Nagata property AN? d?2 and m is not an associated prime of R/I. In this paper, we show that if j1(I) = λ(I/J) + λ[R/(Jd?1 :R I + (Jd?2 :R I + I) :R m∞)] + 1, then I has almost minimal j-multiplicity, G(I) is Cohen-Macaulay and rJ (I) is at most 2, where J = (x1, . . . , xd) is a general minimal reduction of I and Ji = (x1, . . . , xi). In addition, the last theorem is in the spirit of a result of Sally who has studied the depth of associated graded rings and minimal reductions for m-primary ideals.