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http://dx.doi.org/10.4134/BKMS.b190286

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

Mafi, Amir (Department of Mathematics University Of Kurdistan)
Naderi, Dler (Department of Mathematics University Of Kurdistan)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.2, 2020 , pp. 407-417 More about this Journal
Abstract
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 ℓ(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 :RI+(Jd-2 :RI+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.
Keywords
Generalized Hilbert coefficient; minimal reduction; associated graded ring;
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