On the finiteness of the set of Hilbert coefficients

2017/02/25 by Masuti, Shreedevi K., Saloni, Kumari
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.07913

Abstract

Let (R,m) be a Noetherian local ring of dimension d and K,Q be m-primary ideals in R. In this paper we study the finiteness properties of the sets Λ_i^K(R):=\g_i^K(Q): Q is a parameter ideal of R\, where g_i^K(Q) denotes the Hilbert coefficients of Q with respect to K, for 1 ≤ i ≤ d. We prove that Λ_i^K(R) is finite for all 1≤ i ≤ d if and only if R is generalized Cohen-Macaulay. Moreover, we show that if R is unmixed then finiteness of the set Λ₁^K(R) suffices to conclude that R is generalized Cohen-Macaulay. We obtain partial results for R to be Buchsbaum in terms of |Λ_i^K(R)|=1. We also obtain a criterion for the set Δ^K(R):=\g₁^K(I): I is an m-primary ideal of R\ to be finite, generalizing preceding results.

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