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Acta Mathematica Vietnamica

Vanishing and Non-negativity of the First Normal Hilbert Coefficient

Linquan Ma , icon-email Pham Hung Quy

Abstract

Let (R,m) be a Noetherian local ring such that R^ is reduced. We prove that, when R^ is S2, if there exists a parameter ideal QR such that e¯1(Q)=0, then R is regular and ν(m/Q)1. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal [Goto, S., Hong, J., Mandal, M.: The positivity of the first coefficients of normal Hilbert polynomials. Proc. Amer. Math. Soc. 139(7), 2399–2406 (2011)]. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if R^ is equidimensional, then e¯1(Q)0 for all parameter ideals QR, and in characteristic p>0, we actually have e1(Q)0. Our proofs rely on the existence of big Cohen-Macaulay algebras.