Vanishing and Non-negativity of the First Normal Hilbert Coefficient

Let \((R,\mathfrak {m})\) be a Noetherian local ring such that \(\widehat{R}\) is reduced. We prove that, when \(\widehat{R}\) is \(S_2\), if there exists a parameter ideal \(Q\subseteq R\) such that \(\bar{e}_1(Q)=0\), then R is regular and \(\nu (\mathfrak {m}/Q)\le 1\). 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 \(\widehat{R}\) is equidimensional, then \(\bar{e}_1(Q)\ge 0\) for all parameter ideals \(Q\subseteq R\), and in characteristic \(p>0\), we actually have \(e_1^*(Q)\ge 0\). Our proofs rely on the existence of big Cohen-Macaulay algebras.