Vanishing and Non-negativity of the First Normal Hilbert Coefficient

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