Gorenstein Hilbert coefficients (Q357872)

Let \(R\) be the polynomial ring in \(n\) variables over a field \(K\), \(I\) a homogeneous ideal of \(R\) and \(S=R/I\) a quotient ring of dimension \(d\). The Hilbert polynomial of \(S\) can be written in the form NEWLINE\[NEWLINEP_S(x) = \sum_{i=0}^{d-1} (-1)^i e_i \binom{x+d-1-i}{x}.NEWLINE\]NEWLINE The coefficients \(e_i\) are the Hilbert coefficients of \(S\). The coefficient \(e_0\) is known as the multiplicity of \(S\) and is denoted by \(e\).NEWLINENEWLINEThe paper under review establishes lower and upper bounds on these coefficients in terms of information coming from the minimal resolution of \(S\) over \(R\), when \(S\) is a homogeneous Gorenstein algebra with quasi-pure resolution. The results generalize inequalities proved by \textit{H. Srinivasan} involving \(e_0=e\) [J. Algebra 208, No. 2, 425--443 (1998; Zbl 0920.13018)] and also refine, in this setting, previous results of \textit{J. Herzog} and \textit{X. Zheng} [Proc. Am. Math. Soc. 137, No. 2, 487--494 (2009; Zbl 1162.13010)].