A note on the multiplicities of Gorenstein algebras
Let \(R = K[ X_{1},X_{2}, \ldots, X_{n}]\) be a polynomial ring over a field \(K\) and \(S= R/I\) be a homogeneous \(K\)-algebra. Consider a free resolution of \(S\) over \(R\) and denote by \(M_{i}\) and \(m_{i}\) \((1 \leq i \leq s)\) the maximal and the minimal shifts in the resolution. Assume that \(S\) is Cohen-Macaulay, then Herzog, Huneke and the author have conjectured that the multiplicity \(e(S)\) of \(S\) satisfies \[ \prod_{i=1}^{s} m_{i} /s! \leq e(S) \leq \prod_{i=1}^{s} M_{i}/s!. \] This conjecture has been proved whenever \(S\) has a quasi-pure resolution by \textit{J. Herzog} and \textit{H. Srinivasan} [Trans. Am. Math. Soc. 350, No. 7, 2879-2902 (1998; Zbl 0899.13026)]. In this paper, a formula for the multiplicities of homogeneous Gorenstein algebras is given. The formula allows to give a stronger bound than above for \(e(S)\) in the case of Gorenstein algebras with quasi-pure resolution. More explicitly the bound is given by the expression: \[ m_{1} \cdots m_{k}M_{k+1} \cdots M_{s} / s! \leq e(S) \leq M_{1} \cdots M_{k}m_{k+1} \cdots m_{s} / s!, \] where the length of the resolution is \(s=2k\) or \(s=2k+1\).
multiplicities of homogeneous Gorenstein algebras
scientific article; zbMATH DE number 1230345
scientific article