Notes on the multiplicity conjecture
Let \(K\) be a field, let \(S=K[x_1,\dots, x_n]\), let \(I\) be a graded ideal of \(S\) contained in \((x_1,\dots, x_n)\) and let \(R=S/I\). Herzog, Huneke and Srinivasan conjectured that, if \(R\) is Cohen-Macaulay, then the multiplicity of \(R\) is bounded above and below by a function expressed in terms of the maximal and minimal shifts in the graded minimal \(S\)-resolution of \(R\), respectively. This is called Conjecture 1 in the paper. The precise statement of this conjecture is given on the second page of the paper (see also the article of \textit{R. M. Miró-Roig} [J. Algebra 299, No. 2, 714--724 (2006; Zbl 1116.13008)]). It was observed in the article of \textit{J. Herzog} and \textit{H. Srinivasan} [Trans. Am. Math. Soc. 350, No. 7, 2879--2902 (1998; Zbl 0899.13026)] that the lower bound of Conjecture 1 fails in general if \(R\) is not Cohen-Macaulay. This leads to the statement of a conjecture for the upper bound of the multiplicity of \(R\) without the assumption of the Cohen-Macaulayness of \(R\). This is called Conjecture 2 in the article under review. Partial answers to Conjecture 2 are shown in the said article of Herzog and Srinivasan and in the articles of \textit{L. H. Gold} [J. Pure Appl. Algebra 182, No. 2--3, 201--207 (2003; Zbl 1032.13006)] and \textit{T. Römer} [J. Pure Appl. Algebra 195, No. 1, 113--123 (2005; Zbl 1073.13015)]. In this article the authors show that if analogous inequalities of Conjecture 1 hold for an ideal \(I\) of codimension \(s\), not necessarily perfect, then the same inequalities hold when replacing \(I\) by \((I, f_1,\dots, f_m)\), where \(f_1,\dots, f_m\) is a regular sequence modulo \(I\). It is also shown that Conjecture 2 is valid in the limit with respect to taking powers of an ideal. However, this does no imply Conjecture 2 for all sufficiently high powers of \(I\). The following question is also considered: suppose that for a ring \(R\) the lower bound of Conjecture 1, or the upper bound in Conjecture 2 is reached, does this imply that \(R\) is Cohen-Macaulay and has a pure resoluction? The authors show some partial answers to this question.
scientific article