Improving the bounds of the multiplicity conjecture: The codimension 3 level case

Abstract: The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen–Macaulay algebra in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of . All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211–226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if has a pure MFR. Therefore, it seems a reasonable–and useful–idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen–Macaulay algebras. In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose -vector is that of a compressed algebra. In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra , which can be expressed exclusively in terms of the -vector of , and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of is not pure. Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras : namely, when is compressed, or when its -vector ends with . Also, we will prove our lower bound when begins with , where , and our upper bound when ends with , where . [Copyright &y& Elsevier]