The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra A in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of A. 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. Mighore, U. Nagel, T. Romer, 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 A 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 h-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 A, which can be expressed exclusively in terms of the h-vector of A, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of A 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 A: namely, when A is compressed, or when its h-vector h(A) ends with (.., 3, 2). Also, we will prove our lower bound when h(A) begins with (1, 3, h,), where h(2) : 4, and our upper bound when h (A) ends with (.... h(c-1), h(c)), where h(c-1) < h(c) + 1.
2007. Vol. 209, no 1, 79-89 p.