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

The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen–Macaulay algebra AA in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of AA. 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 AA 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 hh-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 AA, which can be expressed exclusively in terms of the hh-vector of AA, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of AA 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 AA: namely, when AA is compressed, or when its hh-vector h(A)h(A) ends with (…,3,2)(…,3,2). Also, we will prove our lower bound when h(A)h(A) begins with (1,3,h2,…)(1,3,h2,…), where h2≤4h2≤4, and our upper bound when h(A)h(A) ends with (…,hc−1,hc)(…,hc−1,hc), where hc−1≤hc+1hc−1≤hc+1.