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 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. 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 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 2 , … ) , 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 .