Improving the bounds of the multiplicity conjecture: The codimension 3 level case
The Multiplicity Conjecture (MC) of Herzog, Huneke and Srinivasan gives a lower bound (resp.\ an upper bound) for the multiplicity of a Cohen-Macaulay algebra \(A\) in terms of the smallest (resp.\ largest) shift appearing in each free module in the minimal free resolution of \(A\). Many papers have appeared in recent years either proving interesting special cases of this conjecture, providing stronger bounds, or extending to the non-Cohen-Macaulay situation. This paper falls into the first two categories, exclusively in the case of codimension 3. First, the author proves the MC for compressed algebras, i.e.\ algebras which have the biggest possible Hilbert function for a given socle type. The second part of the paper is based on the observation that the Hilbert function (and hence the multiplicity) can be read from the graded Betti numbers, and that redundant terms can be cancelled (numerically) and hence do not contribute to the Hilbert function. (By redundant, we mean summands \(R(-t)\) that are repeated in consecutive free modules.) The idea, then, is to formulate a stronger variant of the MC by first formally removing all redundant summands and then looking for the smallest (resp.\ largest) shifts that remain, and bounding the multiplicity in the analogous way with these numbers. A similar idea was put forward by \textit{C. A. Francisco} [J.\ Algebra 299, No. 1, 309--328 (2006; Zbl 1105.13003)]. Unfortunately, in general there are choices for this ``numerical splitting'', and it is not uniquely determined. However, in the current paper the author had the clever idea of restricting to the case of level algebras in codimension 3. In this case, the numerical splitting is uniquely determined, and the author conjectures that the improved MC holds. Furthermore, this stronger bound can be read immediately from the Hilbert function. He proves a few initial cases of his conjecture. In a subsequent paper jointly written by the author with \textit{U. Nagel} and the reviewer, this conjecture was proved for codimension 3 Gorenstein algebras [An improved Multiplicity Conjecture for codimension three Gorenstein algebras, \url{arXiv:math/0604485}].
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