Face Ring Multiplicity via CM-Connectivity Sequences

Abstract

Abstract. The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen-Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all (d −1)-dimensional d-CohenMacaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring k[∆] via the Cohen-Macaulay connectivity of the skeletons of ∆.

The Multiplicity Conjecture for Barycentric Subdivisions

“…Hence it remains to consider dim(∆) = dim(sd(∆)) = 1. By Theorem 4.3 from [22] it follows that equality implies pureness of the minimal free resolution for k[sd ( …”

“…In independent work Novik and Swartz [22] have verified the the multiplicity conjecture for some important classes of Cohen-Macaulay simplicial complexes. The classes of simplicial complexes treated by Novik and Swartz and our Theorem 1.2 overlap in a small fraction of either class.…”

“…[17]). For the cases dim ∆ = 1, 2 the conjecture was settled in [22,Theorem 4.3] except for the equality statement. For dimensions 3 and 4 the result follows from [22] in case the complex is Gorenstein.…”

“…In recent years, this conjecture has attracted attention from commutative algebra and combinatorics (see for example [10], [12], [14], [15], [18], [19], [20], [21], [22], [23]). Here we provide another link to combinatorics and add a large class of rings for which the conjecture holds.…”

On the upper bound of the multiplicity conjecture

“…Both Conjectures 1.1 and 1.2 have been proved for many classes of ideals (see [6,9,10,11,14,17,20,22,24]). For extensions of this conjecture see [15,16,18,23,25].…”