Face ring multiplicity via CM-connectivity sequences

TL;DR

This paper verifies the multiplicity conjecture for face rings of various classes of simplicial complexes, introducing a new interpretation of minimal shifts through CM-connectivity sequences.

Contribution

It provides new proofs of the multiplicity conjecture for specific classes of complexes using CM-connectivity sequences, expanding the conjecture's verified cases.

Findings

Confirmed the conjecture for matroid, 1- and 2-dimensional complexes, and Gorenstein complexes up to dimension four.

Established the lower bound for doubly Cohen-Macaulay complexes with limited connectivity.

Linked minimal shifts in resolutions to CM-connectivity of skeletons.

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-Cohen-Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring via the Cohen-Macaulay connectivity of the skeletons of the complex.