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The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified forthe face rings of the following classes of simplicial complexes: matroidcomplexes, complexes of dimension one and two, and Gorenstein complexes ofdimension at most four. The lower bound part of this conjecture is alsoestablished for the face rings of all doubly Cohen-Macaulay complexes whose1-skeleton's connectivity does not exceed the codimension plus one as well asfor all (d-1)-dimensional d-Cohen-Macaulay complexes. The main ingredient ofthe proofs is a new interpretation of the minimal shifts in the resolution ofthe face ring via the Cohen-Macaulay connectivity of the skeletons of thecomplex.

Face Ring Multiplicity via CM-Connectivity Sequences