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# Face Ring Multiplicity via CM-Connectivity Sequences

Published:2009-08-01
Printed: Aug 2009
• Isabella Novik
• Ed Swartz
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## 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 $\field[\Delta]$ via the Cohen--Macaulay connectivity of the skeletons of $\Delta$.
 MSC Classifications: 13F55 - Stanley-Reisner face rings; simplicial complexes [See also 55U10] 52B05; - unknown classification 52B05;13H15; - unknown classification 13H15;13D02; - unknown classification 13D02;05B35 - Matroids, geometric lattices [See also 52B40, 90C27]

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