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Paper   IPM / M / 12783
School of Mathematics
Title:   A generalization of the Swartz equality
Author(s):
 1 M. R. Pournaki 2 S. A. Seyed Fakhari 3 S. Yassemi
Status:   Published
Journal: Glasg. Math. J.
Vol.:  56
Year:  2014
Pages:   381-386
Supported by:  IPM
Abstract:
For a given (d−1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h0(Γ),h1(Γ),…,hd(Γ)) and set h−1(Γ)=0. The known Swartz equality implies that if ∆ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ id, the inequality ihi(∆)+ (di+1)hi−1(∆) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley-Reisner rings, Hokkaido Math. J. 25 (1996), no. 1, 137-148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18 (2004/05), no. 3, 647-661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen-Macaulay simplicial complexes in co-dimension t.