Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T06:41:05.568Z Has data issue: false hasContentIssue false
  • English
  • Français

A GENERALIZATION OF THE SWARTZ EQUALITY

Published online by Cambridge University Press:  30 August 2013

M. R. POURNAKI ,
S. A. SEYED FAKHARI  and
S. YASSEMI
M. R. POURNAKI
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran e-mail: pournaki@ipm.ir
S. A. SEYED FAKHARI
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran e-mail: fakhari@ipm.ir
S. YASSEMI
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran e-mail: yassemi@ipm.ir
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi(Δ)+(d−i+1)hi−1(Δ) ≥ 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(1) (1996), 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(3) (2004/05), 647–661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.

Keywords

Primary: 13F5505E45Secondary: 13C14
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 381 - 386
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Bruns, W. and Herzog, J., Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39 (Cambridge University Press, Cambridge, UK, 1993).Google Scholar
2.Haghighi, H., Yassemi, S. and Zaare-Nahandi, R., A generalization of k-Cohen–Macaulay simplicial complexes, Ark. Mat. 50 (2) (2012), 279–290.Google Scholar
3.Stanley, R. P., Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, 41 (Birkhäuser, Boston, MA, 1996).Google Scholar
4.Swartz, E., Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18 (3) (2004/05), 647–661.CrossRefGoogle Scholar
5.Terai, N., On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25 (1) (1996), 137–148.Google Scholar