Finite Free Resolutions and 1-Skeletons of Simplicial Complexes
Naoki Terai1
and Takayuki Hibi2
1Department of Mathematics, Faculty of Education Saga University Saga 840 Japan
2Department of Mathematics, Graduate School of Science Osaka University Toyonaka Osaka 560 Japan
2Department of Mathematics, Graduate School of Science Osaka University Toyonaka Osaka 560 Japan
DOI: 10.1023/A:1008648302195
Abstract
A technique of minimal free resolutions of Stanley-Reisner rings enables us to show the following two results: (1) The 1-skeleton of a simplicial ( d-1)-sphere is d-connected, which was first proved by Barnette; (2) The comparability graph of a non-planar distributive lattice of rank d-1 is d-connected.
Pages: 89–93
Keywords: simplicial complex; 1-skeleton; comparability graph; $d$-connected; free resolution
Full Text: PDF
References
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2. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge/New York/Sydney,
1993. P1: rbaP1: rba Journal of Algebraic Combinatorics KL365-05-Terai October 30, 1996 16:1 FINITE FREE RESOLUTIONS AND 1-SKELETONS 93
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2. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge/New York/Sydney,
1993. P1: rbaP1: rba Journal of Algebraic Combinatorics KL365-05-Terai October 30, 1996 16:1 FINITE FREE RESOLUTIONS AND 1-SKELETONS 93
3. T. Hibi, “Level rings and algebras with straightening laws,” J. Algebra 117 (1988), 343-362.
4. T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, N.S.W., Australia, 1992.
5. T. Hibi, “Face number inequalities for matroid complexes and Cohen-Macaulay types of Stanley-Reisner rings of distributive lattices,” Pacific J. Math. 154 (1992), 253-264.
6. M. Hochster, “Cohen-Macaulay rings, combinatorics, and simplicial complexes,” in Ring Theory II, Lect. Notes in Pure and Appl. Math., No. 26, B.R. McDonald and R. Morris (Eds.), pp. 171-223. Dekker, New York, 1977.
7. R.P. Stanley, “Cohen-Macaulay complexes,” in Higher Combinatorics, M. Aigner (Ed.), pp. 51-62. Reidel, Dordrecht/ Boston, 1977.
8. R.P. Stanley, Combinatorics and Commutative Algebra, Birkh\ddot auser, Boston/Basel/Stuttgart, 1983.
9. R.P. Stanley, Enumerative Combinatorics, Volume I, Wadsworth & Brooks/Cole, Monterey, Calif., 1986.