Enriched homology and cohomology modules of simiplicial complexes
Gunnar Fløystad
Matematisk Institutt Johs. Brunsgt. 12 5008 Bergen Norway Johs. Brunsgt. 12 5008 Bergen Norway
DOI: 10.1007/s10801-006-0038-z
Abstract
For a simplicial complex Δ on {1, 2,\cdots , n} we define enriched homology and cohomology modules. They are graded modules over k[ x 1,\cdots , x n ] whose ranks are equal to the dimensions of the reduced homology and cohomology groups.
We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein * complexes Δ , and also orientable homology manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex.
Pages: 285–307
Keywords: keywords simplicial complex; homology; girth; Cohen-Macaulay; simplicial complex; block design; homology manifold; Steiner system
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References
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2. D. Bayer, I. Peeva, and B. Sturmfels, “Monomial resolutions,” Math. Res. Lett. 5(1-2) (1998), 31-46.
3. D. Bayer and B. Sturmfels, “Cellular resolutions of monomial ideals,” J. Reine Angew. Math. 102 (1998), 123-140.
4. W. Bruns and J. Herzog, “Cohen-Macaulay rings,” Cambridge Studies in Advanced Math., Cambridge University Press (1993) vol. 39.
5. D. Eisenbud, G. FlØystad, and F.-O. Schreyer, “Sheaf cohomology and free resolutions over exterior algebras,” Trans. Am. Math. Soc. 355(11) (2003), 4397-4426.
6. G. FlØystad and J.E. Vatne, “(Bi)-Cohen-Macaulay simplicial complexes and their associated coherent sheaves,” Commun. Alg. 33(9) (2005), 3121-3136.
7. G. FlØystad, “Cohen-Macaulay cell complexes,” preprint, math. CO/0502541, to appear in Proceedings, Algebraic and Topological Combinatorics, Anogia,
2005. Springer
8. M. Hochster, “Cohen-Macaulay rings, combinatorics, and simplicial complexes,” in B.R. McDonald and R. Morris (Ed.), Ring Theory II, Proc. Second Oklahoma Conference, Dekker, New York, 1977, pp. 171-223.
9. M. Jungerman and G. Ringel, “Minimal triangulations on orientable surfaces,” Acta Math. 145 (1980), 121-154.
10. D.L. Kreher, “t-Designs, t \geq 3, the CRC handbook of combinatorial designs,” in Charles Colbourn and Jeffrey Dinitz (Eds.), the CRC Handbook of Combinatorial Designs (1996), pp. 47-65.
11. W. K\ddot uhnel, “Triangulations of manifolds with few vertices,” in F. Tricerri (Ed.), Advances in Differential Geometry and Topology, World Scientific, Singapore, 1990, pp. 59-114.
12. F.H. Lutz, “Triangulated manifolds with few vertices: Combinatorial manifolds,” preprint,
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