Koszulness, Krull dimension, and other properties of graph-related algebras
Alexandru Constantinescu
and Matteo Varbaro
DOI: 10.1007/s10801-011-0276-6
Abstract
The algebra of basic covers of a graph G, denoted by [ `( A)]( G) \bar{A}(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of [ `( A)]( G) \bar{A}(G) in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore, we show that if the graph is bipartite, then [ `( A)]( G) \bar{A}(G) is a homogeneous algebra with straightening laws, and thus it is Koszul. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.
Pages: 375–400
Keywords: keywords vertex covers of graphs; cover ideal; edge ideal; fiber cone; Koszul; straightening laws; Krull dimension; arithmetical rank; Cohen-Macaulay; Castelnuovo-Mumford regularity
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References
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3. Birkhoff, G.: Lattice Theory, 3rd edn. Am. Math. Soc. Colloq. Publ., vol.
25. Am. Math. Soc., Providence (1967)
4. Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Am. Math. Soc. 260, 159- 183 (1980)
5. Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics (1993)
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8. Fröberg, R.: On Stanley-Reisner Rings. Topics in Algebra, Banach Center Publications, Part 2, vol. 26, pp. 57-70. Springer, Berlin (1990)
9. Hà, H.T., Van Tuyl, A.: Monomial ideals, edge ideals of Hypergraphs, and their graded Betti numbers. J. Algebr. Comb. 27(2), 215-245 (2008)
10. Hartshorne, R.: Complete intersection and connectedness. Am. J. Math. 84, 497-508 (1962)
11. Hartshorne, R.: A property of A-sequences. Bull. Soc. Math. Fr. 94, 61-65 (1966)
12. Herzog, J., Hibi, T.: Distributive lattices, bipartite graphs and Alexander duality. J. Algebr. Comb. 22(3), 289-302 (2005)
13. Herzog, J., Hibi, T., Trung, N.V.: Symbolic powers of monomial ideals and vertex cover algebras. Adv. Math. 210, 304-322 (2007)
14. Hibi, T.: Distributive lattice, affine semigroup rings and algebras with straightening laws. Adv. Stud. Pure Math., 11 (1987)
15. Kalkbrener, M., Sturmfels, B.: Initial complex of prime ideals. Adv. Math. 116, 365-376 (1995)
16. Katzman, M.: Characteristic-independence of Betti numbers of graph ideals. J. Comb. Theory, Ser. A 113(3), 435-454 (2006)
17. Kummini, M.: Regularity, depth and arithmetic rank of bipartite edge ideals. J. Algebr. Comb. 30(4), 429-445 (2009)
18. Lovász, L., Plummer, M.D.: Matching Theory. North-Holland Mathematics Studies, vol.
121. Annals of Discrete Mathematics, vol.
29. North-Holland, Amsterdam (1986). Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest
19. Lyubeznik, G.: On the local cohomology modules H i (R) for ideals U generated by an R-sequence.
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