Abstract
The algebra of basic covers of a graph G, denoted by \(\bar{A}(G)\), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of \(\bar{A}(G)\) 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 \(\bar{A}(G)\) 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.
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Benedetti, B., Constantinescu, A., Varbaro, M.: Dimension, depth and zero-divisors of the algebra of basic k-covers of a graph. Le Matematiche LXIII(II), 117–156 (2008)
Benedetti, B., Varbaro, M.: Unmixed graphs that are domains. Commun. Algebra (2009, to appear)
Birkhoff, G.: Lattice Theory, 3rd edn. Am. Math. Soc. Colloq. Publ., vol. 25. Am. Math. Soc., Providence (1967)
Björner, A.: Shellable and Cohen–Macaulay partially ordered sets. Trans. Am. Math. Soc. 260, 159–183 (1980)
Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics (1993)
Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Mathematics, vol. 1327 (1980)
De Concini, C., Eisenbud, D., Procesi, C.: Hodge algebras. Astérisque, 91 (1982)
Fröberg, R.: On Stanley-Reisner Rings. Topics in Algebra, Banach Center Publications, Part 2, vol. 26, pp. 57–70. Springer, Berlin (1990)
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)
Hartshorne, R.: Complete intersection and connectedness. Am. J. Math. 84, 497–508 (1962)
Hartshorne, R.: A property of A-sequences. Bull. Soc. Math. Fr. 94, 61–65 (1966)
Herzog, J., Hibi, T.: Distributive lattices, bipartite graphs and Alexander duality. J. Algebr. Comb. 22(3), 289–302 (2005)
Herzog, J., Hibi, T., Trung, N.V.: Symbolic powers of monomial ideals and vertex cover algebras. Adv. Math. 210, 304–322 (2007)
Hibi, T.: Distributive lattice, affine semigroup rings and algebras with straightening laws. Adv. Stud. Pure Math., 11 (1987)
Kalkbrener, M., Sturmfels, B.: Initial complex of prime ideals. Adv. Math. 116, 365–376 (1995)
Katzman, M.: Characteristic-independence of Betti numbers of graph ideals. J. Comb. Theory, Ser. A 113(3), 435–454 (2006)
Kummini, M.: Regularity, depth and arithmetic rank of bipartite edge ideals. J. Algebr. Comb. 30(4), 429–445 (2009)
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
Lyubeznik, G.: On the local cohomology modules \(H_{\mathfrak{U}}^{i}(R)\) for ideals \(\mathfrak{U}\) generated by an R-sequence. In: Complete Intersection. Lect. Notes in Math., vol. 1092, pp. 214–220. Springer, Berlin (1984)
Lyubeznik, G.: On the arithmetical rank of monomial ideals. J. Algebra 112(1), 86–89 (1988)
Morey, S., Reyes, E., Villarreal, R.H.: Cohen–Macaulay, shellable and unmixed clutters with a perfect matching of König type. J. Pure Appl. Algebra 212, 1770–1786 (2008)
Northcott, D.G., Rees, D.: Reduction of ideals in local rings. Proc. Camb. Philos. Soc. 50, 145–158 (1954)
Rinaldo, G.: Koszulness of vertex cover algebras of bipartite graphs. Commun. Algebra (2009, to appear)
Terai, N.: Alexander duality theorem and Stanley–Reisner rings. Free resolutions of projective varieties and related topics. Sūrikaisekikenkyūsho Kōkyūroku 1078, 174–184 (1999)
Varbaro, M.: Gröbner deformations, connectedness and cohomological dimension. J. Algebra 322, 2492–2507 (2009)
Zheng, X.: Resolution of facet ideals. Commun. Algebra 32(6), 2301–2324 (2004)
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Constantinescu, A., Varbaro, M. Koszulness, Krull dimension, and other properties of graph-related algebras. J Algebr Comb 34, 375–400 (2011). https://doi.org/10.1007/s10801-011-0276-6
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DOI: https://doi.org/10.1007/s10801-011-0276-6