×

zbMATH — the first resource for mathematics

Rees algebras of edge ideals. (English) Zbl 0836.13014
In this interesting follow-up on earlier results the author continues observations on several algebraic structures associated with finite graphs \(G\) on vertices \(\{x_1, \dots, x_n\}\) and edges \(\{(x_{i_s}, x_{j_s}) |s = 1, \dots, q\}\) using techniques developed for the description of special ideals in polynomial rings as well as for edge-ideals \(I(G)\) of algebras \(R = k[x_1, \dots, x_n]\) generated by monomials \(f_s = x_{i_s} x_{j_s}\). The algebras \(k[G] = k[f_1, \dots, f_q] = k[T_1, \dots, T_q]/P\), \(P = \ker \{\psi : T_i \to f_i\}\) and \(R(I(G)) = R[Tf_1, \dots, Tf_q] = B/J = R[T_1, \dots, T_q]/J\), \(J = \ker \{\varphi |T_i \to f_iT\}\), \(J = \bigoplus^\infty_{i = 1} J_s\) are described by means of the presentation ideals \(P\) and \(J\) as, e.g., in the main result:
\(J = BJ_1 + B (\bigcup^\infty_{s = 2} P_s)\), where \(P_s = (\{T_\alpha - T_\beta |f_\alpha = f_\beta\) for some \(\alpha, \beta \in I_s\})\), with \(I_s\) the set of all non- decreasing sequences \(\alpha = (i_1, \dots, i_s)\) and \(U_\alpha = U_{i_1} \dots U_{i_s}\).
A sequence of corollaries then produces specific results such as:
(1) if \(G\) is connected, \(J = J_1B\) iff \(G\) is a tree or it contains a unique cycle of odd length, and
(2) if \(G\) is a complete graph, then \(R(I(G))\) is a normal domain with \(J\) generated by binomials of degree 2.
Similarly, \(P = (\{T_w |/,x_w\) is an even closed walk}) and the ideal height of \(P\) is the \(\mathbb{Z}_2\)-dimension of the space of even cycles of \(G\). The incidence matrix is then used to describe \(P\) via vectors \(\alpha \in \mathbb{Z}^q\) such that \(M \alpha = 0\), whence various properties of Gröbner bases of this ideal may then be derived as has been done here.

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
05C05 Trees
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bayer D., Macaulay (1990)
[2] Flaschka H., Pacific J. Math. 149 pp 251– (1991)
[3] Harary F., Graph Theory (1972)
[4] DOI: 10.1007/BF01273309 · Zbl 0211.33801 · doi:10.1007/BF01273309
[5] Herzog J., Commutative Algebra 84 pp 79– (1983)
[6] DOI: 10.2307/1970791 · Zbl 0233.14010 · doi:10.2307/1970791
[7] de Loera J. A., Combinatorica
[8] Matsumura H., Cambridge Studies in Advanced Mathematics 8 (1986)
[9] Rockafellar, R. T. The elementary vectors of a subspace ofRN, inCombinatorial Mathematics and its Applications. Proc. Chapel Hill Conf. pp.104–127. Univ. North Carolina Press.
[10] DOI: 10.1016/0021-8693(79)90331-4 · Zbl 0401.13016 · doi:10.1016/0021-8693(79)90331-4
[11] DOI: 10.1006/jabr.1994.1192 · Zbl 0816.13003 · doi:10.1006/jabr.1994.1192
[12] DOI: 10.2748/tmj/1178227496 · Zbl 0714.14034 · doi:10.2748/tmj/1178227496
[13] Sturmfels B., Toric ideals (1994) · Zbl 0784.14024
[14] Taylor D., Ideals generated by monomials in an R-seguence (1966)
[15] Vasconcelos W. V., London Math. Soc. 195 (1994)
[16] DOI: 10.1007/BF02568497 · Zbl 0737.13003 · doi:10.1007/BF02568497
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.