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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.

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)
Full Text: DOI
[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
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