Rees algebras of edge ideals.

*(English)*Zbl 0836.13014In 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.

\(J = BJ_1 + B (\bigcup^\infty_{s = 2} P_s)\), 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.

Reviewer: J.Neggers (Tuscaloosa)

##### 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) |

##### Keywords:

Rees algebras; algebraic structures associated with finite graphs; ideals in polynomial rings; edge-ideals; incidence matrix; Gröbner bases
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\textit{R. H. Villarreal}, Commun. Algebra 23, No. 9, 3513--3524 (1995; Zbl 0836.13014)

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