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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.

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)
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References:
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