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On the symmetric and Rees algebras of \((n,k)\)-cyclic ideals. (English) Zbl 1056.13003

From the paper: Let \(R=F[X_1,\dots,X_n]\) be a polynomial ring over a field \(F\) and \(I\) an ideal of \(R\). We denote by \(S(I)\) and by \({\mathcal R}(I)\) the symmetric algebra and the Rees algebra of \(I\), respectively. Suppose that \(\Gamma\) is a graph with vertex set \(\underline X=\{X_1,\dots, X_n\}\) and edges \(E\).
R. H. Villarreal [Manuscr. Math. 66, No. 3, 277–293 (1990; Zbl 0737.13003)] defined the graph ideal \(I(\Gamma)\) as the ideal generated by the monomials of the form \(X_iX_j\) where \((X_i,X_j)\in E\). Later A. Conca and E. De Negri [J. Algebra 211, No. 2, 599–624 (1999; Zbl 0924.13012)], generalized this notion considering the ideal \(I_k (\Gamma)\) generated by all monomials \(X_{i_1}\cdots X_{i_k}\) such that \(X_{i_1},\dots, X_{i_k}\) is a path in \(\Gamma\). Further they showed that if \(\Gamma\) is a tree then \(I_k(\Gamma)\) is of linear type and \({\mathcal R}(I_k(\Gamma))\) is normal and Cohen-Macaulay. – Let the graph \(\Gamma\) be a cycle of length \(n\), and let \(1\leq k\leq n\). The authors prove that \(\dim_{\text{Krull}}S(I)=\dim_{\text{Krull}}{\mathcal R}(I)\) when \(\Gamma\) is a cycle and deduce whether the ideal \(I\) is of linear type or not in the following cases:
(1) If \(k=n-1\) then \(I_{n-1}(\Gamma)\) is of linear type and \(S(I_{n-1}(\Gamma))\) is Cohen-Macaulay (theorem 3.2).
(2) If \(n\) is odd and \(k=n-2\) or \(k=\frac {n-1}{2}\) then \(I_k(\Gamma)\) is of linear type and \(S(I_k (\Gamma))\) is Cohen-Macaulay (theorem 3.4 and corollary 3.6).
(3) If \(\text{gcd} (n,k)=r>1\) then \(I_k(\Gamma)\) is not of linear type (theorem 3.7).
(4) If \(\text{gcd}(n,k)=1\) and \([\frac{n-1}{2}]<k\leq n-3\) then \(I_k (\Gamma)\) is not of linear type (theorem 3.8).
(5) If \(\text{gcd} (n,k)=1\), \(kl\equiv 1 \pmod n\) and \(1<k\), \(l\leq[\frac{n-1}{2}]\) then \(I_k(\Gamma)\) is not of linear type (theorem 3.9a).
(6) If \(\text{gcd}(n,k)=1\), \(kl\equiv 1 \pmod n\), \(1<k \leq[\frac{n-1}{2}]\), \([\frac{n-1}{2}]<1<n\) and \(n=t(n-l)+a\), \(2\leq a<n-l\) then \(I_k (\Gamma)\) is not of linear type (theorem 3.9b).

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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