Cohen-Macaulay monomial ideals with given radical
Let \(S = K[x_1,\dots, x_n]\) be a polynomial ring over a field \(K\) and \(I \subset S\) a squarefree Cohen-Macaulay monomial ideal, i.e, \(S/I\) is Cohen-Macaulay, whose minimal set of generators is \(G(I)=\{u_1,\dots, u_m\}\). We consider the following set of monomial ideals \[ {\mathcal F}_I := \{J\subset S\mid J\text{ is a monomial ideal},\; \sqrt{J}=I\} \] which contains the set \({\mathcal J}_I\) of monomial ideals called modification of \(I\): \[ {\mathcal J}_I := \{ J \in{\mathcal F}_I\mid G(J) = \{v_1, \dots, v_m\}\text{ with } \text{supp}(v_i) = \text{supp}(u_i),\; i=1,\dots, m\}. \] A modification \(J\in {\mathcal J}_I\) is called trivial if \(J = \varphi(I)S\) with the \(K\)-homomorphism defined by \(\varphi(x_i) = x_i^{a_i}\), \(i=1,\dots, n\), for some non-negative integers \(a_i\). Trivial modifications are all Cohen-Macaulay ideals. In the paper under review, the authors consider when non-trivial modifications exist and when all the elments of \({\mathcal J}_I\) or \({\mathcal F}_I\) are Cohen-Macaulay. The obtained results are as follows. First of all, all the elements of \({\mathcal J}_I\) is Cohen-Macaulay if and only if \(I\) is a complete intersection (Theorem~1.1). The same result also holds if we replace ``Cohen-Macaulay'' by ``unmixed''. As an easy observation after this result, the authors point out that all the elements of \({\mathcal F}_I\) is Cohen-Macaulay if and only if \(I=(x_1,\dots, x_n)\). The second result is that \(I\) has a non-trivial Cohen-Macaulay modification if and only if \(I\) has an infinitely many Cohen-Macaulay modifications (Theorem~2.4). As an application, they show that the Stanley-Reisner ideal of a natural triangulation of the real projective plane has only trivial Cohen-Macaulay modification, but not all modifications are trivial (Example~2.5). Note: This paper makes a completely wrong citation of the reviewer's paper (citation [5]). The correct one is the following: [\textit{Y. Takayama}, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 48(96), No. 3, 327--344 (2005; Zbl 1092.13020)].
scientific article; zbMATH DE number 5697063
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