Combinatorial characterizations of the Cohen–Macaulayness of the second power of edge ideals

A characterization of triangle-free Gorenstein graphs and Cohen–Macaulayness of second powers of edge ideals

“…Recently, Terai and Trung [21] showed that ∆ is a complete intersection whenever I m ∆ is Cohen-Macaulay for some m 3. By contrast with this situation we have not known a characterization of ∆ for which I 2 ∆ is Cohen-Macaulay yet (see [8], [12], [16], [21]). On the other hand, Vasconcelos (see [23,Conjecture (B)]) suggests that ∆ must be Gorenstein if I 2 ∆ is Cohen-Macaulay.…”

Powers of Monomial Ideals and Combinatorics

“…It is worth to mention that the Cohen-Macaulay property of the second (ordinary or symbolic) power of a Stanley-Resiner ideal is completely different and is still not completely understood, see [34,33,51,59,60].…”

A Remark on the Local Cohomology Modules of a Union of Disjoint Matroids

“…Remark 1 It is note that the question: When S/I (2) is k-Buchsbaum for k = 0; 1 is still open (also see [3]). …”

“…The Cohen-Macaulay (Buchsbaum, generalized Cohen-Macaulay) property of S/I (t) was studied in some works as in [2, 4, 7-9, 11, 12] for t ≥ 3 and in [3] for t = 2. Later, the k-Buchsbaum property of S/I (t) is given in [5,6] for k large enough.…”

“…Here, S/I (r) is called k-Buchsbaum if m k H i m S/I (r) = (0) for all i < dim S/I (r) , where I (r) is the rth symbolic power of I . In [3], we gave a structure of the Z n -graded local cohomology modules of the generalized Cohen-Macaulay ring S/I (2) when is a contractible complex. It is natural to try to find a similar formula for S/I (t) in the case t ≥ 3.…”