Equality of ordinary and symbolic powers of Stanley–Reisner ideals

Abstract

This paper studies properties of simplicial complexes with the equality I (m) = I m for a given m 2. The main results are combinatorial characterizations of such complexes in the twodimensional case. It turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which I m is Cohen-Macaulay for some m 2.In particular, there are complexes with I (2) = I 2 or I (3) = I 3 but I (m) = I m for all m 4 and that if I (m) = I m for some m 4, then I (m) = I m for all m 1. Similarly, there are complexes for which I 2 is Cohen-Macaulay but I m is not Cohen-Macaulay for all m 3 and if I m is Cohen-Macaulay for some m 3, then I is a complete intersection.

On the second powers of Stanley-Reisner ideals

“…Among onedimensional simplicial complexes, the above example is a unique one, as shown in [15]. As for the two-dimensional case, such simplicial complexes are classified in [26]. In [16] a characterization of Cohen-Macaulayness of the second symbolic power I (2) ∆ is given.…”

“…In Section 4 we consider the problem when I (2) = I 2 holds for a Stanley-Reisner ideal I, which is also a necessary condition for Cohen-Macaulayness of I 2 . It is also discussed in [26]. We give a criterion for the second symbolic power to be equal to the ordinary power for Stanley-Reisner ideals in terms of the hypergraph of the generators; see Theorem 4.3.…”

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

“…A result of Hoang and Trung [36,Theorem 4.4] shows that for a graph G without isolated vertices I(G) 2 is Cohen-Macaulay if and only if G is trianglefree and Gorenstein. The Cohen-Macaulay property of I(G) 2 is also studied in [60] in terms of simplicial complexes.…”

“…The case when G is a graph and k = 2 is treated in [7,35,36,60]. The Cohen-Macaulay property of the square of an edge ideal can be expressed in terms of its connected components [25,48] (Corollary 5.10).…”

“…Edge ideals of graphs whose square is Cohen-Macaulay have a rich combinatorial structure and have been classified combinatorially by D. T. Hoang, N. C. Minh and T. N. Trung [35,36]. The Cohen-Macaulay property of I(G) 2 is also studied in [60] in terms of simplicial complexes.…”