Equality of ordinary and symbolic powers of Stanley–Reisner ideals

This paper studies properties of simplicial complexes Δ with the equality for a given m⩾2. The main results are combinatorial characterizations of such complexes in the two-dimensional 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 is Cohen–Macaulay for some m⩾2. In particular, there are complexes with or but for all m⩾4 and that if for some m⩾4, then for all m⩾1. Similarly, there are complexes for which is Cohen–Macaulay but is not Cohen–Macaulay for all m⩾3 and if is Cohen–Macaulay for some m⩾3, then IΔ is a complete intersection.