Cohen–Macaulayness of large powers of Stanley–Reisner ideals

Abstract

We prove that for m⩾3, the symbolic power IΔ(m) of a Stanley–Reisner ideal is Cohen–Macaulay if and only if the simplicial complex Δ is a matroid. Similarly, the ordinary power IΔm is Cohen–Macaulay for some m⩾3 if and only if IΔ is a complete intersection. These results solve several open questions on the Cohen–Macaulayness of ordinary and symbolic powers of Stanley–Reisner ideals. Moreover, they have interesting consequences on the Cohen–Macaulayness of symbolic powers of facet ideals and cover ideals.