Cohen-Macaulayness of large powers of Stanley-Reisner ideals

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.