Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals

We present criteria for the Cohen–Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen–Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen–Macaulayness of the second symbolic power or of all symbolic powers of a Stanley–Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen–Macaulay. In particular, all symbolic powers are Cohen–Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen–Macaulayness can pass from a symbolic power to another symbolic powers in different ways.