We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime p either the i-th Betti number of all high enough powers of a monomial ideal differs in characteristic 0 and in characteristic p or it is the same for all high enough powers. In our main results, we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal allows to spread the characteristic dependence to all powers. For any given prime p, this produces an edge ideal such that all its powers have some Betti numbers that are different over Q and over Zp. Moreover, we show that, for every r≥0 and i≥3 there is a monomial ideal I such that some coefficient in a degree ≥r of the Kodiyalam polynomials P3(I),…,Pi+r(I) depends on the characteristic. We also provide a summary of related results and speculate about the behavior of other combinatorially defined ideals.
