On finite generation of powers of ideals

Following a joint work with Gilmer and Heinzer, we prove that if M is a maximal ideal of an integral domain R such that some power of M is finitely generated, then M is finitely generated under each of the assumptions below: (a) R is coherent. (b) R is seminormal and M is of height 2. (c) R = K[X ; S] is a monoid domain, M = (Xs : s ∈ S), and one of the following conditions holds: • R is seminormal. • ht M = 3 and Q(R) is a finitely generated field extension of K. For each d ≥ 3 we construct counterexamples of d-dimensional monoid domains as described above.