On finite generation of powers of ideals

✍ Scribed by Moshe Roitman

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 132 KB
• Volume: 161
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(00)00107-9

✦ Synopsis

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 ÿnitely generated, then M is ÿnitely 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 = (X s : s ∈ S), and one of the following conditions holds:

• R is seminormal.

• ht M = 3 and Q(R) is a ÿnitely generated ÿeld extension of K. For each d ≥ 3 we construct counterexamples of d-dimensional monoid domains as described above.