On the Rees, Dilworth and Matsumura number
arXiv:1712.03828
Abstract
In this paper we study three numerical invariants of local rings: the Rees number, the Dilworth number, and the Matsumura number. These quantities measure, respectively, the minimal lengths of certain cyclic modules, the maximal numbers of generators of ideals, and a parameter-based analogue in higher dimension. We construct a non-exact local complete-intersection ring over , thereby providing a negative answer to a question of Matsumura on the exactness of zero-dimensional complete intersections. We also present a family of artinian Gorenstein local rings over a field of characteristic zero that are exact, making progress on a question of Watanabe. Finally, answering a further question of Matsumura, we introduce and study the Matsumura number and the class of homologically exact rings, which extend the notion of exactness beyond the one-dimensional case.