Maximal orders over regular local rings of dimension two

Abstract

Introduction. Auslander and Goldman [4] have studied maximal orders over discrete valuation rings. In this paper, relying heavily on their results, we investigate maximal orders over regular local rings of dimension two. The two theories are rather different. We begin ?5 by listing three theorems from [4]. Then we prove a partial generalization of two of them in dimension two (Theorem 5.4) and exhibit two examples which show that the remaining statements do not generalize. In [4] a structure theorem is proved for maximal orders over discrete valuation rings which is the analogue of the Wedderburn structure theorem for simple artin rings. In ?6 we extend this theorem, in a weakened form, to maximal orders over an arbitrary integrally closed noetherian domain R. We sharpen this somewhat when R is regular local of dimension two and the maximal order is well behaved. Various other structure theorems are given in this section. The first four sections are devoted to building up homological machinery, most of which is applied to orders in the last two sections. The setting is fairly general: (R, mn) is a commutative noetherian local ring with maximal ideal m, and A is an R-algebra which is finitely generated as an R-module. The best results (Theorems 1.10 and 2.16) are obtained when A is quasi-local (i.e., A/Rad A is a simple artin ,ring, where Rad A is the Jacobson radical of A). Such rings behave very much like commutative local rings. The author wishes to express his gratitude to Professor Maurice Auslander for his many helpful suggestions and his patient supervision of the research for this paper, which is the major portion of the author's doctoral dissertation. Many thanks also to Professor David Buchsbaum and Dr. Silvio Greco for useful and stimulating conversations. Notations and conventions. Throughout this paper all rings have units and all modules are unitary. We use the abbreviations pd, inj dim, and gl dim for projective, injective, and global dimension, respectively. Only when there is a possible leftright ambiguity will we write l.pd or r.inj dim, etc. R is a commutative noetherian ring and A is an R-algebra which is finitely generated as an R-module. For the first four sections R is local with maximal ideal m. By R& we will mean the completion of R in the m-adic topology, and if M is an