Maximal Orders over Regular Local Rings of Dimension Two

Abstract

MARK RAMRASOIntroduction. 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, m) is a commutative noetherian local ring with maximal ideal m, and A is an /¿-algebra which is finitely generated as an /¿-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 proj'ective, 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 /¿-algebra which is finitely generated as an /¿-module. For the first four sections R is local with maximal ideal m. By R we will mean the completion of R in the ra-adic topology, and if M is an

Maximal orders over regular local rings

“…In this paper various sufficient conditions are given for the maximality of an J?-order in a finite-dimensional central simple K-a.\gebra, where R is a regular local ring whose quotient field is K. Stronger results are obtained when we assume the dimension of R to be three. This work depends upon earlier results of this author [5] for regular local rings of dimension two, and the fundamental work of Auslander and Goldman [1] for dimension one.Introduction. Let (R, 2JÎ) be a regular local ring with maximal ideal 9JI, and K the quotient field of P. Let S be a central simple P-algebra, finite dimensional over K, and A an P-order in X.…”

“…If gl.dim. A<oo, then by [5,Corollary 2.17] A is R-free (and hence /?-reflexive) and gl.dim. A = gl.dim.…”

“…A=l and A is quasi-local (i.e., A has a unique maximal two-sided ideal, necessarily Rad A, the lacobson radical of A). In [5] this author has shown that when gl.dim. P = 2, a sufficient condition for maximality is that (a) gl.dim.…”

“…P = 2, a sufficient condition for maximality is that (a) gl.dim. A<co and (b) A is quasi-local [5,Theorem 5.4]. (Under these circumstances gl.dim.…”

The maximal ideals in quaternion orders

“…A natural question to ask is whether the same is true for maximal orders over arbitrary regular local rings. In this generality, however, the answer is in the negative; Ramras has shown this, in a very recent paper [4], by giving an example of a maximal order over a regular local ring of dimension two whose radical is not the only maximal two-sided ideal. Interestingly enough, his example concerns maximal orders in a quaternion division algebra, essentially the simplest case for orders whose centers are of dimension larger than one.…”