Regular Local Ring Of Dimension Research Articles

Maximal orders over regular local rings

In this paper various sufficient conditions are given for the maximality of an $R$-order in a finite-dimensional central simple $K$-algebra, 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.

Maximal Orders over Regular Local Rings of Dimension Two

Maximal Orders over Regular Local Rings of Dimension Two

Maximal orders over regular local rings of dimension two

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

A note on ideals of homological dimension one in local domains

In this Note I shall give a characterization of ideals of homological dimension one in a local domain, and deduce a result on ideals in a regular local ring of dimension two.

Reduction to multiplicity less thanp in ap-cyclic extension of a two dimensional regular local ring (p = characteristic of the residue field)

Reduction to multiplicity less thanp in ap-cyclic extension of a two dimensional regular local ring (p = characteristic of the residue field)

Some multiplicative properties of complete ideals

1. Let L be a noetherian domain that is integrally closed in its quotient field F. To each ideal A of L is assigned an ideal A,a' the integral closure or completion of A, that consists of all elements x E L for which an equation of the form, xn + a1xn + * + an = 0, ai eAr, holds. If Q is the set of all valuations v of F such that the associated valuation ring R, contains L, then Aa coincides with the ideal Ab = {x; v(x) > v(A), V v E Q}, [2; 3]. An ideal A is said to be complete if A = Aa, and the set of all complete ideals is denoted by F(L). If A and B belong to F(L), the product AB may not belong to F(L), but the completion of the product (AB)a does. Hence a binary composition x is defined on F(L) by the condition, A x B = (AB)a. Under this composition F(L) is a commutative semigroup with an identity in which the cancellation law holds [2]. In case L is a two dimensional regular local ring, Zariski has shown [4, Appendix 5] that (1(L), x) is a Gaussian semigroup, and that the composition x is ordinary product. In this paper we study the case in which L is a two dimensional normal local domain which is subject to conditions less stringent than regularity. (See ?2 below.) It is shown that modulo a simple equivalence relation the semigroup (FO(L), x) is Gaussian, where Fo(L) is the subset of F(L) that consists of primary ideals belonging to the maximal ideal of L. However (FO(L), x) is not Gaussian in an absolute sense for in simple examples it is seen that the maximal ideal M of L is an irreducible element of (F(L), x) that is not prime. (Here we are using the semigroup terminology of Jacobson [1, Chapter IV].) Our methods are direct extensions of those of Zariski. In case L is regular, the form ring associated with L and the sequence of powers of the maximal ideal M is a polynomial ring over a field and Zariski's arguments are based in part on the fact that such a ring is a unique factorization domain. In our case the form ring is an integrally closed noetherian domain, and we obtain results analogous to Zariski's by using the Artin theory of factorization in the sense of quasi-equality that is valid in such domains.