maximal orders over regular local ringso - American Mathematical
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maximal orders over regular local ringso - American Mathematical
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transactions of the american mathematical
Volume 155, Number 2, April 1971
society
MAXIMAL ORDERS OVER REGULAR LOCAL RINGSO BY
MARK RAMRAS Abstract. 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. Under what conditions is A maximal ? Auslander and Goldman [1] have shown that if gl.dim. P=l then A is maximal if and only if 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. A
