The center of an order with finite global dimension

Abstract:

Let $\Lambda$ be a quasi-local ring of global dimension $n < \infty$. Assume that its center $R$ is a noetherian domain, that $\Lambda$ is finitely generated torsion-free as an $R$-module, and that $R$ is an $R$-direct summand of $\Lambda$. Then $R$ is integrally closed in its quotient field $K$ and Macauley of dimension $n$. Furthermore, when $n = 2,\Lambda$ is a maximal $R$-order in the central simple $K$-algebra $\Lambda { \otimes _R}K$. This extends an earlier result of the author, in which $R$ was assumed to have global dimension 2. Examples are given to show that in the above situation $R$ can have infinite global dimension.