Homological properties of fully bounded Noetherian rings and Sklyanin algebras.

Abstract

Let R be a fully bounded Noetherian ring of finite global dimension. Then we prove that Kdim(R) $\le$ gldim(R). If, in addition, R is local--in the sense that R/J(R) is simple Artinian, then we prove that R is Auslander-regular and satisfies a version of the Cohen-Macaulay property. As a consequence, we show that a local fully bounded Noetherian ring of finite global dimension is isomorphic to a matrix ring over a local domain, and a maximal order in its simple Artinian quotient ring. This solves, for FBN rings, two test questions on Noetherian rings (see (GW1, page 286, questions 5 and 6)). In the last chapter, we consider the Sklyanin algebras. Tate and Van den Bergh (TV) have shown that these algebras are Auslander-regular and Cohen-Macaulay. We give an alternative, simpler proof of this fact, thereby answering a question raised in their paper.