HOMOLOGICAL PROPERTIES OF (GRADED) NOETHERIAN PI-RINGS

Let R be a connected, graded, Noetherian PI ring. If injdim(R) = n < infinity, then we prove that R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to n. If gldim(R) = n < infinity, then R is a domain, finitely generated as a module over its centre and a maximal order in its quotient division ring. Similar results hold if R is assumed to be local rather than connected graded. Alternatively, suppose that R is a Noetherian PI ring with gldim(R) < infinity such that hd(R/M(1)) = hd(R/M(2)) for any two maximal ideals M(i) in the same clique. Then, R is a direct sum of prime rings, is integral over its centre, and is Auslander-Gorenstein. If R is a prime ring, then the centre Z(R) of R is a Krull domain and R equals its trace ring TR. Moreover, hd(R/M) = height(M), for every maximal ideal M of R. (C) 1994 Academic Press, Inc.