Gorenstein Rings via Homological Dimensions, and Symmetry in Vanishing of Ext and Tate Cohomology

The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let R be a commutative Noetherian local ring of dimension d. In the 1st part, it is proved that R is Gorenstein if and only if it admits a nonzero CM (Cohen-Macaulay) module M of finite Gorenstein dimension g such that type(M) <= mu(Ext(R)(g)(M,R)) (e.g., type(M) = 1). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero R-module M of depth >= d - 1 such that the injective dimensions of M, Hom(R)(M,M) and Ext(R)(1)(M,M) are finite, then M has finite projective dimension and R is Gorenstein. In the 2nd part, we assume that R is CM with a canonical module omega. For CM R-modules M and N, we show that the vanishing of one of the following implies the same for others: Ext(R)(>> 0)(M,N+), Ext(R)(>> 0)(N,M+) and Tor(>> 0)(R)(M,N), where M+ denotes Ext(R)(d-dim(M))(M,omega). This strengthens a result of Huneke and Jorgensen. Furthermore, we prove a similar result for Tate cohomologies under the additional condition that R is Gorenstein.