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 \(\text {type}(M) \leqslant \mu ( \text {Ext}_R^g(M,R) )\) (e.g., \(\text {type}(M)=1\)). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero R-module M of depth \(\geqslant d - 1\) such that the injective dimensions of M, \(\text {Hom}_R(M,M)\) and \(\text {Ext}_R^1(M,M)\) 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: \(\text {Ext}_R^{\gg 0}(M,N^{+})\), \(\text {Ext}_R^{\gg 0}(N,M^{+})\) and \(\text {Tor}_{\gg 0}^R(M,N)\), where \(M^{+}\) denotes \(\text {Ext}_R^{d-\dim (M)}(M,\omega )\). 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.