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The aim of this article is to consider the spectral sequences induced bytensor-hom adjunction, and provide a number of new results. Let $R$ be acommutative Noetherian local ring of dimension $d$. In the 1st part, it isproved that $R$ is Gorenstein if and only if it admits a nonzero CM(Cohen-Macaulay) module $M$ of finite Gorenstein dimension $g$ such that ${\rmtype}(M) \le \mu( {\rm Ext}_R^g(M,R) )$ (e.g., ${\rm type}(M)=1$). Thisconsiderably strengthens a result of Takahashi. Moreover, we show that if thereis a nonzero $R$-module $M$ of depth $\ge d - 1$ such that the injectivedimensions of $M$, ${\rm Hom}_R(M,M)$ and ${\rm Ext}_R^1(M,M)$ are finite, then$M$ has finite projective dimension and $R$ is Gorenstein. In the 2nd part, weassume 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 samefor others: ${\rm Ext}_R^{\gg 0}(M,N^{+})$, ${\rm Ext}_R^{\gg 0}(N,M^{+})$ and${\rm Tor}_{\gg 0}^R(M,N)$, where $M^{+}$ denotes ${\rmExt}_R^{d-\dim(M)}(M,\omega)$. This strengthens a result of Huneke andJorgensen. Furthermore, we prove a similar result for Tate cohomologies underthe additional condition that $R$ is Gorenstein.

Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology