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 $\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}(M)\le \mu ({\mathrm{E}\mathrm{x}\mathrm{t}}_R^g(M,R))$ (e.g., $\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}(M)=1$). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero $R$-module $M$ of depth $\ge d-1$ such that the injective dimensions of $M$, ${\mathrm{H}\mathrm{o}\mathrm{m}}_R(M,M)$ and ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^1(M,M)$ 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: ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^{\gg 0}(M,N^{+})$, ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^{\gg 0}(N,M^{+})$ and ${\mathrm{T}\mathrm{o}\mathrm{r}}_{\gg 0}^R(M,N)$, where $M^{+}$ denotes ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^{d-\dim (M)}(M,\omega )$. 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.