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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 (e.g., ). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero R-module M of depth such that the injective dimensions of M, and 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 . For CM R-modules M and N, we show that the vanishing of one of the following implies the same for others: , and , where denotes . This …