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- W4320854518 abstract "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 ${rm type}(M) le mu( {rm Ext}_R^g(M,R) )$ (e.g., ${rm type}(M)=1$). 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 $M$, ${rm Hom}_R(M,M)$ and ${rm Ext}_R^1(M,M)$ are finite, then $M$ has finite projective dimension and $R$ is Gorenstein. In the 2nd part, 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: ${rm Ext}_R^{gg 0}(M,N^{+})$, ${rm Ext}_R^{gg 0}(N,M^{+})$ and ${rm Tor}_{gg 0}^R(M,N)$, where $M^{+}$ denotes ${rm Ext}_R^{d-dim(M)}(M,omega)$. 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." @default.
- W4320854518 created "2023-02-16" @default.
- W4320854518 creator A5049108479 @default.
- W4320854518 creator A5088538511 @default.
- W4320854518 date "2023-02-13" @default.
- W4320854518 modified "2023-09-28" @default.
- W4320854518 title "Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology" @default.
- W4320854518 doi "https://doi.org/10.48550/arxiv.2302.06267" @default.
- W4320854518 hasPublicationYear "2023" @default.
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