On top local cohomology modules, Matlis duality and tensor products

Abstract

Let $\mathfrak{a}$ be an ideal of a local ring $(R,\mathfrak{m})$ with $c=\mathrm{cd}(\mathfrak{a},R)$ the cohomological dimension of $\mathfrak{a}$ in $R$. In the case that $c=\dim R$, we first give a bound for depth~$D(H_{\mathfrak{a}}^c(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete. Later, $H_{\mathfrak{a}}^c(R){\otimes }_R{H}_{\mathfrak{a}}^c(R)$, $D(H_{\mathfrak{a}}^c(R)){\otimes }_{R}D(H_{\mathfrak{a}}^c(R))$ and $H_{\mathfrak{a}}^c(R){\otimes }_{R}D(H_{\mathfrak{a}}^c(R))$ are examined. In the case $c=\dim R-1$, the set Att${}_R{H}_{\mathfrak{a}}^c(R)$ is considered.

12 pages, This is the extended version of arXiv:1302.1274v1 and arXiv:1212.0245. accepted to appear in Journal of Algebra and Its Applications