On top local cohomology modules, Matlis duality and tensor products

TL;DR

This paper investigates properties of top local cohomology modules, their Matlis duals, and tensor products in local rings, providing bounds on depth and analyzing associated primes in specific cohomological dimension cases.

Contribution

It offers new bounds for the depth of top local cohomology modules and examines tensor products and associated primes in particular cohomological dimension scenarios.

Findings

Bound for depth of $H^c_rak{a}(R)$ when $c=\dim R$ and $c>2$

Analysis of tensor products of local cohomology modules and their duals

Description of associated primes when $c=\dim R - 1$

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~$D(H^c_\mathfrak{a}(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete. Later, $H^c_\mathfrak{a}(R) \otimes_R H^c_\mathfrak{a}(R)$, $D(H^c_\mathfrak{a}(R)) \otimes_R D(H^c_\mathfrak{a}(R))$ and $H^c_\mathfrak{a}(R) \otimes_R D(H^c_\mathfrak{a}(R))$ are examined. In the case $c=\dim R-1$, the set Att$_R H^c_\mathfrak{a}(R)$ is considered.