Journal of the
Korean Mathematical Society JKMS

Let $(R,\mathfrak{m})$ be a commutative Noetherian local domain, $M$ a non-zero finitely generated $R$-module of dimension $n>0$ and $I$ be an ideal of $R$. In this paper it is shown that if $x_1, \ldots,x_t$ ($1\leq t \leq n$) be a subset of a system of parameters for $M$, then the $R$-module $ H^t_{(x_1, \ldots,x_t)}(R)$ is faithful, i.e., ${\rm Ann}\, H^t_{(x_1, \ldots,x_t)}(R)=0$. Also, it is shown that, if $H^i_I(R)=0$ for all $i> \dim R - \dim R/I$, then the $R$-module $H^{\dim R - \dim R/I}_I(R)$ is faithful. These results provide some partially affirmative answers to the Lynch's conjecture in \cite{L}. Moreover, for an ideal $I$ of an arbitrary Noetherian ring $R$, we calculate the annihilator of the top local cohomology module $H^{1}_{I}(M)$, when $H^i_I(M)=0$ for all integers $i>1$. Also, for such ideals we show that the finitely generated $R$-algebra $D_I(R)$ is a flat $R$-algebra.

Keywords: cohomological dimension, ideal transform, local cohomology, Noetherian ring

MSC numbers: 13D45, 14B15, 13E05