# cohomological dimension, dimension of modules and arithmetic rank

Let $R$ be a noetherian ring and $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$- module. I know two fact. first, dimension of $M$(i.e. krull dimension of $R/{\rm ann}(M)$) is greater than or equal to cohomological dimension of $M$ with respect to $I$. and second, arithmetic rank of $I$(i.e. ${\rm inf}\{r\in \mathbb{N}_0 | \exists x_1, \cdots ,x_r \in R~\mbox{such that}~ \sqrt{<x_1, \cdots,x_r>}=\sqrt{I}\}$) is greater then or equal to cohomological dimension of $M$ with respect to $I$.

I wonder when equalty holds...

-