In the paper, [P, Schenzel, On formal local cohomology and connectedness, J of Alg, 315 (2007), 894--923], he proves the following statement.
(Corollary 2.2) Let $M$ be a finitely generated $R$-module and $I$ an $R$-ideal. Then $$ \operatorname{cd}(I,M) = \max \{ \operatorname{cd}(I, R/p) \mid p \in \operatorname{Min} (M) \}. $$
Here, $\operatorname{cd (I,M)}$ denotes the cohomological dimension of the module $M$; $\sup \{ i \in \mathbb{Z} \mid H^i_I (M) \neq 0 \}$, where $H^i_I$ is the $i$th local cohomogology module with respect to the ideal $I$. In this section of the paper, the ring $R$ is Noetherian.
I am not sure if the notion of cohomogical dimension is used for rings and modules that are not necessarily Noetherian. If so, I would like to ask the following question.
If $R$ is Noetherian, does there exist a module $M$ such that $$ \operatorname{cd}(I,M) \neq \max \{ \operatorname{cd}(I, R/p) \mid p \in \operatorname{Min} (M) \}? $$
Of course, if $M$ is finitely generated, the corollary answers the question negatively. If the answer is positive, then does there exist at least inequality?