For $i \in N_0$, the $i$-th right derived of $Γ_I$ is denoted by $H_I ^i$ and will be referred to as the $i$-th local cohomology functor with respect to $I$.
It is clear that if $(R,\mathfrak m)$ is a Noetherian local ring and $M$ a non-zero finitely generated $R$-module of dimension n. Then $H_{\mathfrak m} ^ n (M)\neq 0$ and so is not a finitely generated $R$-module.
Also if $(R,\mathfrak m)$ is a Noetherian local ring and $M$ a non-zero finitely generated $R$-module of dimension $n > 0$, then $H_{\mathfrak m} ^n (M)$ is not a finitely generated $R$-module.
When $(R,\mathfrak m)$ is a Noetherian local ring and $M$ is a non-zero finitely generated $R$-module, then any integer $i$ for which $H_{\mathfrak m}^i(M)\neq 0$ must satisfy $\operatorname{depth} M ≤ i ≤ \dim M$, while for $i$ at either extremity of this range we do have $H_{\mathfrak m} ^i(M)=0$.
Now I have a question:
Let $(R,\mathfrak m)$ be a Cohen-Macaulay Noetherian local ring, $I$ be an ideal of $R$, and $M$ a finitely generated $R$-module. Let $\operatorname{grade}(I,M)=t$. It is clear that $H_I ^i (M)=0$ for every $i<t$ and also $H_I ^ t(M) \neq 0$. But $H_I ^ t(M)$ is not necessarily a finitely generated $R$-module. When this local cohomology module can not be a finitely generated $R$-module?