$R$ is a *local* Noetherian ring. $f\_I(M)$, the *finiteness dimension* of a module $M$ relative to $I$, is defined in `chapter 9 of the book Local Cohomology. An Algebraic Introduction with Geometric Applications` by `M. P. BRODMANN and R. Y. SHARP` as: $f\_I(M) = \inf\ \{i : H\_I^i(M)$ is not finitely generated$\}$.  

Is there any software that can help to calculate $f\_I(M)$ (*perhaps some special ideals and modules*)?  

I searched *Macaulay2* and *CoCoA* and I think there is not a direct command for this; but I wonder if there is a technique similar to that *Mariano Suárez-Alvarez* used [here](https://math.stackexchange.com/questions/1054354/kta-1-ta-2-ta-3-in-the-form-kx-y-z)?  

thank you.



finiteness dimension
Ok, I can at least explain one way to do this in the case that $I$ is maximal. Since you are hoping for computers to do this for you, I'm going to assume that $R$ has a (normalized) dualizing complex.


Note that then $H^i\_I(M)$ is Matlis dual to $\text{Ext}^{-i}\_R(M, \omega\_R^{\bullet})$. 


Then $H^i\_I(M)$ is finitely generated if and only if the support of $\text{Ext}^{-i}\_R(M, \omega\_R^{\bullet})$ has dimension zero. In particular, the local cohomology is finitely generated if and only if the $\text{Ext}$ group satisfies DCC. That's only going to happen if its zero dimensional.


Regardless, this is something that computer algebra systems can easily compute. Note that if $R = S/J$ where $S$ is regular of dimension $d$, then we also have that
$$
\text{Ext}^{-i}\_R(M, \omega\_R^{\bullet}) = \text{Ext}^{d-i}\_S(M, S)
$$
and Macaulay2 can do this for you. 


