By virtue the Griffith's paper and subsequently e.g. Goto's paper several examples of several desired class of Noetherian normal domains with specific finite length local cohomologies are constructed using the Bourbaki ideals. In order to compute some properties of these rings with Macaulay 2, I need to compute their Bourbaki ideals. does there exist any manuscript or any way to make this computation possible or easier?

$\begingroup$ Just out of curiosity, what is a Bourbaki ideal? $\endgroup$ – abx Nov 29 '14 at 7:41

$\begingroup$ @abx: Let e.g. $R=K[X_1,...,X_n]$. Then the the Bourbaki ideal associated to an $R$module $M$ is an ideal $I$ arising from an exact sequence $0\rightarrow F\rightarrow M\rightarrow I\rightarrow 0$ where $F$ is a finitely generated graded free $R$module $\endgroup$ – Aurora Nov 29 '14 at 7:53

$\begingroup$ I am not so sure if there is an easy way to convert a module into an ideal on Macaulay2. If the module $M$ has a rank, maybe even torsion free, then probably a generic choice of minimal generating set would produce a free submodule $F$ of rank one less than $M$. But here, your ideal $I$ may not be unique since it'd depend on the choice of the generators for $F$. $\endgroup$ – Youngsu Nov 29 '14 at 22:08

$\begingroup$ @Youngsu: Yes, in the Bourbaki exact sequence mentioned in my previous comment, as I know, $M$ should be torsion free and the free module $F$ satisfies $\text{rank}(F)=\text{rank}(M)1$. Would you please kindly explain how can I obtain an ideal $I$ from a generic choice of minimal generating set of $M$ by means of Macaulay 2? $\endgroup$ – Aurora Nov 30 '14 at 0:44

$\begingroup$ Let me try. Would you mind posting an example you have in mind? $\endgroup$ – Youngsu Nov 30 '14 at 1:00